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Properties

Label 1122.2.a.h.1.1
Level $1122$
Weight $2$
Character 1122.1
Self dual yes
Analytic conductor $8.959$
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] = [1122,2,Mod(1,1122)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1122, 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("1122.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1122 = 2 \cdot 3 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1122.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: \(8.95921510679\)
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 \(\chi\) \(=\) 1122.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +4.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} +2.00000 q^{20} -4.00000 q^{21} +1.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} +4.00000 q^{28} -6.00000 q^{29} -2.00000 q^{30} -4.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} +1.00000 q^{34} +8.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} +4.00000 q^{38} +2.00000 q^{39} +2.00000 q^{40} +10.0000 q^{41} -4.00000 q^{42} +4.00000 q^{43} +1.00000 q^{44} +2.00000 q^{45} -4.00000 q^{46} -8.00000 q^{47} -1.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} -1.00000 q^{51} -2.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} +2.00000 q^{55} +4.00000 q^{56} -4.00000 q^{57} -6.00000 q^{58} -4.00000 q^{59} -2.00000 q^{60} -6.00000 q^{61} -4.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} -1.00000 q^{66} +12.0000 q^{67} +1.00000 q^{68} +4.00000 q^{69} +8.00000 q^{70} -12.0000 q^{71} +1.00000 q^{72} +2.00000 q^{73} +2.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} +4.00000 q^{77} +2.00000 q^{78} +4.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} -4.00000 q^{83} -4.00000 q^{84} +2.00000 q^{85} +4.00000 q^{86} +6.00000 q^{87} +1.00000 q^{88} +10.0000 q^{89} +2.00000 q^{90} -8.00000 q^{91} -4.00000 q^{92} +4.00000 q^{93} -8.00000 q^{94} +8.00000 q^{95} -1.00000 q^{96} +10.0000 q^{97} +9.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 − 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) −1.00000 −0.408248
\(7\) 4.00000 1.51186 0.755929 − 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) 1.00000 0.301511
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 4.00000 1.06904
\(15\) −2.00000 −0.516398
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536
\(18\) 1.00000 0.235702
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 2.00000 0.447214
\(21\) −4.00000 −0.872872
\(22\) 1.00000 0.213201
\(23\) −4.00000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) −1.00000 −0.204124
\(25\) −1.00000 −0.200000
\(26\) −2.00000 −0.392232
\(27\) −1.00000 −0.192450
\(28\) 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
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.00000 −0.174078
\(34\) 1.00000 0.171499
\(35\) 8.00000 1.35225
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 4.00000 0.648886
\(39\) 2.00000 0.320256
\(40\) 2.00000 0.316228
\(41\) 10.0000 1.56174 0.780869 − 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) −4.00000 −0.617213
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 1.00000 0.150756
\(45\) 2.00000 0.298142
\(46\) −4.00000 −0.589768
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) −1.00000 −0.144338
\(49\) 9.00000 1.28571
\(50\) −1.00000 −0.141421
\(51\) −1.00000 −0.140028
\(52\) −2.00000 −0.277350
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −1.00000 −0.136083
\(55\) 2.00000 0.269680
\(56\) 4.00000 0.534522
\(57\) −4.00000 −0.529813
\(58\) −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
\(61\) −6.00000 −0.768221 −0.384111 − 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) −4.00000 −0.508001
\(63\) 4.00000 0.503953
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) −1.00000 −0.123091
\(67\) 12.0000 1.46603 0.733017 − 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 1.00000 0.121268
\(69\) 4.00000 0.481543
\(70\) 8.00000 0.956183
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 1.00000 0.117851
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 2.00000 0.232495
\(75\) 1.00000 0.115470
\(76\) 4.00000 0.458831
\(77\) 4.00000 0.455842
\(78\) 2.00000 0.226455
\(79\) 4.00000 0.450035 0.225018 − 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) 10.0000 1.10432
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) −4.00000 −0.436436
\(85\) 2.00000 0.216930
\(86\) 4.00000 0.431331
\(87\) 6.00000 0.643268
\(88\) 1.00000 0.106600
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 2.00000 0.210819
\(91\) −8.00000 −0.838628
\(92\) −4.00000 −0.417029
\(93\) 4.00000 0.414781
\(94\) −8.00000 −0.825137
\(95\) 8.00000 0.820783
\(96\) −1.00000 −0.102062
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 9.00000 0.909137
\(99\) 1.00000 0.100504
\(100\) −1.00000 −0.100000
\(101\) −2.00000 −0.199007 −0.0995037 − 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) −1.00000 −0.0990148
\(103\) −16.0000 −1.57653 −0.788263 − 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) −2.00000 −0.196116
\(105\) −8.00000 −0.780720
\(106\) 6.00000 0.582772
\(107\) −4.00000 −0.386695 −0.193347 − 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 2.00000 0.191565 0.0957826 − 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 2.00000 0.190693
\(111\) −2.00000 −0.189832
\(112\) 4.00000 0.377964
\(113\) 18.0000 1.69330 0.846649 − 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) −4.00000 −0.374634
\(115\) −8.00000 −0.746004
\(116\) −6.00000 −0.557086
\(117\) −2.00000 −0.184900
\(118\) −4.00000 −0.368230
\(119\) 4.00000 0.366679
\(120\) −2.00000 −0.182574
\(121\) 1.00000 0.0909091
\(122\) −6.00000 −0.543214
\(123\) −10.0000 −0.901670
\(124\) −4.00000 −0.359211
\(125\) −12.0000 −1.07331
\(126\) 4.00000 0.356348
\(127\) −16.0000 −1.41977 −0.709885 − 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.00000 −0.352180
\(130\) −4.00000 −0.350823
\(131\) −12.0000 −1.04844 −0.524222 − 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 16.0000 1.38738
\(134\) 12.0000 1.03664
\(135\) −2.00000 −0.172133
\(136\) 1.00000 0.0857493
\(137\) −22.0000 −1.87959 −0.939793 − 0.341743i \(-0.888983\pi\)
−0.939793 + 0.341743i \(0.888983\pi\)
\(138\) 4.00000 0.340503
\(139\) 12.0000 1.01783 0.508913 − 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 8.00000 0.676123
\(141\) 8.00000 0.673722
\(142\) −12.0000 −1.00702
\(143\) −2.00000 −0.167248
\(144\) 1.00000 0.0833333
\(145\) −12.0000 −0.996546
\(146\) 2.00000 0.165521
\(147\) −9.00000 −0.742307
\(148\) 2.00000 0.164399
\(149\) 6.00000 0.491539 0.245770 − 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 1.00000 0.0816497
\(151\) −8.00000 −0.651031 −0.325515 − 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 4.00000 0.324443
\(153\) 1.00000 0.0808452
\(154\) 4.00000 0.322329
\(155\) −8.00000 −0.642575
\(156\) 2.00000 0.160128
\(157\) −2.00000 −0.159617 −0.0798087 − 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 4.00000 0.318223
\(159\) −6.00000 −0.475831
\(160\) 2.00000 0.158114
\(161\) −16.0000 −1.26098
\(162\) 1.00000 0.0785674
\(163\) 12.0000 0.939913 0.469956 − 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) 10.0000 0.780869
\(165\) −2.00000 −0.155700
\(166\) −4.00000 −0.310460
\(167\) −12.0000 −0.928588 −0.464294 − 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) −4.00000 −0.308607
\(169\) −9.00000 −0.692308
\(170\) 2.00000 0.153393
\(171\) 4.00000 0.305888
\(172\) 4.00000 0.304997
\(173\) 10.0000 0.760286 0.380143 − 0.924928i \(-0.375875\pi\)
0.380143 + 0.924928i \(0.375875\pi\)
\(174\) 6.00000 0.454859
\(175\) −4.00000 −0.302372
\(176\) 1.00000 0.0753778
\(177\) 4.00000 0.300658
\(178\) 10.0000 0.749532
\(179\) −20.0000 −1.49487 −0.747435 − 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 2.00000 0.149071
\(181\) −14.0000 −1.04061 −0.520306 − 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) −8.00000 −0.592999
\(183\) 6.00000 0.443533
\(184\) −4.00000 −0.294884
\(185\) 4.00000 0.294086
\(186\) 4.00000 0.293294
\(187\) 1.00000 0.0731272
\(188\) −8.00000 −0.583460
\(189\) −4.00000 −0.290957
\(190\) 8.00000 0.580381
\(191\) −24.0000 −1.73658 −0.868290 − 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −14.0000 −1.00774 −0.503871 − 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 10.0000 0.717958
\(195\) 4.00000 0.286446
\(196\) 9.00000 0.642857
\(197\) −6.00000 −0.427482 −0.213741 − 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 1.00000 0.0710669
\(199\) 12.0000 0.850657 0.425329 − 0.905039i \(-0.360158\pi\)
0.425329 + 0.905039i \(0.360158\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −12.0000 −0.846415
\(202\) −2.00000 −0.140720
\(203\) −24.0000 −1.68447
\(204\) −1.00000 −0.0700140
\(205\) 20.0000 1.39686
\(206\) −16.0000 −1.11477
\(207\) −4.00000 −0.278019
\(208\) −2.00000 −0.138675
\(209\) 4.00000 0.276686
\(210\) −8.00000 −0.552052
\(211\) 20.0000 1.37686 0.688428 − 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 6.00000 0.412082
\(213\) 12.0000 0.822226
\(214\) −4.00000 −0.273434
\(215\) 8.00000 0.545595
\(216\) −1.00000 −0.0680414
\(217\) −16.0000 −1.08615
\(218\) 2.00000 0.135457
\(219\) −2.00000 −0.135147
\(220\) 2.00000 0.134840
\(221\) −2.00000 −0.134535
\(222\) −2.00000 −0.134231
\(223\) 24.0000 1.60716 0.803579 − 0.595198i \(-0.202926\pi\)
0.803579 + 0.595198i \(0.202926\pi\)
\(224\) 4.00000 0.267261
\(225\) −1.00000 −0.0666667
\(226\) 18.0000 1.19734
\(227\) −4.00000 −0.265489 −0.132745 − 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) −4.00000 −0.264906
\(229\) 22.0000 1.45380 0.726900 − 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) −8.00000 −0.527504
\(231\) −4.00000 −0.263181
\(232\) −6.00000 −0.393919
\(233\) −14.0000 −0.917170 −0.458585 − 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) −2.00000 −0.130744
\(235\) −16.0000 −1.04372
\(236\) −4.00000 −0.260378
\(237\) −4.00000 −0.259828
\(238\) 4.00000 0.259281
\(239\) −24.0000 −1.55243 −0.776215 − 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) −2.00000 −0.129099
\(241\) 2.00000 0.128831 0.0644157 − 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 1.00000 0.0642824
\(243\) −1.00000 −0.0641500
\(244\) −6.00000 −0.384111
\(245\) 18.0000 1.14998
\(246\) −10.0000 −0.637577
\(247\) −8.00000 −0.509028
\(248\) −4.00000 −0.254000
\(249\) 4.00000 0.253490
\(250\) −12.0000 −0.758947
\(251\) −12.0000 −0.757433 −0.378717 − 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 4.00000 0.251976
\(253\) −4.00000 −0.251478
\(254\) −16.0000 −1.00393
\(255\) −2.00000 −0.125245
\(256\) 1.00000 0.0625000
\(257\) 2.00000 0.124757 0.0623783 − 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) −4.00000 −0.249029
\(259\) 8.00000 0.497096
\(260\) −4.00000 −0.248069
\(261\) −6.00000 −0.371391
\(262\) −12.0000 −0.741362
\(263\) −16.0000 −0.986602 −0.493301 − 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) −1.00000 −0.0615457
\(265\) 12.0000 0.737154
\(266\) 16.0000 0.981023
\(267\) −10.0000 −0.611990
\(268\) 12.0000 0.733017
\(269\) 10.0000 0.609711 0.304855 − 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) −2.00000 −0.121716
\(271\) −16.0000 −0.971931 −0.485965 − 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 1.00000 0.0606339
\(273\) 8.00000 0.484182
\(274\) −22.0000 −1.32907
\(275\) −1.00000 −0.0603023
\(276\) 4.00000 0.240772
\(277\) 10.0000 0.600842 0.300421 − 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 12.0000 0.719712
\(279\) −4.00000 −0.239474
\(280\) 8.00000 0.478091
\(281\) 10.0000 0.596550 0.298275 − 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 8.00000 0.476393
\(283\) −20.0000 −1.18888 −0.594438 − 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) −12.0000 −0.712069
\(285\) −8.00000 −0.473879
\(286\) −2.00000 −0.118262
\(287\) 40.0000 2.36113
\(288\) 1.00000 0.0589256
\(289\) 1.00000 0.0588235
\(290\) −12.0000 −0.704664
\(291\) −10.0000 −0.586210
\(292\) 2.00000 0.117041
\(293\) 6.00000 0.350524 0.175262 − 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) −9.00000 −0.524891
\(295\) −8.00000 −0.465778
\(296\) 2.00000 0.116248
\(297\) −1.00000 −0.0580259
\(298\) 6.00000 0.347571
\(299\) 8.00000 0.462652
\(300\) 1.00000 0.0577350
\(301\) 16.0000 0.922225
\(302\) −8.00000 −0.460348
\(303\) 2.00000 0.114897
\(304\) 4.00000 0.229416
\(305\) −12.0000 −0.687118
\(306\) 1.00000 0.0571662
\(307\) 28.0000 1.59804 0.799022 − 0.601302i \(-0.205351\pi\)
0.799022 + 0.601302i \(0.205351\pi\)
\(308\) 4.00000 0.227921
\(309\) 16.0000 0.910208
\(310\) −8.00000 −0.454369
\(311\) 12.0000 0.680458 0.340229 − 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 2.00000 0.113228
\(313\) 2.00000 0.113047 0.0565233 − 0.998401i \(-0.481998\pi\)
0.0565233 + 0.998401i \(0.481998\pi\)
\(314\) −2.00000 −0.112867
\(315\) 8.00000 0.450749
\(316\) 4.00000 0.225018
\(317\) 18.0000 1.01098 0.505490 − 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) −6.00000 −0.336463
\(319\) −6.00000 −0.335936
\(320\) 2.00000 0.111803
\(321\) 4.00000 0.223258
\(322\) −16.0000 −0.891645
\(323\) 4.00000 0.222566
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) 12.0000 0.664619
\(327\) −2.00000 −0.110600
\(328\) 10.0000 0.552158
\(329\) −32.0000 −1.76422
\(330\) −2.00000 −0.110096
\(331\) −20.0000 −1.09930 −0.549650 − 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) −4.00000 −0.219529
\(333\) 2.00000 0.109599
\(334\) −12.0000 −0.656611
\(335\) 24.0000 1.31126
\(336\) −4.00000 −0.218218
\(337\) 26.0000 1.41631 0.708155 − 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) −9.00000 −0.489535
\(339\) −18.0000 −0.977626
\(340\) 2.00000 0.108465
\(341\) −4.00000 −0.216612
\(342\) 4.00000 0.216295
\(343\) 8.00000 0.431959
\(344\) 4.00000 0.215666
\(345\) 8.00000 0.430706
\(346\) 10.0000 0.537603
\(347\) −28.0000 −1.50312 −0.751559 − 0.659665i \(-0.770698\pi\)
−0.751559 + 0.659665i \(0.770698\pi\)
\(348\) 6.00000 0.321634
\(349\) 14.0000 0.749403 0.374701 − 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) −4.00000 −0.213809
\(351\) 2.00000 0.106752
\(352\) 1.00000 0.0533002
\(353\) −14.0000 −0.745145 −0.372572 − 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 4.00000 0.212598
\(355\) −24.0000 −1.27379
\(356\) 10.0000 0.529999
\(357\) −4.00000 −0.211702
\(358\) −20.0000 −1.05703
\(359\) −24.0000 −1.26667 −0.633336 − 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 2.00000 0.105409
\(361\) −3.00000 −0.157895
\(362\) −14.0000 −0.735824
\(363\) −1.00000 −0.0524864
\(364\) −8.00000 −0.419314
\(365\) 4.00000 0.209370
\(366\) 6.00000 0.313625
\(367\) 28.0000 1.46159 0.730794 − 0.682598i \(-0.239150\pi\)
0.730794 + 0.682598i \(0.239150\pi\)
\(368\) −4.00000 −0.208514
\(369\) 10.0000 0.520579
\(370\) 4.00000 0.207950
\(371\) 24.0000 1.24602
\(372\) 4.00000 0.207390
\(373\) 22.0000 1.13912 0.569558 − 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) 1.00000 0.0517088
\(375\) 12.0000 0.619677
\(376\) −8.00000 −0.412568
\(377\) 12.0000 0.618031
\(378\) −4.00000 −0.205738
\(379\) 20.0000 1.02733 0.513665 − 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 8.00000 0.410391
\(381\) 16.0000 0.819705
\(382\) −24.0000 −1.22795
\(383\) 24.0000 1.22634 0.613171 − 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 8.00000 0.407718
\(386\) −14.0000 −0.712581
\(387\) 4.00000 0.203331
\(388\) 10.0000 0.507673
\(389\) −18.0000 −0.912636 −0.456318 − 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 4.00000 0.202548
\(391\) −4.00000 −0.202289
\(392\) 9.00000 0.454569
\(393\) 12.0000 0.605320
\(394\) −6.00000 −0.302276
\(395\) 8.00000 0.402524
\(396\) 1.00000 0.0502519
\(397\) −22.0000 −1.10415 −0.552074 − 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) 12.0000 0.601506
\(399\) −16.0000 −0.801002
\(400\) −1.00000 −0.0500000
\(401\) −6.00000 −0.299626 −0.149813 − 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) −12.0000 −0.598506
\(403\) 8.00000 0.398508
\(404\) −2.00000 −0.0995037
\(405\) 2.00000 0.0993808
\(406\) −24.0000 −1.19110
\(407\) 2.00000 0.0991363
\(408\) −1.00000 −0.0495074
\(409\) −6.00000 −0.296681 −0.148340 − 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 20.0000 0.987730
\(411\) 22.0000 1.08518
\(412\) −16.0000 −0.788263
\(413\) −16.0000 −0.787309
\(414\) −4.00000 −0.196589
\(415\) −8.00000 −0.392705
\(416\) −2.00000 −0.0980581
\(417\) −12.0000 −0.587643
\(418\) 4.00000 0.195646
\(419\) 28.0000 1.36789 0.683945 − 0.729534i \(-0.260263\pi\)
0.683945 + 0.729534i \(0.260263\pi\)
\(420\) −8.00000 −0.390360
\(421\) −2.00000 −0.0974740 −0.0487370 − 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 20.0000 0.973585
\(423\) −8.00000 −0.388973
\(424\) 6.00000 0.291386
\(425\) −1.00000 −0.0485071
\(426\) 12.0000 0.581402
\(427\) −24.0000 −1.16144
\(428\) −4.00000 −0.193347
\(429\) 2.00000 0.0965609
\(430\) 8.00000 0.385794
\(431\) 12.0000 0.578020 0.289010 − 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 2.00000 0.0961139 0.0480569 − 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) −16.0000 −0.768025
\(435\) 12.0000 0.575356
\(436\) 2.00000 0.0957826
\(437\) −16.0000 −0.765384
\(438\) −2.00000 −0.0955637
\(439\) 20.0000 0.954548 0.477274 − 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 2.00000 0.0953463
\(441\) 9.00000 0.428571
\(442\) −2.00000 −0.0951303
\(443\) 20.0000 0.950229 0.475114 − 0.879924i \(-0.342407\pi\)
0.475114 + 0.879924i \(0.342407\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 20.0000 0.948091
\(446\) 24.0000 1.13643
\(447\) −6.00000 −0.283790
\(448\) 4.00000 0.188982
\(449\) 10.0000 0.471929 0.235965 − 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 10.0000 0.470882
\(452\) 18.0000 0.846649
\(453\) 8.00000 0.375873
\(454\) −4.00000 −0.187729
\(455\) −16.0000 −0.750092
\(456\) −4.00000 −0.187317
\(457\) −22.0000 −1.02912 −0.514558 − 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 22.0000 1.02799
\(459\) −1.00000 −0.0466760
\(460\) −8.00000 −0.373002
\(461\) −10.0000 −0.465746 −0.232873 − 0.972507i \(-0.574813\pi\)
−0.232873 + 0.972507i \(0.574813\pi\)
\(462\) −4.00000 −0.186097
\(463\) 8.00000 0.371792 0.185896 − 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) −6.00000 −0.278543
\(465\) 8.00000 0.370991
\(466\) −14.0000 −0.648537
\(467\) −20.0000 −0.925490 −0.462745 − 0.886492i \(-0.653135\pi\)
−0.462745 + 0.886492i \(0.653135\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 48.0000 2.21643
\(470\) −16.0000 −0.738025
\(471\) 2.00000 0.0921551
\(472\) −4.00000 −0.184115
\(473\) 4.00000 0.183920
\(474\) −4.00000 −0.183726
\(475\) −4.00000 −0.183533
\(476\) 4.00000 0.183340
\(477\) 6.00000 0.274721
\(478\) −24.0000 −1.09773
\(479\) 20.0000 0.913823 0.456912 − 0.889512i \(-0.348956\pi\)
0.456912 + 0.889512i \(0.348956\pi\)
\(480\) −2.00000 −0.0912871
\(481\) −4.00000 −0.182384
\(482\) 2.00000 0.0910975
\(483\) 16.0000 0.728025
\(484\) 1.00000 0.0454545
\(485\) 20.0000 0.908153
\(486\) −1.00000 −0.0453609
\(487\) −28.0000 −1.26880 −0.634401 − 0.773004i \(-0.718753\pi\)
−0.634401 + 0.773004i \(0.718753\pi\)
\(488\) −6.00000 −0.271607
\(489\) −12.0000 −0.542659
\(490\) 18.0000 0.813157
\(491\) 12.0000 0.541552 0.270776 − 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) −10.0000 −0.450835
\(493\) −6.00000 −0.270226
\(494\) −8.00000 −0.359937
\(495\) 2.00000 0.0898933
\(496\) −4.00000 −0.179605
\(497\) −48.0000 −2.15309
\(498\) 4.00000 0.179244
\(499\) 20.0000 0.895323 0.447661 − 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) −12.0000 −0.536656
\(501\) 12.0000 0.536120
\(502\) −12.0000 −0.535586
\(503\) −12.0000 −0.535054 −0.267527 − 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 4.00000 0.178174
\(505\) −4.00000 −0.177998
\(506\) −4.00000 −0.177822
\(507\) 9.00000 0.399704
\(508\) −16.0000 −0.709885
\(509\) 6.00000 0.265945 0.132973 − 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) −2.00000 −0.0885615
\(511\) 8.00000 0.353899
\(512\) 1.00000 0.0441942
\(513\) −4.00000 −0.176604
\(514\) 2.00000 0.0882162
\(515\) −32.0000 −1.41009
\(516\) −4.00000 −0.176090
\(517\) −8.00000 −0.351840
\(518\) 8.00000 0.351500
\(519\) −10.0000 −0.438951
\(520\) −4.00000 −0.175412
\(521\) 18.0000 0.788594 0.394297 − 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) −6.00000 −0.262613
\(523\) −4.00000 −0.174908 −0.0874539 − 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) −12.0000 −0.524222
\(525\) 4.00000 0.174574
\(526\) −16.0000 −0.697633
\(527\) −4.00000 −0.174243
\(528\) −1.00000 −0.0435194
\(529\) −7.00000 −0.304348
\(530\) 12.0000 0.521247
\(531\) −4.00000 −0.173585
\(532\) 16.0000 0.693688
\(533\) −20.0000 −0.866296
\(534\) −10.0000 −0.432742
\(535\) −8.00000 −0.345870
\(536\) 12.0000 0.518321
\(537\) 20.0000 0.863064
\(538\) 10.0000 0.431131
\(539\) 9.00000 0.387657
\(540\) −2.00000 −0.0860663
\(541\) 10.0000 0.429934 0.214967 − 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) −16.0000 −0.687259
\(543\) 14.0000 0.600798
\(544\) 1.00000 0.0428746
\(545\) 4.00000 0.171341
\(546\) 8.00000 0.342368
\(547\) −28.0000 −1.19719 −0.598597 − 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) −22.0000 −0.939793
\(549\) −6.00000 −0.256074
\(550\) −1.00000 −0.0426401
\(551\) −24.0000 −1.02243
\(552\) 4.00000 0.170251
\(553\) 16.0000 0.680389
\(554\) 10.0000 0.424859
\(555\) −4.00000 −0.169791
\(556\) 12.0000 0.508913
\(557\) 6.00000 0.254228 0.127114 − 0.991888i \(-0.459429\pi\)
0.127114 + 0.991888i \(0.459429\pi\)
\(558\) −4.00000 −0.169334
\(559\) −8.00000 −0.338364
\(560\) 8.00000 0.338062
\(561\) −1.00000 −0.0422200
\(562\) 10.0000 0.421825
\(563\) −4.00000 −0.168580 −0.0842900 − 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 8.00000 0.336861
\(565\) 36.0000 1.51453
\(566\) −20.0000 −0.840663
\(567\) 4.00000 0.167984
\(568\) −12.0000 −0.503509
\(569\) −22.0000 −0.922288 −0.461144 − 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) −8.00000 −0.335083
\(571\) 4.00000 0.167395 0.0836974 − 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) −2.00000 −0.0836242
\(573\) 24.0000 1.00261
\(574\) 40.0000 1.66957
\(575\) 4.00000 0.166812
\(576\) 1.00000 0.0416667
\(577\) 2.00000 0.0832611 0.0416305 − 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 1.00000 0.0415945
\(579\) 14.0000 0.581820
\(580\) −12.0000 −0.498273
\(581\) −16.0000 −0.663792
\(582\) −10.0000 −0.414513
\(583\) 6.00000 0.248495
\(584\) 2.00000 0.0827606
\(585\) −4.00000 −0.165380
\(586\) 6.00000 0.247858
\(587\) 36.0000 1.48588 0.742940 − 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) −9.00000 −0.371154
\(589\) −16.0000 −0.659269
\(590\) −8.00000 −0.329355
\(591\) 6.00000 0.246807
\(592\) 2.00000 0.0821995
\(593\) 18.0000 0.739171 0.369586 − 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 8.00000 0.327968
\(596\) 6.00000 0.245770
\(597\) −12.0000 −0.491127
\(598\) 8.00000 0.327144
\(599\) −40.0000 −1.63436 −0.817178 − 0.576386i \(-0.804463\pi\)
−0.817178 + 0.576386i \(0.804463\pi\)
\(600\) 1.00000 0.0408248
\(601\) −30.0000 −1.22373 −0.611863 − 0.790964i \(-0.709580\pi\)
−0.611863 + 0.790964i \(0.709580\pi\)
\(602\) 16.0000 0.652111
\(603\) 12.0000 0.488678
\(604\) −8.00000 −0.325515
\(605\) 2.00000 0.0813116
\(606\) 2.00000 0.0812444
\(607\) −12.0000 −0.487065 −0.243532 − 0.969893i \(-0.578306\pi\)
−0.243532 + 0.969893i \(0.578306\pi\)
\(608\) 4.00000 0.162221
\(609\) 24.0000 0.972529
\(610\) −12.0000 −0.485866
\(611\) 16.0000 0.647291
\(612\) 1.00000 0.0404226
\(613\) −2.00000 −0.0807792 −0.0403896 − 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) 28.0000 1.12999
\(615\) −20.0000 −0.806478
\(616\) 4.00000 0.161165
\(617\) 26.0000 1.04672 0.523360 − 0.852111i \(-0.324678\pi\)
0.523360 + 0.852111i \(0.324678\pi\)
\(618\) 16.0000 0.643614
\(619\) −28.0000 −1.12542 −0.562708 − 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) −8.00000 −0.321288
\(621\) 4.00000 0.160514
\(622\) 12.0000 0.481156
\(623\) 40.0000 1.60257
\(624\) 2.00000 0.0800641
\(625\) −19.0000 −0.760000
\(626\) 2.00000 0.0799361
\(627\) −4.00000 −0.159745
\(628\) −2.00000 −0.0798087
\(629\) 2.00000 0.0797452
\(630\) 8.00000 0.318728
\(631\) −48.0000 −1.91085 −0.955425 − 0.295234i \(-0.904602\pi\)
−0.955425 + 0.295234i \(0.904602\pi\)
\(632\) 4.00000 0.159111
\(633\) −20.0000 −0.794929
\(634\) 18.0000 0.714871
\(635\) −32.0000 −1.26988
\(636\) −6.00000 −0.237915
\(637\) −18.0000 −0.713186
\(638\) −6.00000 −0.237542
\(639\) −12.0000 −0.474713
\(640\) 2.00000 0.0790569
\(641\) −6.00000 −0.236986 −0.118493 − 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) 4.00000 0.157867
\(643\) 12.0000 0.473234 0.236617 − 0.971603i \(-0.423961\pi\)
0.236617 + 0.971603i \(0.423961\pi\)
\(644\) −16.0000 −0.630488
\(645\) −8.00000 −0.315000
\(646\) 4.00000 0.157378
\(647\) 48.0000 1.88707 0.943537 − 0.331266i \(-0.107476\pi\)
0.943537 + 0.331266i \(0.107476\pi\)
\(648\) 1.00000 0.0392837
\(649\) −4.00000 −0.157014
\(650\) 2.00000 0.0784465
\(651\) 16.0000 0.627089
\(652\) 12.0000 0.469956
\(653\) −22.0000 −0.860927 −0.430463 − 0.902608i \(-0.641650\pi\)
−0.430463 + 0.902608i \(0.641650\pi\)
\(654\) −2.00000 −0.0782062
\(655\) −24.0000 −0.937758
\(656\) 10.0000 0.390434
\(657\) 2.00000 0.0780274
\(658\) −32.0000 −1.24749
\(659\) −20.0000 −0.779089 −0.389545 − 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) −2.00000 −0.0778499
\(661\) 30.0000 1.16686 0.583432 − 0.812162i \(-0.301709\pi\)
0.583432 + 0.812162i \(0.301709\pi\)
\(662\) −20.0000 −0.777322
\(663\) 2.00000 0.0776736
\(664\) −4.00000 −0.155230
\(665\) 32.0000 1.24091
\(666\) 2.00000 0.0774984
\(667\) 24.0000 0.929284
\(668\) −12.0000 −0.464294
\(669\) −24.0000 −0.927894
\(670\) 24.0000 0.927201
\(671\) −6.00000 −0.231627
\(672\) −4.00000 −0.154303
\(673\) 34.0000 1.31060 0.655302 − 0.755367i \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) 26.0000 1.00148
\(675\) 1.00000 0.0384900
\(676\) −9.00000 −0.346154
\(677\) −38.0000 −1.46046 −0.730229 − 0.683202i \(-0.760587\pi\)
−0.730229 + 0.683202i \(0.760587\pi\)
\(678\) −18.0000 −0.691286
\(679\) 40.0000 1.53506
\(680\) 2.00000 0.0766965
\(681\) 4.00000 0.153280
\(682\) −4.00000 −0.153168
\(683\) 36.0000 1.37750 0.688751 − 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 4.00000 0.152944
\(685\) −44.0000 −1.68115
\(686\) 8.00000 0.305441
\(687\) −22.0000 −0.839352
\(688\) 4.00000 0.152499
\(689\) −12.0000 −0.457164
\(690\) 8.00000 0.304555
\(691\) 20.0000 0.760836 0.380418 − 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 10.0000 0.380143
\(693\) 4.00000 0.151947
\(694\) −28.0000 −1.06287
\(695\) 24.0000 0.910372
\(696\) 6.00000 0.227429
\(697\) 10.0000 0.378777
\(698\) 14.0000 0.529908
\(699\) 14.0000 0.529529
\(700\) −4.00000 −0.151186
\(701\) −10.0000 −0.377695 −0.188847 − 0.982006i \(-0.560475\pi\)
−0.188847 + 0.982006i \(0.560475\pi\)
\(702\) 2.00000 0.0754851
\(703\) 8.00000 0.301726
\(704\) 1.00000 0.0376889
\(705\) 16.0000 0.602595
\(706\) −14.0000 −0.526897
\(707\) −8.00000 −0.300871
\(708\) 4.00000 0.150329
\(709\) 10.0000 0.375558 0.187779 − 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) −24.0000 −0.900704
\(711\) 4.00000 0.150012
\(712\) 10.0000 0.374766
\(713\) 16.0000 0.599205
\(714\) −4.00000 −0.149696
\(715\) −4.00000 −0.149592
\(716\) −20.0000 −0.747435
\(717\) 24.0000 0.896296
\(718\) −24.0000 −0.895672
\(719\) 28.0000 1.04422 0.522112 − 0.852877i \(-0.325144\pi\)
0.522112 + 0.852877i \(0.325144\pi\)
\(720\) 2.00000 0.0745356
\(721\) −64.0000 −2.38348
\(722\) −3.00000 −0.111648
\(723\) −2.00000 −0.0743808
\(724\) −14.0000 −0.520306
\(725\) 6.00000 0.222834
\(726\) −1.00000 −0.0371135
\(727\) 8.00000 0.296704 0.148352 − 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) −8.00000 −0.296500
\(729\) 1.00000 0.0370370
\(730\) 4.00000 0.148047
\(731\) 4.00000 0.147945
\(732\) 6.00000 0.221766
\(733\) 30.0000 1.10808 0.554038 − 0.832492i \(-0.313086\pi\)
0.554038 + 0.832492i \(0.313086\pi\)
\(734\) 28.0000 1.03350
\(735\) −18.0000 −0.663940
\(736\) −4.00000 −0.147442
\(737\) 12.0000 0.442026
\(738\) 10.0000 0.368105
\(739\) 4.00000 0.147142 0.0735712 − 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 4.00000 0.147043
\(741\) 8.00000 0.293887
\(742\) 24.0000 0.881068
\(743\) 20.0000 0.733729 0.366864 − 0.930274i \(-0.380431\pi\)
0.366864 + 0.930274i \(0.380431\pi\)
\(744\) 4.00000 0.146647
\(745\) 12.0000 0.439646
\(746\) 22.0000 0.805477
\(747\) −4.00000 −0.146352
\(748\) 1.00000 0.0365636
\(749\) −16.0000 −0.584627
\(750\) 12.0000 0.438178
\(751\) 20.0000 0.729810 0.364905 − 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) −8.00000 −0.291730
\(753\) 12.0000 0.437304
\(754\) 12.0000 0.437014
\(755\) −16.0000 −0.582300
\(756\) −4.00000 −0.145479
\(757\) −10.0000 −0.363456 −0.181728 − 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 20.0000 0.726433
\(759\) 4.00000 0.145191
\(760\) 8.00000 0.290191
\(761\) 10.0000 0.362500 0.181250 − 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) 16.0000 0.579619
\(763\) 8.00000 0.289619
\(764\) −24.0000 −0.868290
\(765\) 2.00000 0.0723102
\(766\) 24.0000 0.867155
\(767\) 8.00000 0.288863
\(768\) −1.00000 −0.0360844
\(769\) −46.0000 −1.65880 −0.829401 − 0.558653i \(-0.811318\pi\)
−0.829401 + 0.558653i \(0.811318\pi\)
\(770\) 8.00000 0.288300
\(771\) −2.00000 −0.0720282
\(772\) −14.0000 −0.503871
\(773\) 46.0000 1.65451 0.827253 − 0.561830i \(-0.189903\pi\)
0.827253 + 0.561830i \(0.189903\pi\)
\(774\) 4.00000 0.143777
\(775\) 4.00000 0.143684
\(776\) 10.0000 0.358979
\(777\) −8.00000 −0.286998
\(778\) −18.0000 −0.645331
\(779\) 40.0000 1.43315
\(780\) 4.00000 0.143223
\(781\) −12.0000 −0.429394
\(782\) −4.00000 −0.143040
\(783\) 6.00000 0.214423
\(784\) 9.00000 0.321429
\(785\) −4.00000 −0.142766
\(786\) 12.0000 0.428026
\(787\) 4.00000 0.142585 0.0712923 − 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) −6.00000 −0.213741
\(789\) 16.0000 0.569615
\(790\) 8.00000 0.284627
\(791\) 72.0000 2.56003
\(792\) 1.00000 0.0355335
\(793\) 12.0000 0.426132
\(794\) −22.0000 −0.780751
\(795\) −12.0000 −0.425596
\(796\) 12.0000 0.425329
\(797\) 30.0000 1.06265 0.531327 − 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) −16.0000 −0.566394
\(799\) −8.00000 −0.283020
\(800\) −1.00000 −0.0353553
\(801\) 10.0000 0.353333
\(802\) −6.00000 −0.211867
\(803\) 2.00000 0.0705785
\(804\) −12.0000 −0.423207
\(805\) −32.0000 −1.12785
\(806\) 8.00000 0.281788
\(807\) −10.0000 −0.352017
\(808\) −2.00000 −0.0703598
\(809\) 10.0000 0.351581 0.175791 − 0.984428i \(-0.443752\pi\)
0.175791 + 0.984428i \(0.443752\pi\)
\(810\) 2.00000 0.0702728
\(811\) −28.0000 −0.983213 −0.491606 − 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) −24.0000 −0.842235
\(813\) 16.0000 0.561144
\(814\) 2.00000 0.0701000
\(815\) 24.0000 0.840683
\(816\) −1.00000 −0.0350070
\(817\) 16.0000 0.559769
\(818\) −6.00000 −0.209785
\(819\) −8.00000 −0.279543
\(820\) 20.0000 0.698430
\(821\) −6.00000 −0.209401 −0.104701 − 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) 22.0000 0.767338
\(823\) −44.0000 −1.53374 −0.766872 − 0.641800i \(-0.778188\pi\)
−0.766872 + 0.641800i \(0.778188\pi\)
\(824\) −16.0000 −0.557386
\(825\) 1.00000 0.0348155
\(826\) −16.0000 −0.556711
\(827\) 12.0000 0.417281 0.208640 − 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) −4.00000 −0.139010
\(829\) −50.0000 −1.73657 −0.868286 − 0.496064i \(-0.834778\pi\)
−0.868286 + 0.496064i \(0.834778\pi\)
\(830\) −8.00000 −0.277684
\(831\) −10.0000 −0.346896
\(832\) −2.00000 −0.0693375
\(833\) 9.00000 0.311832
\(834\) −12.0000 −0.415526
\(835\) −24.0000 −0.830554
\(836\) 4.00000 0.138343
\(837\) 4.00000 0.138260
\(838\) 28.0000 0.967244
\(839\) 4.00000 0.138095 0.0690477 − 0.997613i \(-0.478004\pi\)
0.0690477 + 0.997613i \(0.478004\pi\)
\(840\) −8.00000 −0.276026
\(841\) 7.00000 0.241379
\(842\) −2.00000 −0.0689246
\(843\) −10.0000 −0.344418
\(844\) 20.0000 0.688428
\(845\) −18.0000 −0.619219
\(846\) −8.00000 −0.275046
\(847\) 4.00000 0.137442
\(848\) 6.00000 0.206041
\(849\) 20.0000 0.686398
\(850\) −1.00000 −0.0342997
\(851\) −8.00000 −0.274236
\(852\) 12.0000 0.411113
\(853\) 58.0000 1.98588 0.992941 − 0.118609i \(-0.0378434\pi\)
0.992941 + 0.118609i \(0.0378434\pi\)
\(854\) −24.0000 −0.821263
\(855\) 8.00000 0.273594
\(856\) −4.00000 −0.136717
\(857\) 42.0000 1.43469 0.717346 − 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 2.00000 0.0682789
\(859\) 44.0000 1.50126 0.750630 − 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) 8.00000 0.272798
\(861\) −40.0000 −1.36320
\(862\) 12.0000 0.408722
\(863\) 48.0000 1.63394 0.816970 − 0.576681i \(-0.195652\pi\)
0.816970 + 0.576681i \(0.195652\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 20.0000 0.680020
\(866\) 2.00000 0.0679628
\(867\) −1.00000 −0.0339618
\(868\) −16.0000 −0.543075
\(869\) 4.00000 0.135691
\(870\) 12.0000 0.406838
\(871\) −24.0000 −0.813209
\(872\) 2.00000 0.0677285
\(873\) 10.0000 0.338449
\(874\) −16.0000 −0.541208
\(875\) −48.0000 −1.62270
\(876\) −2.00000 −0.0675737
\(877\) 50.0000 1.68838 0.844190 − 0.536044i \(-0.180082\pi\)
0.844190 + 0.536044i \(0.180082\pi\)
\(878\) 20.0000 0.674967
\(879\) −6.00000 −0.202375
\(880\) 2.00000 0.0674200
\(881\) 34.0000 1.14549 0.572745 − 0.819734i \(-0.305879\pi\)
0.572745 + 0.819734i \(0.305879\pi\)
\(882\) 9.00000 0.303046
\(883\) −52.0000 −1.74994 −0.874970 − 0.484178i \(-0.839119\pi\)
−0.874970 + 0.484178i \(0.839119\pi\)
\(884\) −2.00000 −0.0672673
\(885\) 8.00000 0.268917
\(886\) 20.0000 0.671913
\(887\) −4.00000 −0.134307 −0.0671534 − 0.997743i \(-0.521392\pi\)
−0.0671534 + 0.997743i \(0.521392\pi\)
\(888\) −2.00000 −0.0671156
\(889\) −64.0000 −2.14649
\(890\) 20.0000 0.670402
\(891\) 1.00000 0.0335013
\(892\) 24.0000 0.803579
\(893\) −32.0000 −1.07084
\(894\) −6.00000 −0.200670
\(895\) −40.0000 −1.33705
\(896\) 4.00000 0.133631
\(897\) −8.00000 −0.267112
\(898\) 10.0000 0.333704
\(899\) 24.0000 0.800445
\(900\) −1.00000 −0.0333333
\(901\) 6.00000 0.199889
\(902\) 10.0000 0.332964
\(903\) −16.0000 −0.532447
\(904\) 18.0000 0.598671
\(905\) −28.0000 −0.930751
\(906\) 8.00000 0.265782
\(907\) −4.00000 −0.132818 −0.0664089 − 0.997792i \(-0.521154\pi\)
−0.0664089 + 0.997792i \(0.521154\pi\)
\(908\) −4.00000 −0.132745
\(909\) −2.00000 −0.0663358
\(910\) −16.0000 −0.530395
\(911\) −52.0000 −1.72284 −0.861418 − 0.507896i \(-0.830423\pi\)
−0.861418 + 0.507896i \(0.830423\pi\)
\(912\) −4.00000 −0.132453
\(913\) −4.00000 −0.132381
\(914\) −22.0000 −0.727695
\(915\) 12.0000 0.396708
\(916\) 22.0000 0.726900
\(917\) −48.0000 −1.58510
\(918\) −1.00000 −0.0330049
\(919\) −56.0000 −1.84727 −0.923635 − 0.383274i \(-0.874797\pi\)
−0.923635 + 0.383274i \(0.874797\pi\)
\(920\) −8.00000 −0.263752
\(921\) −28.0000 −0.922631
\(922\) −10.0000 −0.329332
\(923\) 24.0000 0.789970
\(924\) −4.00000 −0.131590
\(925\) −2.00000 −0.0657596
\(926\) 8.00000 0.262896
\(927\) −16.0000 −0.525509
\(928\) −6.00000 −0.196960
\(929\) −6.00000 −0.196854 −0.0984268 − 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 8.00000 0.262330
\(931\) 36.0000 1.17985
\(932\) −14.0000 −0.458585
\(933\) −12.0000 −0.392862
\(934\) −20.0000 −0.654420
\(935\) 2.00000 0.0654070
\(936\) −2.00000 −0.0653720
\(937\) −22.0000 −0.718709 −0.359354 − 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) 48.0000 1.56726
\(939\) −2.00000 −0.0652675
\(940\) −16.0000 −0.521862
\(941\) −30.0000 −0.977972 −0.488986 − 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) 2.00000 0.0651635
\(943\) −40.0000 −1.30258
\(944\) −4.00000 −0.130189
\(945\) −8.00000 −0.260240
\(946\) 4.00000 0.130051
\(947\) 52.0000 1.68977 0.844886 − 0.534946i \(-0.179668\pi\)
0.844886 + 0.534946i \(0.179668\pi\)
\(948\) −4.00000 −0.129914
\(949\) −4.00000 −0.129845
\(950\) −4.00000 −0.129777
\(951\) −18.0000 −0.583690
\(952\) 4.00000 0.129641
\(953\) 26.0000 0.842223 0.421111 − 0.907009i \(-0.361640\pi\)
0.421111 + 0.907009i \(0.361640\pi\)
\(954\) 6.00000 0.194257
\(955\) −48.0000 −1.55324
\(956\) −24.0000 −0.776215
\(957\) 6.00000 0.193952
\(958\) 20.0000 0.646171
\(959\) −88.0000 −2.84167
\(960\) −2.00000 −0.0645497
\(961\) −15.0000 −0.483871
\(962\) −4.00000 −0.128965
\(963\) −4.00000 −0.128898
\(964\) 2.00000 0.0644157
\(965\) −28.0000 −0.901352
\(966\) 16.0000 0.514792
\(967\) −16.0000 −0.514525 −0.257263 − 0.966342i \(-0.582821\pi\)
−0.257263 + 0.966342i \(0.582821\pi\)
\(968\) 1.00000 0.0321412
\(969\) −4.00000 −0.128499
\(970\) 20.0000 0.642161
\(971\) −36.0000 −1.15529 −0.577647 − 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 48.0000 1.53881
\(974\) −28.0000 −0.897178
\(975\) −2.00000 −0.0640513
\(976\) −6.00000 −0.192055
\(977\) 50.0000 1.59964 0.799821 − 0.600239i \(-0.204928\pi\)
0.799821 + 0.600239i \(0.204928\pi\)
\(978\) −12.0000 −0.383718
\(979\) 10.0000 0.319601
\(980\) 18.0000 0.574989
\(981\) 2.00000 0.0638551
\(982\) 12.0000 0.382935
\(983\) −28.0000 −0.893061 −0.446531 − 0.894768i \(-0.647341\pi\)
−0.446531 + 0.894768i \(0.647341\pi\)
\(984\) −10.0000 −0.318788
\(985\) −12.0000 −0.382352
\(986\) −6.00000 −0.191079
\(987\) 32.0000 1.01857
\(988\) −8.00000 −0.254514
\(989\) −16.0000 −0.508770
\(990\) 2.00000 0.0635642
\(991\) −12.0000 −0.381193 −0.190596 − 0.981669i \(-0.561042\pi\)
−0.190596 + 0.981669i \(0.561042\pi\)
\(992\) −4.00000 −0.127000
\(993\) 20.0000 0.634681
\(994\) −48.0000 −1.52247
\(995\) 24.0000 0.760851
\(996\) 4.00000 0.126745
\(997\) −46.0000 −1.45683 −0.728417 − 0.685134i \(-0.759744\pi\)
−0.728417 + 0.685134i \(0.759744\pi\)
\(998\) 20.0000 0.633089
\(999\) −2.00000 −0.0632772
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 1122.2.a.h.1.1 ✓ 1
3.2 odd 2 3366.2.a.d.1.1 1
4.3 odd 2 8976.2.a.z.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1122.2.a.h.1.1 ✓ 1 1.1 even 1 trivial
3366.2.a.d.1.1 1 3.2 odd 2
8976.2.a.z.1.1 1 4.3 odd 2