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Properties

Label 1155.2.a.d.1.1
Level $1155$
Weight $2$
Character 1155.1
Self dual yes
Analytic conductor $9.223$
Analytic rank $1$
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] = [1155,2,Mod(1,1155)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1155, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1155.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1155.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: \(9.22272143346\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 1155.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{15} -1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} +1.00000 q^{21} +1.00000 q^{22} -3.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} -6.00000 q^{29} -1.00000 q^{30} -5.00000 q^{32} +1.00000 q^{33} -6.00000 q^{34} +1.00000 q^{35} -1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} +2.00000 q^{39} -3.00000 q^{40} -10.0000 q^{41} -1.00000 q^{42} +4.00000 q^{43} +1.00000 q^{44} -1.00000 q^{45} +8.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -6.00000 q^{51} +2.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} +1.00000 q^{55} -3.00000 q^{56} -4.00000 q^{57} +6.00000 q^{58} -8.00000 q^{59} -1.00000 q^{60} +2.00000 q^{61} -1.00000 q^{63} +7.00000 q^{64} +2.00000 q^{65} -1.00000 q^{66} -8.00000 q^{67} -6.00000 q^{68} -1.00000 q^{70} -8.00000 q^{71} +3.00000 q^{72} -14.0000 q^{73} -6.00000 q^{74} -1.00000 q^{75} -4.00000 q^{76} +1.00000 q^{77} -2.00000 q^{78} +4.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} -16.0000 q^{83} -1.00000 q^{84} -6.00000 q^{85} -4.00000 q^{86} +6.00000 q^{87} -3.00000 q^{88} -10.0000 q^{89} +1.00000 q^{90} +2.00000 q^{91} -8.00000 q^{94} -4.00000 q^{95} +5.00000 q^{96} -2.00000 q^{97} -1.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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 −0.353553 − 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) −1.00000 −0.447214
\(6\) 1.00000 0.408248
\(7\) −1.00000 −0.377964
\(8\) 3.00000 1.06066
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −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 \(0.589456\pi\)
\(14\) 1.00000 0.267261
\(15\) 1.00000 0.258199
\(16\) −1.00000 −0.250000
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) −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
\(21\) 1.00000 0.218218
\(22\) 1.00000 0.213201
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −3.00000 −0.612372
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) −1.00000 −0.182574
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −5.00000 −0.883883
\(33\) 1.00000 0.174078
\(34\) −6.00000 −1.02899
\(35\) 1.00000 0.169031
\(36\) −1.00000 −0.166667
\(37\) 6.00000 0.986394 0.493197 − 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) −4.00000 −0.648886
\(39\) 2.00000 0.320256
\(40\) −3.00000 −0.474342
\(41\) −10.0000 −1.56174 −0.780869 − 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) −1.00000 −0.154303
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 1.00000 0.150756
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −6.00000 −0.840168
\(52\) 2.00000 0.277350
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 1.00000 0.136083
\(55\) 1.00000 0.134840
\(56\) −3.00000 −0.400892
\(57\) −4.00000 −0.529813
\(58\) 6.00000 0.787839
\(59\) −8.00000 −1.04151 −0.520756 − 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) −1.00000 −0.129099
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 7.00000 0.875000
\(65\) 2.00000 0.248069
\(66\) −1.00000 −0.123091
\(67\) −8.00000 −0.977356 −0.488678 − 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) −1.00000 −0.119523
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 3.00000 0.353553
\(73\) −14.0000 −1.63858 −0.819288 − 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) −6.00000 −0.697486
\(75\) −1.00000 −0.115470
\(76\) −4.00000 −0.458831
\(77\) 1.00000 0.113961
\(78\) −2.00000 −0.226455
\(79\) 4.00000 0.450035 0.225018 − 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 10.0000 1.10432
\(83\) −16.0000 −1.75623 −0.878114 − 0.478451i \(-0.841198\pi\)
−0.878114 + 0.478451i \(0.841198\pi\)
\(84\) −1.00000 −0.109109
\(85\) −6.00000 −0.650791
\(86\) −4.00000 −0.431331
\(87\) 6.00000 0.643268
\(88\) −3.00000 −0.319801
\(89\) −10.0000 −1.06000 −0.529999 − 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 1.00000 0.105409
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) −4.00000 −0.410391
\(96\) 5.00000 0.510310
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −1.00000 −0.101015
\(99\) −1.00000 −0.100504
\(100\) −1.00000 −0.100000
\(101\) −6.00000 −0.597022 −0.298511 − 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 6.00000 0.594089
\(103\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) −6.00000 −0.588348
\(105\) −1.00000 −0.0975900
\(106\) 6.00000 0.582772
\(107\) 20.0000 1.93347 0.966736 − 0.255774i \(-0.0823304\pi\)
0.966736 + 0.255774i \(0.0823304\pi\)
\(108\) 1.00000 0.0962250
\(109\) −10.0000 −0.957826 −0.478913 − 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) −1.00000 −0.0953463
\(111\) −6.00000 −0.569495
\(112\) 1.00000 0.0944911
\(113\) −2.00000 −0.188144 −0.0940721 − 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) −2.00000 −0.184900
\(118\) 8.00000 0.736460
\(119\) −6.00000 −0.550019
\(120\) 3.00000 0.273861
\(121\) 1.00000 0.0909091
\(122\) −2.00000 −0.181071
\(123\) 10.0000 0.901670
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 1.00000 0.0890871
\(127\) 16.0000 1.41977 0.709885 − 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 3.00000 0.265165
\(129\) −4.00000 −0.352180
\(130\) −2.00000 −0.175412
\(131\) −20.0000 −1.74741 −0.873704 − 0.486458i \(-0.838289\pi\)
−0.873704 + 0.486458i \(0.838289\pi\)
\(132\) −1.00000 −0.0870388
\(133\) −4.00000 −0.346844
\(134\) 8.00000 0.691095
\(135\) 1.00000 0.0860663
\(136\) 18.0000 1.54349
\(137\) 6.00000 0.512615 0.256307 − 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 − 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −8.00000 −0.673722
\(142\) 8.00000 0.671345
\(143\) 2.00000 0.167248
\(144\) −1.00000 −0.0833333
\(145\) 6.00000 0.498273
\(146\) 14.0000 1.15865
\(147\) −1.00000 −0.0824786
\(148\) −6.00000 −0.493197
\(149\) −14.0000 −1.14692 −0.573462 − 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) 1.00000 0.0816497
\(151\) −12.0000 −0.976546 −0.488273 − 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 12.0000 0.973329
\(153\) 6.00000 0.485071
\(154\) −1.00000 −0.0805823
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) −6.00000 −0.478852 −0.239426 − 0.970915i \(-0.576959\pi\)
−0.239426 + 0.970915i \(0.576959\pi\)
\(158\) −4.00000 −0.318223
\(159\) 6.00000 0.475831
\(160\) 5.00000 0.395285
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 16.0000 1.25322 0.626608 − 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) 10.0000 0.780869
\(165\) −1.00000 −0.0778499
\(166\) 16.0000 1.24184
\(167\) −4.00000 −0.309529 −0.154765 − 0.987951i \(-0.549462\pi\)
−0.154765 + 0.987951i \(0.549462\pi\)
\(168\) 3.00000 0.231455
\(169\) −9.00000 −0.692308
\(170\) 6.00000 0.460179
\(171\) 4.00000 0.305888
\(172\) −4.00000 −0.304997
\(173\) −6.00000 −0.456172 −0.228086 − 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) −6.00000 −0.454859
\(175\) −1.00000 −0.0755929
\(176\) 1.00000 0.0753778
\(177\) 8.00000 0.601317
\(178\) 10.0000 0.749532
\(179\) −12.0000 −0.896922 −0.448461 − 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 1.00000 0.0745356
\(181\) −2.00000 −0.148659 −0.0743294 − 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) −2.00000 −0.148250
\(183\) −2.00000 −0.147844
\(184\) 0 0
\(185\) −6.00000 −0.441129
\(186\) 0 0
\(187\) −6.00000 −0.438763
\(188\) −8.00000 −0.583460
\(189\) 1.00000 0.0727393
\(190\) 4.00000 0.290191
\(191\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −7.00000 −0.505181
\(193\) 18.0000 1.29567 0.647834 − 0.761781i \(-0.275675\pi\)
0.647834 + 0.761781i \(0.275675\pi\)
\(194\) 2.00000 0.143592
\(195\) −2.00000 −0.143223
\(196\) −1.00000 −0.0714286
\(197\) −26.0000 −1.85242 −0.926212 − 0.377004i \(-0.876954\pi\)
−0.926212 + 0.377004i \(0.876954\pi\)
\(198\) 1.00000 0.0710669
\(199\) −16.0000 −1.13421 −0.567105 − 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 3.00000 0.212132
\(201\) 8.00000 0.564276
\(202\) 6.00000 0.422159
\(203\) 6.00000 0.421117
\(204\) 6.00000 0.420084
\(205\) 10.0000 0.698430
\(206\) 0 0
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) −4.00000 −0.276686
\(210\) 1.00000 0.0690066
\(211\) 16.0000 1.10149 0.550743 − 0.834675i \(-0.314345\pi\)
0.550743 + 0.834675i \(0.314345\pi\)
\(212\) 6.00000 0.412082
\(213\) 8.00000 0.548151
\(214\) −20.0000 −1.36717
\(215\) −4.00000 −0.272798
\(216\) −3.00000 −0.204124
\(217\) 0 0
\(218\) 10.0000 0.677285
\(219\) 14.0000 0.946032
\(220\) −1.00000 −0.0674200
\(221\) −12.0000 −0.807207
\(222\) 6.00000 0.402694
\(223\) 8.00000 0.535720 0.267860 − 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 5.00000 0.334077
\(225\) 1.00000 0.0666667
\(226\) 2.00000 0.133038
\(227\) 24.0000 1.59294 0.796468 − 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) 4.00000 0.264906
\(229\) −10.0000 −0.660819 −0.330409 − 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) −1.00000 −0.0657952
\(232\) −18.0000 −1.18176
\(233\) −6.00000 −0.393073 −0.196537 − 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 2.00000 0.130744
\(235\) −8.00000 −0.521862
\(236\) 8.00000 0.520756
\(237\) −4.00000 −0.259828
\(238\) 6.00000 0.388922
\(239\) 24.0000 1.55243 0.776215 − 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 22.0000 1.41714 0.708572 − 0.705638i \(-0.249340\pi\)
0.708572 + 0.705638i \(0.249340\pi\)
\(242\) −1.00000 −0.0642824
\(243\) −1.00000 −0.0641500
\(244\) −2.00000 −0.128037
\(245\) −1.00000 −0.0638877
\(246\) −10.0000 −0.637577
\(247\) −8.00000 −0.509028
\(248\) 0 0
\(249\) 16.0000 1.01396
\(250\) 1.00000 0.0632456
\(251\) −16.0000 −1.00991 −0.504956 − 0.863145i \(-0.668491\pi\)
−0.504956 + 0.863145i \(0.668491\pi\)
\(252\) 1.00000 0.0629941
\(253\) 0 0
\(254\) −16.0000 −1.00393
\(255\) 6.00000 0.375735
\(256\) −17.0000 −1.06250
\(257\) 30.0000 1.87135 0.935674 − 0.352865i \(-0.114792\pi\)
0.935674 + 0.352865i \(0.114792\pi\)
\(258\) 4.00000 0.249029
\(259\) −6.00000 −0.372822
\(260\) −2.00000 −0.124035
\(261\) −6.00000 −0.371391
\(262\) 20.0000 1.23560
\(263\) −24.0000 −1.47990 −0.739952 − 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 3.00000 0.184637
\(265\) 6.00000 0.368577
\(266\) 4.00000 0.245256
\(267\) 10.0000 0.611990
\(268\) 8.00000 0.488678
\(269\) 18.0000 1.09748 0.548740 − 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −16.0000 −0.971931 −0.485965 − 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) −6.00000 −0.363803
\(273\) −2.00000 −0.121046
\(274\) −6.00000 −0.362473
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) 22.0000 1.32185 0.660926 − 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) −4.00000 −0.239904
\(279\) 0 0
\(280\) 3.00000 0.179284
\(281\) 30.0000 1.78965 0.894825 − 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 8.00000 0.476393
\(283\) 4.00000 0.237775 0.118888 − 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 8.00000 0.474713
\(285\) 4.00000 0.236940
\(286\) −2.00000 −0.118262
\(287\) 10.0000 0.590281
\(288\) −5.00000 −0.294628
\(289\) 19.0000 1.11765
\(290\) −6.00000 −0.352332
\(291\) 2.00000 0.117242
\(292\) 14.0000 0.819288
\(293\) −30.0000 −1.75262 −0.876309 − 0.481749i \(-0.840002\pi\)
−0.876309 + 0.481749i \(0.840002\pi\)
\(294\) 1.00000 0.0583212
\(295\) 8.00000 0.465778
\(296\) 18.0000 1.04623
\(297\) 1.00000 0.0580259
\(298\) 14.0000 0.810998
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) −4.00000 −0.230556
\(302\) 12.0000 0.690522
\(303\) 6.00000 0.344691
\(304\) −4.00000 −0.229416
\(305\) −2.00000 −0.114520
\(306\) −6.00000 −0.342997
\(307\) 4.00000 0.228292 0.114146 − 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 0 0
\(310\) 0 0
\(311\) 4.00000 0.226819 0.113410 − 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) 6.00000 0.339683
\(313\) −26.0000 −1.46961 −0.734803 − 0.678280i \(-0.762726\pi\)
−0.734803 + 0.678280i \(0.762726\pi\)
\(314\) 6.00000 0.338600
\(315\) 1.00000 0.0563436
\(316\) −4.00000 −0.225018
\(317\) −22.0000 −1.23564 −0.617822 − 0.786318i \(-0.711985\pi\)
−0.617822 + 0.786318i \(0.711985\pi\)
\(318\) −6.00000 −0.336463
\(319\) 6.00000 0.335936
\(320\) −7.00000 −0.391312
\(321\) −20.0000 −1.11629
\(322\) 0 0
\(323\) 24.0000 1.33540
\(324\) −1.00000 −0.0555556
\(325\) −2.00000 −0.110940
\(326\) −16.0000 −0.886158
\(327\) 10.0000 0.553001
\(328\) −30.0000 −1.65647
\(329\) −8.00000 −0.441054
\(330\) 1.00000 0.0550482
\(331\) −20.0000 −1.09930 −0.549650 − 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 16.0000 0.878114
\(333\) 6.00000 0.328798
\(334\) 4.00000 0.218870
\(335\) 8.00000 0.437087
\(336\) −1.00000 −0.0545545
\(337\) −14.0000 −0.762629 −0.381314 − 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 9.00000 0.489535
\(339\) 2.00000 0.108625
\(340\) 6.00000 0.325396
\(341\) 0 0
\(342\) −4.00000 −0.216295
\(343\) −1.00000 −0.0539949
\(344\) 12.0000 0.646997
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) 12.0000 0.644194 0.322097 − 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) −6.00000 −0.321634
\(349\) −14.0000 −0.749403 −0.374701 − 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 1.00000 0.0534522
\(351\) 2.00000 0.106752
\(352\) 5.00000 0.266501
\(353\) −18.0000 −0.958043 −0.479022 − 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) −8.00000 −0.425195
\(355\) 8.00000 0.424596
\(356\) 10.0000 0.529999
\(357\) 6.00000 0.317554
\(358\) 12.0000 0.634220
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) −3.00000 −0.158114
\(361\) −3.00000 −0.157895
\(362\) 2.00000 0.105118
\(363\) −1.00000 −0.0524864
\(364\) −2.00000 −0.104828
\(365\) 14.0000 0.732793
\(366\) 2.00000 0.104542
\(367\) −32.0000 −1.67039 −0.835193 − 0.549957i \(-0.814644\pi\)
−0.835193 + 0.549957i \(0.814644\pi\)
\(368\) 0 0
\(369\) −10.0000 −0.520579
\(370\) 6.00000 0.311925
\(371\) 6.00000 0.311504
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 − 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 6.00000 0.310253
\(375\) 1.00000 0.0516398
\(376\) 24.0000 1.23771
\(377\) 12.0000 0.618031
\(378\) −1.00000 −0.0514344
\(379\) 20.0000 1.02733 0.513665 − 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 4.00000 0.205196
\(381\) −16.0000 −0.819705
\(382\) 0 0
\(383\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) −3.00000 −0.153093
\(385\) −1.00000 −0.0509647
\(386\) −18.0000 −0.916176
\(387\) 4.00000 0.203331
\(388\) 2.00000 0.101535
\(389\) −10.0000 −0.507020 −0.253510 − 0.967333i \(-0.581585\pi\)
−0.253510 + 0.967333i \(0.581585\pi\)
\(390\) 2.00000 0.101274
\(391\) 0 0
\(392\) 3.00000 0.151523
\(393\) 20.0000 1.00887
\(394\) 26.0000 1.30986
\(395\) −4.00000 −0.201262
\(396\) 1.00000 0.0502519
\(397\) −14.0000 −0.702640 −0.351320 − 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 16.0000 0.802008
\(399\) 4.00000 0.200250
\(400\) −1.00000 −0.0500000
\(401\) 2.00000 0.0998752 0.0499376 − 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) −8.00000 −0.399004
\(403\) 0 0
\(404\) 6.00000 0.298511
\(405\) −1.00000 −0.0496904
\(406\) −6.00000 −0.297775
\(407\) −6.00000 −0.297409
\(408\) −18.0000 −0.891133
\(409\) −2.00000 −0.0988936 −0.0494468 − 0.998777i \(-0.515746\pi\)
−0.0494468 + 0.998777i \(0.515746\pi\)
\(410\) −10.0000 −0.493865
\(411\) −6.00000 −0.295958
\(412\) 0 0
\(413\) 8.00000 0.393654
\(414\) 0 0
\(415\) 16.0000 0.785409
\(416\) 10.0000 0.490290
\(417\) −4.00000 −0.195881
\(418\) 4.00000 0.195646
\(419\) −16.0000 −0.781651 −0.390826 − 0.920465i \(-0.627810\pi\)
−0.390826 + 0.920465i \(0.627810\pi\)
\(420\) 1.00000 0.0487950
\(421\) −26.0000 −1.26716 −0.633581 − 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) −16.0000 −0.778868
\(423\) 8.00000 0.388973
\(424\) −18.0000 −0.874157
\(425\) 6.00000 0.291043
\(426\) −8.00000 −0.387601
\(427\) −2.00000 −0.0967868
\(428\) −20.0000 −0.966736
\(429\) −2.00000 −0.0965609
\(430\) 4.00000 0.192897
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 1.00000 0.0481125
\(433\) −10.0000 −0.480569 −0.240285 − 0.970702i \(-0.577241\pi\)
−0.240285 + 0.970702i \(0.577241\pi\)
\(434\) 0 0
\(435\) −6.00000 −0.287678
\(436\) 10.0000 0.478913
\(437\) 0 0
\(438\) −14.0000 −0.668946
\(439\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 3.00000 0.143019
\(441\) 1.00000 0.0476190
\(442\) 12.0000 0.570782
\(443\) −12.0000 −0.570137 −0.285069 − 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 6.00000 0.284747
\(445\) 10.0000 0.474045
\(446\) −8.00000 −0.378811
\(447\) 14.0000 0.662177
\(448\) −7.00000 −0.330719
\(449\) −14.0000 −0.660701 −0.330350 − 0.943858i \(-0.607167\pi\)
−0.330350 + 0.943858i \(0.607167\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 10.0000 0.470882
\(452\) 2.00000 0.0940721
\(453\) 12.0000 0.563809
\(454\) −24.0000 −1.12638
\(455\) −2.00000 −0.0937614
\(456\) −12.0000 −0.561951
\(457\) 26.0000 1.21623 0.608114 − 0.793849i \(-0.291926\pi\)
0.608114 + 0.793849i \(0.291926\pi\)
\(458\) 10.0000 0.467269
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) −30.0000 −1.39724 −0.698620 − 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 1.00000 0.0465242
\(463\) 4.00000 0.185896 0.0929479 − 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) −28.0000 −1.29569 −0.647843 − 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) 2.00000 0.0924500
\(469\) 8.00000 0.369406
\(470\) 8.00000 0.369012
\(471\) 6.00000 0.276465
\(472\) −24.0000 −1.10469
\(473\) −4.00000 −0.183920
\(474\) 4.00000 0.183726
\(475\) 4.00000 0.183533
\(476\) 6.00000 0.275010
\(477\) −6.00000 −0.274721
\(478\) −24.0000 −1.09773
\(479\) 24.0000 1.09659 0.548294 − 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) −5.00000 −0.228218
\(481\) −12.0000 −0.547153
\(482\) −22.0000 −1.00207
\(483\) 0 0
\(484\) −1.00000 −0.0454545
\(485\) 2.00000 0.0908153
\(486\) 1.00000 0.0453609
\(487\) 4.00000 0.181257 0.0906287 − 0.995885i \(-0.471112\pi\)
0.0906287 + 0.995885i \(0.471112\pi\)
\(488\) 6.00000 0.271607
\(489\) −16.0000 −0.723545
\(490\) 1.00000 0.0451754
\(491\) 12.0000 0.541552 0.270776 − 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) −10.0000 −0.450835
\(493\) −36.0000 −1.62136
\(494\) 8.00000 0.359937
\(495\) 1.00000 0.0449467
\(496\) 0 0
\(497\) 8.00000 0.358849
\(498\) −16.0000 −0.716977
\(499\) 20.0000 0.895323 0.447661 − 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 1.00000 0.0447214
\(501\) 4.00000 0.178707
\(502\) 16.0000 0.714115
\(503\) 20.0000 0.891756 0.445878 − 0.895094i \(-0.352892\pi\)
0.445878 + 0.895094i \(0.352892\pi\)
\(504\) −3.00000 −0.133631
\(505\) 6.00000 0.266996
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) −16.0000 −0.709885
\(509\) −14.0000 −0.620539 −0.310270 − 0.950649i \(-0.600419\pi\)
−0.310270 + 0.950649i \(0.600419\pi\)
\(510\) −6.00000 −0.265684
\(511\) 14.0000 0.619324
\(512\) 11.0000 0.486136
\(513\) −4.00000 −0.176604
\(514\) −30.0000 −1.32324
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) −8.00000 −0.351840
\(518\) 6.00000 0.263625
\(519\) 6.00000 0.263371
\(520\) 6.00000 0.263117
\(521\) 14.0000 0.613351 0.306676 − 0.951814i \(-0.400783\pi\)
0.306676 + 0.951814i \(0.400783\pi\)
\(522\) 6.00000 0.262613
\(523\) 20.0000 0.874539 0.437269 − 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 20.0000 0.873704
\(525\) 1.00000 0.0436436
\(526\) 24.0000 1.04645
\(527\) 0 0
\(528\) −1.00000 −0.0435194
\(529\) −23.0000 −1.00000
\(530\) −6.00000 −0.260623
\(531\) −8.00000 −0.347170
\(532\) 4.00000 0.173422
\(533\) 20.0000 0.866296
\(534\) −10.0000 −0.432742
\(535\) −20.0000 −0.864675
\(536\) −24.0000 −1.03664
\(537\) 12.0000 0.517838
\(538\) −18.0000 −0.776035
\(539\) −1.00000 −0.0430730
\(540\) −1.00000 −0.0430331
\(541\) 14.0000 0.601907 0.300954 − 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) 16.0000 0.687259
\(543\) 2.00000 0.0858282
\(544\) −30.0000 −1.28624
\(545\) 10.0000 0.428353
\(546\) 2.00000 0.0855921
\(547\) −20.0000 −0.855138 −0.427569 − 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) −6.00000 −0.256307
\(549\) 2.00000 0.0853579
\(550\) 1.00000 0.0426401
\(551\) −24.0000 −1.02243
\(552\) 0 0
\(553\) −4.00000 −0.170097
\(554\) −22.0000 −0.934690
\(555\) 6.00000 0.254686
\(556\) −4.00000 −0.169638
\(557\) −18.0000 −0.762684 −0.381342 − 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) −1.00000 −0.0422577
\(561\) 6.00000 0.253320
\(562\) −30.0000 −1.26547
\(563\) −8.00000 −0.337160 −0.168580 − 0.985688i \(-0.553918\pi\)
−0.168580 + 0.985688i \(0.553918\pi\)
\(564\) 8.00000 0.336861
\(565\) 2.00000 0.0841406
\(566\) −4.00000 −0.168133
\(567\) −1.00000 −0.0419961
\(568\) −24.0000 −1.00702
\(569\) 30.0000 1.25767 0.628833 − 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) −4.00000 −0.167542
\(571\) 16.0000 0.669579 0.334790 − 0.942293i \(-0.391335\pi\)
0.334790 + 0.942293i \(0.391335\pi\)
\(572\) −2.00000 −0.0836242
\(573\) 0 0
\(574\) −10.0000 −0.417392
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) −42.0000 −1.74848 −0.874241 − 0.485491i \(-0.838641\pi\)
−0.874241 + 0.485491i \(0.838641\pi\)
\(578\) −19.0000 −0.790296
\(579\) −18.0000 −0.748054
\(580\) −6.00000 −0.249136
\(581\) 16.0000 0.663792
\(582\) −2.00000 −0.0829027
\(583\) 6.00000 0.248495
\(584\) −42.0000 −1.73797
\(585\) 2.00000 0.0826898
\(586\) 30.0000 1.23929
\(587\) −12.0000 −0.495293 −0.247647 − 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 1.00000 0.0412393
\(589\) 0 0
\(590\) −8.00000 −0.329355
\(591\) 26.0000 1.06950
\(592\) −6.00000 −0.246598
\(593\) 6.00000 0.246390 0.123195 − 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 6.00000 0.245976
\(596\) 14.0000 0.573462
\(597\) 16.0000 0.654836
\(598\) 0 0
\(599\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) −3.00000 −0.122474
\(601\) −34.0000 −1.38689 −0.693444 − 0.720510i \(-0.743908\pi\)
−0.693444 + 0.720510i \(0.743908\pi\)
\(602\) 4.00000 0.163028
\(603\) −8.00000 −0.325785
\(604\) 12.0000 0.488273
\(605\) −1.00000 −0.0406558
\(606\) −6.00000 −0.243733
\(607\) −8.00000 −0.324710 −0.162355 − 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) −20.0000 −0.811107
\(609\) −6.00000 −0.243132
\(610\) 2.00000 0.0809776
\(611\) −16.0000 −0.647291
\(612\) −6.00000 −0.242536
\(613\) 38.0000 1.53481 0.767403 − 0.641165i \(-0.221549\pi\)
0.767403 + 0.641165i \(0.221549\pi\)
\(614\) −4.00000 −0.161427
\(615\) −10.0000 −0.403239
\(616\) 3.00000 0.120873
\(617\) 30.0000 1.20775 0.603877 − 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 − 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −4.00000 −0.160385
\(623\) 10.0000 0.400642
\(624\) −2.00000 −0.0800641
\(625\) 1.00000 0.0400000
\(626\) 26.0000 1.03917
\(627\) 4.00000 0.159745
\(628\) 6.00000 0.239426
\(629\) 36.0000 1.43541
\(630\) −1.00000 −0.0398410
\(631\) −8.00000 −0.318475 −0.159237 − 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 12.0000 0.477334
\(633\) −16.0000 −0.635943
\(634\) 22.0000 0.873732
\(635\) −16.0000 −0.634941
\(636\) −6.00000 −0.237915
\(637\) −2.00000 −0.0792429
\(638\) −6.00000 −0.237542
\(639\) −8.00000 −0.316475
\(640\) −3.00000 −0.118585
\(641\) −14.0000 −0.552967 −0.276483 − 0.961019i \(-0.589169\pi\)
−0.276483 + 0.961019i \(0.589169\pi\)
\(642\) 20.0000 0.789337
\(643\) −36.0000 −1.41970 −0.709851 − 0.704352i \(-0.751238\pi\)
−0.709851 + 0.704352i \(0.751238\pi\)
\(644\) 0 0
\(645\) 4.00000 0.157500
\(646\) −24.0000 −0.944267
\(647\) −24.0000 −0.943537 −0.471769 − 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 3.00000 0.117851
\(649\) 8.00000 0.314027
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) −16.0000 −0.626608
\(653\) −30.0000 −1.17399 −0.586995 − 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) −10.0000 −0.391031
\(655\) 20.0000 0.781465
\(656\) 10.0000 0.390434
\(657\) −14.0000 −0.546192
\(658\) 8.00000 0.311872
\(659\) 20.0000 0.779089 0.389545 − 0.921008i \(-0.372632\pi\)
0.389545 + 0.921008i \(0.372632\pi\)
\(660\) 1.00000 0.0389249
\(661\) −26.0000 −1.01128 −0.505641 − 0.862744i \(-0.668744\pi\)
−0.505641 + 0.862744i \(0.668744\pi\)
\(662\) 20.0000 0.777322
\(663\) 12.0000 0.466041
\(664\) −48.0000 −1.86276
\(665\) 4.00000 0.155113
\(666\) −6.00000 −0.232495
\(667\) 0 0
\(668\) 4.00000 0.154765
\(669\) −8.00000 −0.309298
\(670\) −8.00000 −0.309067
\(671\) −2.00000 −0.0772091
\(672\) −5.00000 −0.192879
\(673\) 2.00000 0.0770943 0.0385472 − 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) 14.0000 0.539260
\(675\) −1.00000 −0.0384900
\(676\) 9.00000 0.346154
\(677\) 18.0000 0.691796 0.345898 − 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) −2.00000 −0.0768095
\(679\) 2.00000 0.0767530
\(680\) −18.0000 −0.690268
\(681\) −24.0000 −0.919682
\(682\) 0 0
\(683\) −4.00000 −0.153056 −0.0765279 − 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) −4.00000 −0.152944
\(685\) −6.00000 −0.229248
\(686\) 1.00000 0.0381802
\(687\) 10.0000 0.381524
\(688\) −4.00000 −0.152499
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 − 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 6.00000 0.228086
\(693\) 1.00000 0.0379869
\(694\) −12.0000 −0.455514
\(695\) −4.00000 −0.151729
\(696\) 18.0000 0.682288
\(697\) −60.0000 −2.27266
\(698\) 14.0000 0.529908
\(699\) 6.00000 0.226941
\(700\) 1.00000 0.0377964
\(701\) 2.00000 0.0755390 0.0377695 − 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 24.0000 0.905177
\(704\) −7.00000 −0.263822
\(705\) 8.00000 0.301297
\(706\) 18.0000 0.677439
\(707\) 6.00000 0.225653
\(708\) −8.00000 −0.300658
\(709\) −26.0000 −0.976450 −0.488225 − 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) −8.00000 −0.300235
\(711\) 4.00000 0.150012
\(712\) −30.0000 −1.12430
\(713\) 0 0
\(714\) −6.00000 −0.224544
\(715\) −2.00000 −0.0747958
\(716\) 12.0000 0.448461
\(717\) −24.0000 −0.896296
\(718\) 0 0
\(719\) −12.0000 −0.447524 −0.223762 − 0.974644i \(-0.571834\pi\)
−0.223762 + 0.974644i \(0.571834\pi\)
\(720\) 1.00000 0.0372678
\(721\) 0 0
\(722\) 3.00000 0.111648
\(723\) −22.0000 −0.818189
\(724\) 2.00000 0.0743294
\(725\) −6.00000 −0.222834
\(726\) 1.00000 0.0371135
\(727\) 32.0000 1.18681 0.593407 − 0.804902i \(-0.297782\pi\)
0.593407 + 0.804902i \(0.297782\pi\)
\(728\) 6.00000 0.222375
\(729\) 1.00000 0.0370370
\(730\) −14.0000 −0.518163
\(731\) 24.0000 0.887672
\(732\) 2.00000 0.0739221
\(733\) −26.0000 −0.960332 −0.480166 − 0.877178i \(-0.659424\pi\)
−0.480166 + 0.877178i \(0.659424\pi\)
\(734\) 32.0000 1.18114
\(735\) 1.00000 0.0368856
\(736\) 0 0
\(737\) 8.00000 0.294684
\(738\) 10.0000 0.368105
\(739\) 16.0000 0.588570 0.294285 − 0.955718i \(-0.404919\pi\)
0.294285 + 0.955718i \(0.404919\pi\)
\(740\) 6.00000 0.220564
\(741\) 8.00000 0.293887
\(742\) −6.00000 −0.220267
\(743\) 8.00000 0.293492 0.146746 − 0.989174i \(-0.453120\pi\)
0.146746 + 0.989174i \(0.453120\pi\)
\(744\) 0 0
\(745\) 14.0000 0.512920
\(746\) 10.0000 0.366126
\(747\) −16.0000 −0.585409
\(748\) 6.00000 0.219382
\(749\) −20.0000 −0.730784
\(750\) −1.00000 −0.0365148
\(751\) 48.0000 1.75154 0.875772 − 0.482724i \(-0.160353\pi\)
0.875772 + 0.482724i \(0.160353\pi\)
\(752\) −8.00000 −0.291730
\(753\) 16.0000 0.583072
\(754\) −12.0000 −0.437014
\(755\) 12.0000 0.436725
\(756\) −1.00000 −0.0363696
\(757\) −34.0000 −1.23575 −0.617876 − 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) −20.0000 −0.726433
\(759\) 0 0
\(760\) −12.0000 −0.435286
\(761\) 22.0000 0.797499 0.398750 − 0.917060i \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) 16.0000 0.579619
\(763\) 10.0000 0.362024
\(764\) 0 0
\(765\) −6.00000 −0.216930
\(766\) 0 0
\(767\) 16.0000 0.577727
\(768\) 17.0000 0.613435
\(769\) −18.0000 −0.649097 −0.324548 − 0.945869i \(-0.605212\pi\)
−0.324548 + 0.945869i \(0.605212\pi\)
\(770\) 1.00000 0.0360375
\(771\) −30.0000 −1.08042
\(772\) −18.0000 −0.647834
\(773\) 26.0000 0.935155 0.467578 − 0.883952i \(-0.345127\pi\)
0.467578 + 0.883952i \(0.345127\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) −6.00000 −0.215387
\(777\) 6.00000 0.215249
\(778\) 10.0000 0.358517
\(779\) −40.0000 −1.43315
\(780\) 2.00000 0.0716115
\(781\) 8.00000 0.286263
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) −1.00000 −0.0357143
\(785\) 6.00000 0.214149
\(786\) −20.0000 −0.713376
\(787\) 12.0000 0.427754 0.213877 − 0.976861i \(-0.431391\pi\)
0.213877 + 0.976861i \(0.431391\pi\)
\(788\) 26.0000 0.926212
\(789\) 24.0000 0.854423
\(790\) 4.00000 0.142314
\(791\) 2.00000 0.0711118
\(792\) −3.00000 −0.106600
\(793\) −4.00000 −0.142044
\(794\) 14.0000 0.496841
\(795\) −6.00000 −0.212798
\(796\) 16.0000 0.567105
\(797\) 2.00000 0.0708436 0.0354218 − 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) −4.00000 −0.141598
\(799\) 48.0000 1.69812
\(800\) −5.00000 −0.176777
\(801\) −10.0000 −0.353333
\(802\) −2.00000 −0.0706225
\(803\) 14.0000 0.494049
\(804\) −8.00000 −0.282138
\(805\) 0 0
\(806\) 0 0
\(807\) −18.0000 −0.633630
\(808\) −18.0000 −0.633238
\(809\) 6.00000 0.210949 0.105474 − 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 1.00000 0.0351364
\(811\) 20.0000 0.702295 0.351147 − 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) −6.00000 −0.210559
\(813\) 16.0000 0.561144
\(814\) 6.00000 0.210300
\(815\) −16.0000 −0.560456
\(816\) 6.00000 0.210042
\(817\) 16.0000 0.559769
\(818\) 2.00000 0.0699284
\(819\) 2.00000 0.0698857
\(820\) −10.0000 −0.349215
\(821\) −14.0000 −0.488603 −0.244302 − 0.969699i \(-0.578559\pi\)
−0.244302 + 0.969699i \(0.578559\pi\)
\(822\) 6.00000 0.209274
\(823\) 20.0000 0.697156 0.348578 − 0.937280i \(-0.386665\pi\)
0.348578 + 0.937280i \(0.386665\pi\)
\(824\) 0 0
\(825\) 1.00000 0.0348155
\(826\) −8.00000 −0.278356
\(827\) −36.0000 −1.25184 −0.625921 − 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 0 0
\(829\) −2.00000 −0.0694629 −0.0347314 − 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) −16.0000 −0.555368
\(831\) −22.0000 −0.763172
\(832\) −14.0000 −0.485363
\(833\) 6.00000 0.207888
\(834\) 4.00000 0.138509
\(835\) 4.00000 0.138426
\(836\) 4.00000 0.138343
\(837\) 0 0
\(838\) 16.0000 0.552711
\(839\) 12.0000 0.414286 0.207143 − 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) −3.00000 −0.103510
\(841\) 7.00000 0.241379
\(842\) 26.0000 0.896019
\(843\) −30.0000 −1.03325
\(844\) −16.0000 −0.550743
\(845\) 9.00000 0.309609
\(846\) −8.00000 −0.275046
\(847\) −1.00000 −0.0343604
\(848\) 6.00000 0.206041
\(849\) −4.00000 −0.137280
\(850\) −6.00000 −0.205798
\(851\) 0 0
\(852\) −8.00000 −0.274075
\(853\) 22.0000 0.753266 0.376633 − 0.926363i \(-0.377082\pi\)
0.376633 + 0.926363i \(0.377082\pi\)
\(854\) 2.00000 0.0684386
\(855\) −4.00000 −0.136797
\(856\) 60.0000 2.05076
\(857\) 22.0000 0.751506 0.375753 − 0.926720i \(-0.377384\pi\)
0.375753 + 0.926720i \(0.377384\pi\)
\(858\) 2.00000 0.0682789
\(859\) 44.0000 1.50126 0.750630 − 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) 4.00000 0.136399
\(861\) −10.0000 −0.340799
\(862\) 0 0
\(863\) −48.0000 −1.63394 −0.816970 − 0.576681i \(-0.804348\pi\)
−0.816970 + 0.576681i \(0.804348\pi\)
\(864\) 5.00000 0.170103
\(865\) 6.00000 0.204006
\(866\) 10.0000 0.339814
\(867\) −19.0000 −0.645274
\(868\) 0 0
\(869\) −4.00000 −0.135691
\(870\) 6.00000 0.203419
\(871\) 16.0000 0.542139
\(872\) −30.0000 −1.01593
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) −14.0000 −0.473016
\(877\) 14.0000 0.472746 0.236373 − 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) 0 0
\(879\) 30.0000 1.01187
\(880\) −1.00000 −0.0337100
\(881\) 54.0000 1.81931 0.909653 − 0.415369i \(-0.136347\pi\)
0.909653 + 0.415369i \(0.136347\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 32.0000 1.07689 0.538443 − 0.842662i \(-0.319013\pi\)
0.538443 + 0.842662i \(0.319013\pi\)
\(884\) 12.0000 0.403604
\(885\) −8.00000 −0.268917
\(886\) 12.0000 0.403148
\(887\) −12.0000 −0.402921 −0.201460 − 0.979497i \(-0.564569\pi\)
−0.201460 + 0.979497i \(0.564569\pi\)
\(888\) −18.0000 −0.604040
\(889\) −16.0000 −0.536623
\(890\) −10.0000 −0.335201
\(891\) −1.00000 −0.0335013
\(892\) −8.00000 −0.267860
\(893\) 32.0000 1.07084
\(894\) −14.0000 −0.468230
\(895\) 12.0000 0.401116
\(896\) −3.00000 −0.100223
\(897\) 0 0
\(898\) 14.0000 0.467186
\(899\) 0 0
\(900\) −1.00000 −0.0333333
\(901\) −36.0000 −1.19933
\(902\) −10.0000 −0.332964
\(903\) 4.00000 0.133112
\(904\) −6.00000 −0.199557
\(905\) 2.00000 0.0664822
\(906\) −12.0000 −0.398673
\(907\) 16.0000 0.531271 0.265636 − 0.964073i \(-0.414418\pi\)
0.265636 + 0.964073i \(0.414418\pi\)
\(908\) −24.0000 −0.796468
\(909\) −6.00000 −0.199007
\(910\) 2.00000 0.0662994
\(911\) 24.0000 0.795155 0.397578 − 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) 4.00000 0.132453
\(913\) 16.0000 0.529523
\(914\) −26.0000 −0.860004
\(915\) 2.00000 0.0661180
\(916\) 10.0000 0.330409
\(917\) 20.0000 0.660458
\(918\) 6.00000 0.198030
\(919\) 52.0000 1.71532 0.857661 − 0.514216i \(-0.171917\pi\)
0.857661 + 0.514216i \(0.171917\pi\)
\(920\) 0 0
\(921\) −4.00000 −0.131804
\(922\) 30.0000 0.987997
\(923\) 16.0000 0.526646
\(924\) 1.00000 0.0328976
\(925\) 6.00000 0.197279
\(926\) −4.00000 −0.131448
\(927\) 0 0
\(928\) 30.0000 0.984798
\(929\) −34.0000 −1.11550 −0.557752 − 0.830008i \(-0.688336\pi\)
−0.557752 + 0.830008i \(0.688336\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) 6.00000 0.196537
\(933\) −4.00000 −0.130954
\(934\) 28.0000 0.916188
\(935\) 6.00000 0.196221
\(936\) −6.00000 −0.196116
\(937\) 2.00000 0.0653372 0.0326686 − 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) −8.00000 −0.261209
\(939\) 26.0000 0.848478
\(940\) 8.00000 0.260931
\(941\) −46.0000 −1.49956 −0.749779 − 0.661689i \(-0.769840\pi\)
−0.749779 + 0.661689i \(0.769840\pi\)
\(942\) −6.00000 −0.195491
\(943\) 0 0
\(944\) 8.00000 0.260378
\(945\) −1.00000 −0.0325300
\(946\) 4.00000 0.130051
\(947\) 36.0000 1.16984 0.584921 − 0.811090i \(-0.301125\pi\)
0.584921 + 0.811090i \(0.301125\pi\)
\(948\) 4.00000 0.129914
\(949\) 28.0000 0.908918
\(950\) −4.00000 −0.129777
\(951\) 22.0000 0.713399
\(952\) −18.0000 −0.583383
\(953\) −46.0000 −1.49009 −0.745043 − 0.667016i \(-0.767571\pi\)
−0.745043 + 0.667016i \(0.767571\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) −24.0000 −0.776215
\(957\) −6.00000 −0.193952
\(958\) −24.0000 −0.775405
\(959\) −6.00000 −0.193750
\(960\) 7.00000 0.225924
\(961\) −31.0000 −1.00000
\(962\) 12.0000 0.386896
\(963\) 20.0000 0.644491
\(964\) −22.0000 −0.708572
\(965\) −18.0000 −0.579441
\(966\) 0 0
\(967\) −56.0000 −1.80084 −0.900419 − 0.435023i \(-0.856740\pi\)
−0.900419 + 0.435023i \(0.856740\pi\)
\(968\) 3.00000 0.0964237
\(969\) −24.0000 −0.770991
\(970\) −2.00000 −0.0642161
\(971\) −16.0000 −0.513464 −0.256732 − 0.966483i \(-0.582646\pi\)
−0.256732 + 0.966483i \(0.582646\pi\)
\(972\) 1.00000 0.0320750
\(973\) −4.00000 −0.128234
\(974\) −4.00000 −0.128168
\(975\) 2.00000 0.0640513
\(976\) −2.00000 −0.0640184
\(977\) −42.0000 −1.34370 −0.671850 − 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) 16.0000 0.511624
\(979\) 10.0000 0.319601
\(980\) 1.00000 0.0319438
\(981\) −10.0000 −0.319275
\(982\) −12.0000 −0.382935
\(983\) 24.0000 0.765481 0.382741 − 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 30.0000 0.956365
\(985\) 26.0000 0.828429
\(986\) 36.0000 1.14647
\(987\) 8.00000 0.254643
\(988\) 8.00000 0.254514
\(989\) 0 0
\(990\) −1.00000 −0.0317821
\(991\) 32.0000 1.01651 0.508257 − 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 0 0
\(993\) 20.0000 0.634681
\(994\) −8.00000 −0.253745
\(995\) 16.0000 0.507234
\(996\) −16.0000 −0.506979
\(997\) −58.0000 −1.83688 −0.918439 − 0.395562i \(-0.870550\pi\)
−0.918439 + 0.395562i \(0.870550\pi\)
\(998\) −20.0000 −0.633089
\(999\) −6.00000 −0.189832
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 1155.2.a.d.1.1 ✓ 1
3.2 odd 2 3465.2.a.q.1.1 1
5.4 even 2 5775.2.a.u.1.1 1
7.6 odd 2 8085.2.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1155.2.a.d.1.1 ✓ 1 1.1 even 1 trivial
3465.2.a.q.1.1 1 3.2 odd 2
5775.2.a.u.1.1 1 5.4 even 2
8085.2.a.h.1.1 1 7.6 odd 2