↑

Properties

Label 1122.2.a.g.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

Downloads

Learn more

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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} +4.00000 q^{13} -2.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} -2.00000 q^{19} +2.00000 q^{20} +2.00000 q^{21} +1.00000 q^{22} +2.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} +4.00000 q^{26} -1.00000 q^{27} -2.00000 q^{28} +6.00000 q^{29} -2.00000 q^{30} +8.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} +1.00000 q^{34} -4.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} -2.00000 q^{38} -4.00000 q^{39} +2.00000 q^{40} -2.00000 q^{41} +2.00000 q^{42} -2.00000 q^{43} +1.00000 q^{44} +2.00000 q^{45} +2.00000 q^{46} +4.00000 q^{47} -1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} -1.00000 q^{51} +4.00000 q^{52} +12.0000 q^{53} -1.00000 q^{54} +2.00000 q^{55} -2.00000 q^{56} +2.00000 q^{57} +6.00000 q^{58} -10.0000 q^{59} -2.00000 q^{60} +6.00000 q^{61} +8.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} +8.00000 q^{65} -1.00000 q^{66} -12.0000 q^{67} +1.00000 q^{68} -2.00000 q^{69} -4.00000 q^{70} +6.00000 q^{71} +1.00000 q^{72} +8.00000 q^{73} +2.00000 q^{74} +1.00000 q^{75} -2.00000 q^{76} -2.00000 q^{77} -4.00000 q^{78} -2.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -4.00000 q^{83} +2.00000 q^{84} +2.00000 q^{85} -2.00000 q^{86} -6.00000 q^{87} +1.00000 q^{88} -2.00000 q^{89} +2.00000 q^{90} -8.00000 q^{91} +2.00000 q^{92} -8.00000 q^{93} +4.00000 q^{94} -4.00000 q^{95} -1.00000 q^{96} +10.0000 q^{97} -3.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) 1.00000 0.301511
\(12\) −1.00000 −0.288675
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) −2.00000 −0.534522
\(15\) −2.00000 −0.516398
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536
\(18\) 1.00000 0.235702
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 2.00000 0.447214
\(21\) 2.00000 0.436436
\(22\) 1.00000 0.213201
\(23\) 2.00000 0.417029 0.208514 − 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) −1.00000 −0.204124
\(25\) −1.00000 −0.200000
\(26\) 4.00000 0.784465
\(27\) −1.00000 −0.192450
\(28\) −2.00000 −0.377964
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) −2.00000 −0.365148
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.00000 −0.174078
\(34\) 1.00000 0.171499
\(35\) −4.00000 −0.676123
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −2.00000 −0.324443
\(39\) −4.00000 −0.640513
\(40\) 2.00000 0.316228
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 2.00000 0.308607
\(43\) −2.00000 −0.304997 −0.152499 − 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 1.00000 0.150756
\(45\) 2.00000 0.298142
\(46\) 2.00000 0.294884
\(47\) 4.00000 0.583460 0.291730 − 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.00000 −0.428571
\(50\) −1.00000 −0.141421
\(51\) −1.00000 −0.140028
\(52\) 4.00000 0.554700
\(53\) 12.0000 1.64833 0.824163 − 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) −1.00000 −0.136083
\(55\) 2.00000 0.269680
\(56\) −2.00000 −0.267261
\(57\) 2.00000 0.264906
\(58\) 6.00000 0.787839
\(59\) −10.0000 −1.30189 −0.650945 − 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) −2.00000 −0.258199
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 8.00000 1.01600
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) 8.00000 0.992278
\(66\) −1.00000 −0.123091
\(67\) −12.0000 −1.46603 −0.733017 − 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 1.00000 0.121268
\(69\) −2.00000 −0.240772
\(70\) −4.00000 −0.478091
\(71\) 6.00000 0.712069 0.356034 − 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 1.00000 0.117851
\(73\) 8.00000 0.936329 0.468165 − 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) 2.00000 0.232495
\(75\) 1.00000 0.115470
\(76\) −2.00000 −0.229416
\(77\) −2.00000 −0.227921
\(78\) −4.00000 −0.452911
\(79\) −2.00000 −0.225018 −0.112509 − 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 2.00000 0.218218
\(85\) 2.00000 0.216930
\(86\) −2.00000 −0.215666
\(87\) −6.00000 −0.643268
\(88\) 1.00000 0.106600
\(89\) −2.00000 −0.212000 −0.106000 − 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 2.00000 0.210819
\(91\) −8.00000 −0.838628
\(92\) 2.00000 0.208514
\(93\) −8.00000 −0.829561
\(94\) 4.00000 0.412568
\(95\) −4.00000 −0.410391
\(96\) −1.00000 −0.102062
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) −3.00000 −0.303046
\(99\) 1.00000 0.100504
\(100\) −1.00000 −0.100000
\(101\) −14.0000 −1.39305 −0.696526 − 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) −1.00000 −0.0990148
\(103\) 8.00000 0.788263 0.394132 − 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 4.00000 0.392232
\(105\) 4.00000 0.390360
\(106\) 12.0000 1.16554
\(107\) −4.00000 −0.386695 −0.193347 − 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) −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
\(112\) −2.00000 −0.188982
\(113\) −12.0000 −1.12887 −0.564433 − 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) 2.00000 0.187317
\(115\) 4.00000 0.373002
\(116\) 6.00000 0.557086
\(117\) 4.00000 0.369800
\(118\) −10.0000 −0.920575
\(119\) −2.00000 −0.183340
\(120\) −2.00000 −0.182574
\(121\) 1.00000 0.0909091
\(122\) 6.00000 0.543214
\(123\) 2.00000 0.180334
\(124\) 8.00000 0.718421
\(125\) −12.0000 −1.07331
\(126\) −2.00000 −0.178174
\(127\) −4.00000 −0.354943 −0.177471 − 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) 1.00000 0.0883883
\(129\) 2.00000 0.176090
\(130\) 8.00000 0.701646
\(131\) −12.0000 −1.04844 −0.524222 − 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 4.00000 0.346844
\(134\) −12.0000 −1.03664
\(135\) −2.00000 −0.172133
\(136\) 1.00000 0.0857493
\(137\) 2.00000 0.170872 0.0854358 − 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) −2.00000 −0.170251
\(139\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) −4.00000 −0.338062
\(141\) −4.00000 −0.336861
\(142\) 6.00000 0.503509
\(143\) 4.00000 0.334497
\(144\) 1.00000 0.0833333
\(145\) 12.0000 0.996546
\(146\) 8.00000 0.662085
\(147\) 3.00000 0.247436
\(148\) 2.00000 0.164399
\(149\) −18.0000 −1.47462 −0.737309 − 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 1.00000 0.0816497
\(151\) −8.00000 −0.651031 −0.325515 − 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) −2.00000 −0.162221
\(153\) 1.00000 0.0808452
\(154\) −2.00000 −0.161165
\(155\) 16.0000 1.28515
\(156\) −4.00000 −0.320256
\(157\) 10.0000 0.798087 0.399043 − 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) −2.00000 −0.159111
\(159\) −12.0000 −0.951662
\(160\) 2.00000 0.158114
\(161\) −4.00000 −0.315244
\(162\) 1.00000 0.0785674
\(163\) 12.0000 0.939913 0.469956 − 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) −2.00000 −0.156174
\(165\) −2.00000 −0.155700
\(166\) −4.00000 −0.310460
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 2.00000 0.154303
\(169\) 3.00000 0.230769
\(170\) 2.00000 0.153393
\(171\) −2.00000 −0.152944
\(172\) −2.00000 −0.152499
\(173\) −14.0000 −1.06440 −0.532200 − 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) −6.00000 −0.454859
\(175\) 2.00000 0.151186
\(176\) 1.00000 0.0753778
\(177\) 10.0000 0.751646
\(178\) −2.00000 −0.149906
\(179\) 10.0000 0.747435 0.373718 − 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 2.00000 0.149071
\(181\) 10.0000 0.743294 0.371647 − 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) −8.00000 −0.592999
\(183\) −6.00000 −0.443533
\(184\) 2.00000 0.147442
\(185\) 4.00000 0.294086
\(186\) −8.00000 −0.586588
\(187\) 1.00000 0.0731272
\(188\) 4.00000 0.291730
\(189\) 2.00000 0.145479
\(190\) −4.00000 −0.290191
\(191\) −24.0000 −1.73658 −0.868290 − 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −20.0000 −1.43963 −0.719816 − 0.694165i \(-0.755774\pi\)
−0.719816 + 0.694165i \(0.755774\pi\)
\(194\) 10.0000 0.717958
\(195\) −8.00000 −0.572892
\(196\) −3.00000 −0.214286
\(197\) 6.00000 0.427482 0.213741 − 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 1.00000 0.0710669
\(199\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 12.0000 0.846415
\(202\) −14.0000 −0.985037
\(203\) −12.0000 −0.842235
\(204\) −1.00000 −0.0700140
\(205\) −4.00000 −0.279372
\(206\) 8.00000 0.557386
\(207\) 2.00000 0.139010
\(208\) 4.00000 0.277350
\(209\) −2.00000 −0.138343
\(210\) 4.00000 0.276026
\(211\) −16.0000 −1.10149 −0.550743 − 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 12.0000 0.824163
\(213\) −6.00000 −0.411113
\(214\) −4.00000 −0.273434
\(215\) −4.00000 −0.272798
\(216\) −1.00000 −0.0680414
\(217\) −16.0000 −1.08615
\(218\) 2.00000 0.135457
\(219\) −8.00000 −0.540590
\(220\) 2.00000 0.134840
\(221\) 4.00000 0.269069
\(222\) −2.00000 −0.134231
\(223\) −24.0000 −1.60716 −0.803579 − 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) −2.00000 −0.133631
\(225\) −1.00000 −0.0666667
\(226\) −12.0000 −0.798228
\(227\) −28.0000 −1.85843 −0.929213 − 0.369546i \(-0.879513\pi\)
−0.929213 + 0.369546i \(0.879513\pi\)
\(228\) 2.00000 0.132453
\(229\) −14.0000 −0.925146 −0.462573 − 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 4.00000 0.263752
\(231\) 2.00000 0.131590
\(232\) 6.00000 0.393919
\(233\) −2.00000 −0.131024 −0.0655122 − 0.997852i \(-0.520868\pi\)
−0.0655122 + 0.997852i \(0.520868\pi\)
\(234\) 4.00000 0.261488
\(235\) 8.00000 0.521862
\(236\) −10.0000 −0.650945
\(237\) 2.00000 0.129914
\(238\) −2.00000 −0.129641
\(239\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) −2.00000 −0.129099
\(241\) −4.00000 −0.257663 −0.128831 − 0.991667i \(-0.541123\pi\)
−0.128831 + 0.991667i \(0.541123\pi\)
\(242\) 1.00000 0.0642824
\(243\) −1.00000 −0.0641500
\(244\) 6.00000 0.384111
\(245\) −6.00000 −0.383326
\(246\) 2.00000 0.127515
\(247\) −8.00000 −0.509028
\(248\) 8.00000 0.508001
\(249\) 4.00000 0.253490
\(250\) −12.0000 −0.758947
\(251\) 18.0000 1.13615 0.568075 − 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) −2.00000 −0.125988
\(253\) 2.00000 0.125739
\(254\) −4.00000 −0.250982
\(255\) −2.00000 −0.125245
\(256\) 1.00000 0.0625000
\(257\) −22.0000 −1.37232 −0.686161 − 0.727450i \(-0.740706\pi\)
−0.686161 + 0.727450i \(0.740706\pi\)
\(258\) 2.00000 0.124515
\(259\) −4.00000 −0.248548
\(260\) 8.00000 0.496139
\(261\) 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
\(265\) 24.0000 1.47431
\(266\) 4.00000 0.245256
\(267\) 2.00000 0.122398
\(268\) −12.0000 −0.733017
\(269\) −14.0000 −0.853595 −0.426798 − 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) −2.00000 −0.121716
\(271\) 20.0000 1.21491 0.607457 − 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 1.00000 0.0606339
\(273\) 8.00000 0.484182
\(274\) 2.00000 0.120824
\(275\) −1.00000 −0.0603023
\(276\) −2.00000 −0.120386
\(277\) 10.0000 0.600842 0.300421 − 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 0 0
\(279\) 8.00000 0.478947
\(280\) −4.00000 −0.239046
\(281\) −26.0000 −1.55103 −0.775515 − 0.631329i \(-0.782510\pi\)
−0.775515 + 0.631329i \(0.782510\pi\)
\(282\) −4.00000 −0.238197
\(283\) 16.0000 0.951101 0.475551 − 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) 6.00000 0.356034
\(285\) 4.00000 0.236940
\(286\) 4.00000 0.236525
\(287\) 4.00000 0.236113
\(288\) 1.00000 0.0589256
\(289\) 1.00000 0.0588235
\(290\) 12.0000 0.704664
\(291\) −10.0000 −0.586210
\(292\) 8.00000 0.468165
\(293\) 6.00000 0.350524 0.175262 − 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 3.00000 0.174964
\(295\) −20.0000 −1.16445
\(296\) 2.00000 0.116248
\(297\) −1.00000 −0.0580259
\(298\) −18.0000 −1.04271
\(299\) 8.00000 0.462652
\(300\) 1.00000 0.0577350
\(301\) 4.00000 0.230556
\(302\) −8.00000 −0.460348
\(303\) 14.0000 0.804279
\(304\) −2.00000 −0.114708
\(305\) 12.0000 0.687118
\(306\) 1.00000 0.0571662
\(307\) −14.0000 −0.799022 −0.399511 − 0.916728i \(-0.630820\pi\)
−0.399511 + 0.916728i \(0.630820\pi\)
\(308\) −2.00000 −0.113961
\(309\) −8.00000 −0.455104
\(310\) 16.0000 0.908739
\(311\) −18.0000 −1.02069 −0.510343 − 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) −4.00000 −0.226455
\(313\) 26.0000 1.46961 0.734803 − 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) 10.0000 0.564333
\(315\) −4.00000 −0.225374
\(316\) −2.00000 −0.112509
\(317\) 6.00000 0.336994 0.168497 − 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) −12.0000 −0.672927
\(319\) 6.00000 0.335936
\(320\) 2.00000 0.111803
\(321\) 4.00000 0.223258
\(322\) −4.00000 −0.222911
\(323\) −2.00000 −0.111283
\(324\) 1.00000 0.0555556
\(325\) −4.00000 −0.221880
\(326\) 12.0000 0.664619
\(327\) −2.00000 −0.110600
\(328\) −2.00000 −0.110432
\(329\) −8.00000 −0.441054
\(330\) −2.00000 −0.110096
\(331\) 28.0000 1.53902 0.769510 − 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) −4.00000 −0.219529
\(333\) 2.00000 0.109599
\(334\) 0 0
\(335\) −24.0000 −1.31126
\(336\) 2.00000 0.109109
\(337\) 8.00000 0.435788 0.217894 − 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) 3.00000 0.163178
\(339\) 12.0000 0.651751
\(340\) 2.00000 0.108465
\(341\) 8.00000 0.433224
\(342\) −2.00000 −0.108148
\(343\) 20.0000 1.07990
\(344\) −2.00000 −0.107833
\(345\) −4.00000 −0.215353
\(346\) −14.0000 −0.752645
\(347\) −4.00000 −0.214731 −0.107366 − 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) −6.00000 −0.321634
\(349\) −4.00000 −0.214115 −0.107058 − 0.994253i \(-0.534143\pi\)
−0.107058 + 0.994253i \(0.534143\pi\)
\(350\) 2.00000 0.106904
\(351\) −4.00000 −0.213504
\(352\) 1.00000 0.0533002
\(353\) 34.0000 1.80964 0.904819 − 0.425797i \(-0.140006\pi\)
0.904819 + 0.425797i \(0.140006\pi\)
\(354\) 10.0000 0.531494
\(355\) 12.0000 0.636894
\(356\) −2.00000 −0.106000
\(357\) 2.00000 0.105851
\(358\) 10.0000 0.528516
\(359\) 24.0000 1.26667 0.633336 − 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 2.00000 0.105409
\(361\) −15.0000 −0.789474
\(362\) 10.0000 0.525588
\(363\) −1.00000 −0.0524864
\(364\) −8.00000 −0.419314
\(365\) 16.0000 0.837478
\(366\) −6.00000 −0.313625
\(367\) 16.0000 0.835193 0.417597 − 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) 2.00000 0.104257
\(369\) −2.00000 −0.104116
\(370\) 4.00000 0.207950
\(371\) −24.0000 −1.24602
\(372\) −8.00000 −0.414781
\(373\) 4.00000 0.207112 0.103556 − 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) 1.00000 0.0517088
\(375\) 12.0000 0.619677
\(376\) 4.00000 0.206284
\(377\) 24.0000 1.23606
\(378\) 2.00000 0.102869
\(379\) −4.00000 −0.205466 −0.102733 − 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) −4.00000 −0.205196
\(381\) 4.00000 0.204926
\(382\) −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
\(385\) −4.00000 −0.203859
\(386\) −20.0000 −1.01797
\(387\) −2.00000 −0.101666
\(388\) 10.0000 0.507673
\(389\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) −8.00000 −0.405096
\(391\) 2.00000 0.101144
\(392\) −3.00000 −0.151523
\(393\) 12.0000 0.605320
\(394\) 6.00000 0.302276
\(395\) −4.00000 −0.201262
\(396\) 1.00000 0.0502519
\(397\) 14.0000 0.702640 0.351320 − 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 0 0
\(399\) −4.00000 −0.200250
\(400\) −1.00000 −0.0500000
\(401\) −12.0000 −0.599251 −0.299626 − 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) 12.0000 0.598506
\(403\) 32.0000 1.59403
\(404\) −14.0000 −0.696526
\(405\) 2.00000 0.0993808
\(406\) −12.0000 −0.595550
\(407\) 2.00000 0.0991363
\(408\) −1.00000 −0.0495074
\(409\) 6.00000 0.296681 0.148340 − 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) −4.00000 −0.197546
\(411\) −2.00000 −0.0986527
\(412\) 8.00000 0.394132
\(413\) 20.0000 0.984136
\(414\) 2.00000 0.0982946
\(415\) −8.00000 −0.392705
\(416\) 4.00000 0.196116
\(417\) 0 0
\(418\) −2.00000 −0.0978232
\(419\) 28.0000 1.36789 0.683945 − 0.729534i \(-0.260263\pi\)
0.683945 + 0.729534i \(0.260263\pi\)
\(420\) 4.00000 0.195180
\(421\) −2.00000 −0.0974740 −0.0487370 − 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) −16.0000 −0.778868
\(423\) 4.00000 0.194487
\(424\) 12.0000 0.582772
\(425\) −1.00000 −0.0485071
\(426\) −6.00000 −0.290701
\(427\) −12.0000 −0.580721
\(428\) −4.00000 −0.193347
\(429\) −4.00000 −0.193122
\(430\) −4.00000 −0.192897
\(431\) −36.0000 −1.73406 −0.867029 − 0.498257i \(-0.833974\pi\)
−0.867029 + 0.498257i \(0.833974\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 14.0000 0.672797 0.336399 − 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) −16.0000 −0.768025
\(435\) −12.0000 −0.575356
\(436\) 2.00000 0.0957826
\(437\) −4.00000 −0.191346
\(438\) −8.00000 −0.382255
\(439\) 14.0000 0.668184 0.334092 − 0.942541i \(-0.391570\pi\)
0.334092 + 0.942541i \(0.391570\pi\)
\(440\) 2.00000 0.0953463
\(441\) −3.00000 −0.142857
\(442\) 4.00000 0.190261
\(443\) 14.0000 0.665160 0.332580 − 0.943075i \(-0.392081\pi\)
0.332580 + 0.943075i \(0.392081\pi\)
\(444\) −2.00000 −0.0949158
\(445\) −4.00000 −0.189618
\(446\) −24.0000 −1.13643
\(447\) 18.0000 0.851371
\(448\) −2.00000 −0.0944911
\(449\) −8.00000 −0.377543 −0.188772 − 0.982021i \(-0.560451\pi\)
−0.188772 + 0.982021i \(0.560451\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −2.00000 −0.0941763
\(452\) −12.0000 −0.564433
\(453\) 8.00000 0.375873
\(454\) −28.0000 −1.31411
\(455\) −16.0000 −0.750092
\(456\) 2.00000 0.0936586
\(457\) 14.0000 0.654892 0.327446 − 0.944870i \(-0.393812\pi\)
0.327446 + 0.944870i \(0.393812\pi\)
\(458\) −14.0000 −0.654177
\(459\) −1.00000 −0.0466760
\(460\) 4.00000 0.186501
\(461\) −10.0000 −0.465746 −0.232873 − 0.972507i \(-0.574813\pi\)
−0.232873 + 0.972507i \(0.574813\pi\)
\(462\) 2.00000 0.0930484
\(463\) 32.0000 1.48717 0.743583 − 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) 6.00000 0.278543
\(465\) −16.0000 −0.741982
\(466\) −2.00000 −0.0926482
\(467\) −2.00000 −0.0925490 −0.0462745 − 0.998929i \(-0.514735\pi\)
−0.0462745 + 0.998929i \(0.514735\pi\)
\(468\) 4.00000 0.184900
\(469\) 24.0000 1.10822
\(470\) 8.00000 0.369012
\(471\) −10.0000 −0.460776
\(472\) −10.0000 −0.460287
\(473\) −2.00000 −0.0919601
\(474\) 2.00000 0.0918630
\(475\) 2.00000 0.0917663
\(476\) −2.00000 −0.0916698
\(477\) 12.0000 0.549442
\(478\) 0 0
\(479\) 8.00000 0.365529 0.182765 − 0.983157i \(-0.441495\pi\)
0.182765 + 0.983157i \(0.441495\pi\)
\(480\) −2.00000 −0.0912871
\(481\) 8.00000 0.364769
\(482\) −4.00000 −0.182195
\(483\) 4.00000 0.182006
\(484\) 1.00000 0.0454545
\(485\) 20.0000 0.908153
\(486\) −1.00000 −0.0453609
\(487\) −40.0000 −1.81257 −0.906287 − 0.422664i \(-0.861095\pi\)
−0.906287 + 0.422664i \(0.861095\pi\)
\(488\) 6.00000 0.271607
\(489\) −12.0000 −0.542659
\(490\) −6.00000 −0.271052
\(491\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 2.00000 0.0901670
\(493\) 6.00000 0.270226
\(494\) −8.00000 −0.359937
\(495\) 2.00000 0.0898933
\(496\) 8.00000 0.359211
\(497\) −12.0000 −0.538274
\(498\) 4.00000 0.179244
\(499\) −28.0000 −1.25345 −0.626726 − 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) −12.0000 −0.536656
\(501\) 0 0
\(502\) 18.0000 0.803379
\(503\) 36.0000 1.60516 0.802580 − 0.596544i \(-0.203460\pi\)
0.802580 + 0.596544i \(0.203460\pi\)
\(504\) −2.00000 −0.0890871
\(505\) −28.0000 −1.24598
\(506\) 2.00000 0.0889108
\(507\) −3.00000 −0.133235
\(508\) −4.00000 −0.177471
\(509\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) −2.00000 −0.0885615
\(511\) −16.0000 −0.707798
\(512\) 1.00000 0.0441942
\(513\) 2.00000 0.0883022
\(514\) −22.0000 −0.970378
\(515\) 16.0000 0.705044
\(516\) 2.00000 0.0880451
\(517\) 4.00000 0.175920
\(518\) −4.00000 −0.175750
\(519\) 14.0000 0.614532
\(520\) 8.00000 0.350823
\(521\) 12.0000 0.525730 0.262865 − 0.964833i \(-0.415333\pi\)
0.262865 + 0.964833i \(0.415333\pi\)
\(522\) 6.00000 0.262613
\(523\) −22.0000 −0.961993 −0.480996 − 0.876723i \(-0.659725\pi\)
−0.480996 + 0.876723i \(0.659725\pi\)
\(524\) −12.0000 −0.524222
\(525\) −2.00000 −0.0872872
\(526\) −16.0000 −0.697633
\(527\) 8.00000 0.348485
\(528\) −1.00000 −0.0435194
\(529\) −19.0000 −0.826087
\(530\) 24.0000 1.04249
\(531\) −10.0000 −0.433963
\(532\) 4.00000 0.173422
\(533\) −8.00000 −0.346518
\(534\) 2.00000 0.0865485
\(535\) −8.00000 −0.345870
\(536\) −12.0000 −0.518321
\(537\) −10.0000 −0.431532
\(538\) −14.0000 −0.603583
\(539\) −3.00000 −0.129219
\(540\) −2.00000 −0.0860663
\(541\) 22.0000 0.945854 0.472927 − 0.881102i \(-0.343197\pi\)
0.472927 + 0.881102i \(0.343197\pi\)
\(542\) 20.0000 0.859074
\(543\) −10.0000 −0.429141
\(544\) 1.00000 0.0428746
\(545\) 4.00000 0.171341
\(546\) 8.00000 0.342368
\(547\) 20.0000 0.855138 0.427569 − 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) 2.00000 0.0854358
\(549\) 6.00000 0.256074
\(550\) −1.00000 −0.0426401
\(551\) −12.0000 −0.511217
\(552\) −2.00000 −0.0851257
\(553\) 4.00000 0.170097
\(554\) 10.0000 0.424859
\(555\) −4.00000 −0.169791
\(556\) 0 0
\(557\) −42.0000 −1.77960 −0.889799 − 0.456354i \(-0.849155\pi\)
−0.889799 + 0.456354i \(0.849155\pi\)
\(558\) 8.00000 0.338667
\(559\) −8.00000 −0.338364
\(560\) −4.00000 −0.169031
\(561\) −1.00000 −0.0422200
\(562\) −26.0000 −1.09674
\(563\) −4.00000 −0.168580 −0.0842900 − 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) −4.00000 −0.168430
\(565\) −24.0000 −1.00969
\(566\) 16.0000 0.672530
\(567\) −2.00000 −0.0839921
\(568\) 6.00000 0.251754
\(569\) −22.0000 −0.922288 −0.461144 − 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) 4.00000 0.167542
\(571\) 40.0000 1.67395 0.836974 − 0.547243i \(-0.184323\pi\)
0.836974 + 0.547243i \(0.184323\pi\)
\(572\) 4.00000 0.167248
\(573\) 24.0000 1.00261
\(574\) 4.00000 0.166957
\(575\) −2.00000 −0.0834058
\(576\) 1.00000 0.0416667
\(577\) −34.0000 −1.41544 −0.707719 − 0.706494i \(-0.750276\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(578\) 1.00000 0.0415945
\(579\) 20.0000 0.831172
\(580\) 12.0000 0.498273
\(581\) 8.00000 0.331896
\(582\) −10.0000 −0.414513
\(583\) 12.0000 0.496989
\(584\) 8.00000 0.331042
\(585\) 8.00000 0.330759
\(586\) 6.00000 0.247858
\(587\) 30.0000 1.23823 0.619116 − 0.785299i \(-0.287491\pi\)
0.619116 + 0.785299i \(0.287491\pi\)
\(588\) 3.00000 0.123718
\(589\) −16.0000 −0.659269
\(590\) −20.0000 −0.823387
\(591\) −6.00000 −0.246807
\(592\) 2.00000 0.0821995
\(593\) 30.0000 1.23195 0.615976 − 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) −1.00000 −0.0410305
\(595\) −4.00000 −0.163984
\(596\) −18.0000 −0.737309
\(597\) 0 0
\(598\) 8.00000 0.327144
\(599\) −16.0000 −0.653742 −0.326871 − 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) 1.00000 0.0408248
\(601\) 24.0000 0.978980 0.489490 − 0.872009i \(-0.337183\pi\)
0.489490 + 0.872009i \(0.337183\pi\)
\(602\) 4.00000 0.163028
\(603\) −12.0000 −0.488678
\(604\) −8.00000 −0.325515
\(605\) 2.00000 0.0813116
\(606\) 14.0000 0.568711
\(607\) 6.00000 0.243532 0.121766 − 0.992559i \(-0.461144\pi\)
0.121766 + 0.992559i \(0.461144\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 12.0000 0.486265
\(610\) 12.0000 0.485866
\(611\) 16.0000 0.647291
\(612\) 1.00000 0.0404226
\(613\) −20.0000 −0.807792 −0.403896 − 0.914805i \(-0.632344\pi\)
−0.403896 + 0.914805i \(0.632344\pi\)
\(614\) −14.0000 −0.564994
\(615\) 4.00000 0.161296
\(616\) −2.00000 −0.0805823
\(617\) 8.00000 0.322068 0.161034 − 0.986949i \(-0.448517\pi\)
0.161034 + 0.986949i \(0.448517\pi\)
\(618\) −8.00000 −0.321807
\(619\) −28.0000 −1.12542 −0.562708 − 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 16.0000 0.642575
\(621\) −2.00000 −0.0802572
\(622\) −18.0000 −0.721734
\(623\) 4.00000 0.160257
\(624\) −4.00000 −0.160128
\(625\) −19.0000 −0.760000
\(626\) 26.0000 1.03917
\(627\) 2.00000 0.0798723
\(628\) 10.0000 0.399043
\(629\) 2.00000 0.0797452
\(630\) −4.00000 −0.159364
\(631\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) −2.00000 −0.0795557
\(633\) 16.0000 0.635943
\(634\) 6.00000 0.238290
\(635\) −8.00000 −0.317470
\(636\) −12.0000 −0.475831
\(637\) −12.0000 −0.475457
\(638\) 6.00000 0.237542
\(639\) 6.00000 0.237356
\(640\) 2.00000 0.0790569
\(641\) −12.0000 −0.473972 −0.236986 − 0.971513i \(-0.576159\pi\)
−0.236986 + 0.971513i \(0.576159\pi\)
\(642\) 4.00000 0.157867
\(643\) 36.0000 1.41970 0.709851 − 0.704352i \(-0.248762\pi\)
0.709851 + 0.704352i \(0.248762\pi\)
\(644\) −4.00000 −0.157622
\(645\) 4.00000 0.157500
\(646\) −2.00000 −0.0786889
\(647\) −12.0000 −0.471769 −0.235884 − 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 1.00000 0.0392837
\(649\) −10.0000 −0.392534
\(650\) −4.00000 −0.156893
\(651\) 16.0000 0.627089
\(652\) 12.0000 0.469956
\(653\) 26.0000 1.01746 0.508729 − 0.860927i \(-0.330115\pi\)
0.508729 + 0.860927i \(0.330115\pi\)
\(654\) −2.00000 −0.0782062
\(655\) −24.0000 −0.937758
\(656\) −2.00000 −0.0780869
\(657\) 8.00000 0.312110
\(658\) −8.00000 −0.311872
\(659\) −8.00000 −0.311636 −0.155818 − 0.987786i \(-0.549801\pi\)
−0.155818 + 0.987786i \(0.549801\pi\)
\(660\) −2.00000 −0.0778499
\(661\) 42.0000 1.63361 0.816805 − 0.576913i \(-0.195743\pi\)
0.816805 + 0.576913i \(0.195743\pi\)
\(662\) 28.0000 1.08825
\(663\) −4.00000 −0.155347
\(664\) −4.00000 −0.155230
\(665\) 8.00000 0.310227
\(666\) 2.00000 0.0774984
\(667\) 12.0000 0.464642
\(668\) 0 0
\(669\) 24.0000 0.927894
\(670\) −24.0000 −0.927201
\(671\) 6.00000 0.231627
\(672\) 2.00000 0.0771517
\(673\) −20.0000 −0.770943 −0.385472 − 0.922720i \(-0.625961\pi\)
−0.385472 + 0.922720i \(0.625961\pi\)
\(674\) 8.00000 0.308148
\(675\) 1.00000 0.0384900
\(676\) 3.00000 0.115385
\(677\) 22.0000 0.845529 0.422764 − 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) 12.0000 0.460857
\(679\) −20.0000 −0.767530
\(680\) 2.00000 0.0766965
\(681\) 28.0000 1.07296
\(682\) 8.00000 0.306336
\(683\) 36.0000 1.37750 0.688751 − 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) −2.00000 −0.0764719
\(685\) 4.00000 0.152832
\(686\) 20.0000 0.763604
\(687\) 14.0000 0.534133
\(688\) −2.00000 −0.0762493
\(689\) 48.0000 1.82865
\(690\) −4.00000 −0.152277
\(691\) −28.0000 −1.06517 −0.532585 − 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) −14.0000 −0.532200
\(693\) −2.00000 −0.0759737
\(694\) −4.00000 −0.151838
\(695\) 0 0
\(696\) −6.00000 −0.227429
\(697\) −2.00000 −0.0757554
\(698\) −4.00000 −0.151402
\(699\) 2.00000 0.0756469
\(700\) 2.00000 0.0755929
\(701\) −34.0000 −1.28416 −0.642081 − 0.766637i \(-0.721929\pi\)
−0.642081 + 0.766637i \(0.721929\pi\)
\(702\) −4.00000 −0.150970
\(703\) −4.00000 −0.150863
\(704\) 1.00000 0.0376889
\(705\) −8.00000 −0.301297
\(706\) 34.0000 1.27961
\(707\) 28.0000 1.05305
\(708\) 10.0000 0.375823
\(709\) 46.0000 1.72757 0.863783 − 0.503864i \(-0.168089\pi\)
0.863783 + 0.503864i \(0.168089\pi\)
\(710\) 12.0000 0.450352
\(711\) −2.00000 −0.0750059
\(712\) −2.00000 −0.0749532
\(713\) 16.0000 0.599205
\(714\) 2.00000 0.0748481
\(715\) 8.00000 0.299183
\(716\) 10.0000 0.373718
\(717\) 0 0
\(718\) 24.0000 0.895672
\(719\) −14.0000 −0.522112 −0.261056 − 0.965324i \(-0.584071\pi\)
−0.261056 + 0.965324i \(0.584071\pi\)
\(720\) 2.00000 0.0745356
\(721\) −16.0000 −0.595871
\(722\) −15.0000 −0.558242
\(723\) 4.00000 0.148762
\(724\) 10.0000 0.371647
\(725\) −6.00000 −0.222834
\(726\) −1.00000 −0.0371135
\(727\) −40.0000 −1.48352 −0.741759 − 0.670667i \(-0.766008\pi\)
−0.741759 + 0.670667i \(0.766008\pi\)
\(728\) −8.00000 −0.296500
\(729\) 1.00000 0.0370370
\(730\) 16.0000 0.592187
\(731\) −2.00000 −0.0739727
\(732\) −6.00000 −0.221766
\(733\) 12.0000 0.443230 0.221615 − 0.975134i \(-0.428867\pi\)
0.221615 + 0.975134i \(0.428867\pi\)
\(734\) 16.0000 0.590571
\(735\) 6.00000 0.221313
\(736\) 2.00000 0.0737210
\(737\) −12.0000 −0.442026
\(738\) −2.00000 −0.0736210
\(739\) −50.0000 −1.83928 −0.919640 − 0.392763i \(-0.871519\pi\)
−0.919640 + 0.392763i \(0.871519\pi\)
\(740\) 4.00000 0.147043
\(741\) 8.00000 0.293887
\(742\) −24.0000 −0.881068
\(743\) −4.00000 −0.146746 −0.0733729 − 0.997305i \(-0.523376\pi\)
−0.0733729 + 0.997305i \(0.523376\pi\)
\(744\) −8.00000 −0.293294
\(745\) −36.0000 −1.31894
\(746\) 4.00000 0.146450
\(747\) −4.00000 −0.146352
\(748\) 1.00000 0.0365636
\(749\) 8.00000 0.292314
\(750\) 12.0000 0.438178
\(751\) −4.00000 −0.145962 −0.0729810 − 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 4.00000 0.145865
\(753\) −18.0000 −0.655956
\(754\) 24.0000 0.874028
\(755\) −16.0000 −0.582300
\(756\) 2.00000 0.0727393
\(757\) −10.0000 −0.363456 −0.181728 − 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) −4.00000 −0.145287
\(759\) −2.00000 −0.0725954
\(760\) −4.00000 −0.145095
\(761\) −50.0000 −1.81250 −0.906249 − 0.422744i \(-0.861067\pi\)
−0.906249 + 0.422744i \(0.861067\pi\)
\(762\) 4.00000 0.144905
\(763\) −4.00000 −0.144810
\(764\) −24.0000 −0.868290
\(765\) 2.00000 0.0723102
\(766\) 24.0000 0.867155
\(767\) −40.0000 −1.44432
\(768\) −1.00000 −0.0360844
\(769\) 26.0000 0.937584 0.468792 − 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) −4.00000 −0.144150
\(771\) 22.0000 0.792311
\(772\) −20.0000 −0.719816
\(773\) 40.0000 1.43870 0.719350 − 0.694648i \(-0.244440\pi\)
0.719350 + 0.694648i \(0.244440\pi\)
\(774\) −2.00000 −0.0718885
\(775\) −8.00000 −0.287368
\(776\) 10.0000 0.358979
\(777\) 4.00000 0.143499
\(778\) 0 0
\(779\) 4.00000 0.143315
\(780\) −8.00000 −0.286446
\(781\) 6.00000 0.214697
\(782\) 2.00000 0.0715199
\(783\) −6.00000 −0.214423
\(784\) −3.00000 −0.107143
\(785\) 20.0000 0.713831
\(786\) 12.0000 0.428026
\(787\) −20.0000 −0.712923 −0.356462 − 0.934310i \(-0.616017\pi\)
−0.356462 + 0.934310i \(0.616017\pi\)
\(788\) 6.00000 0.213741
\(789\) 16.0000 0.569615
\(790\) −4.00000 −0.142314
\(791\) 24.0000 0.853342
\(792\) 1.00000 0.0355335
\(793\) 24.0000 0.852265
\(794\) 14.0000 0.496841
\(795\) −24.0000 −0.851192
\(796\) 0 0
\(797\) 36.0000 1.27519 0.637593 − 0.770374i \(-0.279930\pi\)
0.637593 + 0.770374i \(0.279930\pi\)
\(798\) −4.00000 −0.141598
\(799\) 4.00000 0.141510
\(800\) −1.00000 −0.0353553
\(801\) −2.00000 −0.0706665
\(802\) −12.0000 −0.423735
\(803\) 8.00000 0.282314
\(804\) 12.0000 0.423207
\(805\) −8.00000 −0.281963
\(806\) 32.0000 1.12715
\(807\) 14.0000 0.492823
\(808\) −14.0000 −0.492518
\(809\) −38.0000 −1.33601 −0.668004 − 0.744157i \(-0.732851\pi\)
−0.668004 + 0.744157i \(0.732851\pi\)
\(810\) 2.00000 0.0702728
\(811\) 44.0000 1.54505 0.772524 − 0.634985i \(-0.218994\pi\)
0.772524 + 0.634985i \(0.218994\pi\)
\(812\) −12.0000 −0.421117
\(813\) −20.0000 −0.701431
\(814\) 2.00000 0.0701000
\(815\) 24.0000 0.840683
\(816\) −1.00000 −0.0350070
\(817\) 4.00000 0.139942
\(818\) 6.00000 0.209785
\(819\) −8.00000 −0.279543
\(820\) −4.00000 −0.139686
\(821\) −6.00000 −0.209401 −0.104701 − 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) −2.00000 −0.0697580
\(823\) −44.0000 −1.53374 −0.766872 − 0.641800i \(-0.778188\pi\)
−0.766872 + 0.641800i \(0.778188\pi\)
\(824\) 8.00000 0.278693
\(825\) 1.00000 0.0348155
\(826\) 20.0000 0.695889
\(827\) 12.0000 0.417281 0.208640 − 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 2.00000 0.0695048
\(829\) 22.0000 0.764092 0.382046 − 0.924143i \(-0.375220\pi\)
0.382046 + 0.924143i \(0.375220\pi\)
\(830\) −8.00000 −0.277684
\(831\) −10.0000 −0.346896
\(832\) 4.00000 0.138675
\(833\) −3.00000 −0.103944
\(834\) 0 0
\(835\) 0 0
\(836\) −2.00000 −0.0691714
\(837\) −8.00000 −0.276520
\(838\) 28.0000 0.967244
\(839\) −2.00000 −0.0690477 −0.0345238 − 0.999404i \(-0.510991\pi\)
−0.0345238 + 0.999404i \(0.510991\pi\)
\(840\) 4.00000 0.138013
\(841\) 7.00000 0.241379
\(842\) −2.00000 −0.0689246
\(843\) 26.0000 0.895488
\(844\) −16.0000 −0.550743
\(845\) 6.00000 0.206406
\(846\) 4.00000 0.137523
\(847\) −2.00000 −0.0687208
\(848\) 12.0000 0.412082
\(849\) −16.0000 −0.549119
\(850\) −1.00000 −0.0342997
\(851\) 4.00000 0.137118
\(852\) −6.00000 −0.205557
\(853\) −38.0000 −1.30110 −0.650548 − 0.759465i \(-0.725461\pi\)
−0.650548 + 0.759465i \(0.725461\pi\)
\(854\) −12.0000 −0.410632
\(855\) −4.00000 −0.136797
\(856\) −4.00000 −0.136717
\(857\) −30.0000 −1.02478 −0.512390 − 0.858753i \(-0.671240\pi\)
−0.512390 + 0.858753i \(0.671240\pi\)
\(858\) −4.00000 −0.136558
\(859\) −28.0000 −0.955348 −0.477674 − 0.878537i \(-0.658520\pi\)
−0.477674 + 0.878537i \(0.658520\pi\)
\(860\) −4.00000 −0.136399
\(861\) −4.00000 −0.136320
\(862\) −36.0000 −1.22616
\(863\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −28.0000 −0.952029
\(866\) 14.0000 0.475739
\(867\) −1.00000 −0.0339618
\(868\) −16.0000 −0.543075
\(869\) −2.00000 −0.0678454
\(870\) −12.0000 −0.406838
\(871\) −48.0000 −1.62642
\(872\) 2.00000 0.0677285
\(873\) 10.0000 0.338449
\(874\) −4.00000 −0.135302
\(875\) 24.0000 0.811348
\(876\) −8.00000 −0.270295
\(877\) 26.0000 0.877958 0.438979 − 0.898497i \(-0.355340\pi\)
0.438979 + 0.898497i \(0.355340\pi\)
\(878\) 14.0000 0.472477
\(879\) −6.00000 −0.202375
\(880\) 2.00000 0.0674200
\(881\) −8.00000 −0.269527 −0.134763 − 0.990878i \(-0.543027\pi\)
−0.134763 + 0.990878i \(0.543027\pi\)
\(882\) −3.00000 −0.101015
\(883\) −16.0000 −0.538443 −0.269221 − 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) 4.00000 0.134535
\(885\) 20.0000 0.672293
\(886\) 14.0000 0.470339
\(887\) 8.00000 0.268614 0.134307 − 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 8.00000 0.268311
\(890\) −4.00000 −0.134080
\(891\) 1.00000 0.0335013
\(892\) −24.0000 −0.803579
\(893\) −8.00000 −0.267710
\(894\) 18.0000 0.602010
\(895\) 20.0000 0.668526
\(896\) −2.00000 −0.0668153
\(897\) −8.00000 −0.267112
\(898\) −8.00000 −0.266963
\(899\) 48.0000 1.60089
\(900\) −1.00000 −0.0333333
\(901\) 12.0000 0.399778
\(902\) −2.00000 −0.0665927
\(903\) −4.00000 −0.133112
\(904\) −12.0000 −0.399114
\(905\) 20.0000 0.664822
\(906\) 8.00000 0.265782
\(907\) −28.0000 −0.929725 −0.464862 − 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) −28.0000 −0.929213
\(909\) −14.0000 −0.464351
\(910\) −16.0000 −0.530395
\(911\) 38.0000 1.25900 0.629498 − 0.777002i \(-0.283261\pi\)
0.629498 + 0.777002i \(0.283261\pi\)
\(912\) 2.00000 0.0662266
\(913\) −4.00000 −0.132381
\(914\) 14.0000 0.463079
\(915\) −12.0000 −0.396708
\(916\) −14.0000 −0.462573
\(917\) 24.0000 0.792550
\(918\) −1.00000 −0.0330049
\(919\) 40.0000 1.31948 0.659739 − 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 4.00000 0.131876
\(921\) 14.0000 0.461316
\(922\) −10.0000 −0.329332
\(923\) 24.0000 0.789970
\(924\) 2.00000 0.0657952
\(925\) −2.00000 −0.0657596
\(926\) 32.0000 1.05159
\(927\) 8.00000 0.262754
\(928\) 6.00000 0.196960
\(929\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) −16.0000 −0.524661
\(931\) 6.00000 0.196642
\(932\) −2.00000 −0.0655122
\(933\) 18.0000 0.589294
\(934\) −2.00000 −0.0654420
\(935\) 2.00000 0.0654070
\(936\) 4.00000 0.130744
\(937\) −58.0000 −1.89478 −0.947389 − 0.320085i \(-0.896288\pi\)
−0.947389 + 0.320085i \(0.896288\pi\)
\(938\) 24.0000 0.783628
\(939\) −26.0000 −0.848478
\(940\) 8.00000 0.260931
\(941\) −30.0000 −0.977972 −0.488986 − 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) −10.0000 −0.325818
\(943\) −4.00000 −0.130258
\(944\) −10.0000 −0.325472
\(945\) 4.00000 0.130120
\(946\) −2.00000 −0.0650256
\(947\) −8.00000 −0.259965 −0.129983 − 0.991516i \(-0.541492\pi\)
−0.129983 + 0.991516i \(0.541492\pi\)
\(948\) 2.00000 0.0649570
\(949\) 32.0000 1.03876
\(950\) 2.00000 0.0648886
\(951\) −6.00000 −0.194563
\(952\) −2.00000 −0.0648204
\(953\) 14.0000 0.453504 0.226752 − 0.973952i \(-0.427189\pi\)
0.226752 + 0.973952i \(0.427189\pi\)
\(954\) 12.0000 0.388514
\(955\) −48.0000 −1.55324
\(956\) 0 0
\(957\) −6.00000 −0.193952
\(958\) 8.00000 0.258468
\(959\) −4.00000 −0.129167
\(960\) −2.00000 −0.0645497
\(961\) 33.0000 1.06452
\(962\) 8.00000 0.257930
\(963\) −4.00000 −0.128898
\(964\) −4.00000 −0.128831
\(965\) −40.0000 −1.28765
\(966\) 4.00000 0.128698
\(967\) −28.0000 −0.900419 −0.450210 − 0.892923i \(-0.648651\pi\)
−0.450210 + 0.892923i \(0.648651\pi\)
\(968\) 1.00000 0.0321412
\(969\) 2.00000 0.0642493
\(970\) 20.0000 0.642161
\(971\) 30.0000 0.962746 0.481373 − 0.876516i \(-0.340138\pi\)
0.481373 + 0.876516i \(0.340138\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −40.0000 −1.28168
\(975\) 4.00000 0.128103
\(976\) 6.00000 0.192055
\(977\) 2.00000 0.0639857 0.0319928 − 0.999488i \(-0.489815\pi\)
0.0319928 + 0.999488i \(0.489815\pi\)
\(978\) −12.0000 −0.383718
\(979\) −2.00000 −0.0639203
\(980\) −6.00000 −0.191663
\(981\) 2.00000 0.0638551
\(982\) 0 0
\(983\) 26.0000 0.829271 0.414636 − 0.909988i \(-0.363909\pi\)
0.414636 + 0.909988i \(0.363909\pi\)
\(984\) 2.00000 0.0637577
\(985\) 12.0000 0.382352
\(986\) 6.00000 0.191079
\(987\) 8.00000 0.254643
\(988\) −8.00000 −0.254514
\(989\) −4.00000 −0.127193
\(990\) 2.00000 0.0635642
\(991\) −12.0000 −0.381193 −0.190596 − 0.981669i \(-0.561042\pi\)
−0.190596 + 0.981669i \(0.561042\pi\)
\(992\) 8.00000 0.254000
\(993\) −28.0000 −0.888553
\(994\) −12.0000 −0.380617
\(995\) 0 0
\(996\) 4.00000 0.126745
\(997\) 14.0000 0.443384 0.221692 − 0.975117i \(-0.428842\pi\)
0.221692 + 0.975117i \(0.428842\pi\)
\(998\) −28.0000 −0.886325
\(999\) −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.g.1.1 ✓ 1
3.2 odd 2 3366.2.a.b.1.1 1
4.3 odd 2 8976.2.a.bb.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1122.2.a.g.1.1 ✓ 1 1.1 even 1 trivial
3366.2.a.b.1.1 1 3.2 odd 2
8976.2.a.bb.1.1 1 4.3 odd 2