Properties

Label 1014.4.b.g.337.1
Level $1014$
Weight $4$
Character 1014.337
Analytic conductor $59.828$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1014,4,Mod(337,1014)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1014, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1014.337");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1014.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(59.8279367458\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1014.337
Dual form 1014.4.b.g.337.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000i q^{2} +3.00000 q^{3} -4.00000 q^{4} -4.00000i q^{5} -6.00000i q^{6} +4.00000i q^{7} +8.00000i q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-2.00000i q^{2} +3.00000 q^{3} -4.00000 q^{4} -4.00000i q^{5} -6.00000i q^{6} +4.00000i q^{7} +8.00000i q^{8} +9.00000 q^{9} -8.00000 q^{10} +2.00000i q^{11} -12.0000 q^{12} +8.00000 q^{14} -12.0000i q^{15} +16.0000 q^{16} +6.00000 q^{17} -18.0000i q^{18} +36.0000i q^{19} +16.0000i q^{20} +12.0000i q^{21} +4.00000 q^{22} +20.0000 q^{23} +24.0000i q^{24} +109.000 q^{25} +27.0000 q^{27} -16.0000i q^{28} -14.0000 q^{29} -24.0000 q^{30} +152.000i q^{31} -32.0000i q^{32} +6.00000i q^{33} -12.0000i q^{34} +16.0000 q^{35} -36.0000 q^{36} -258.000i q^{37} +72.0000 q^{38} +32.0000 q^{40} -84.0000i q^{41} +24.0000 q^{42} +188.000 q^{43} -8.00000i q^{44} -36.0000i q^{45} -40.0000i q^{46} +254.000i q^{47} +48.0000 q^{48} +327.000 q^{49} -218.000i q^{50} +18.0000 q^{51} +366.000 q^{53} -54.0000i q^{54} +8.00000 q^{55} -32.0000 q^{56} +108.000i q^{57} +28.0000i q^{58} +550.000i q^{59} +48.0000i q^{60} -14.0000 q^{61} +304.000 q^{62} +36.0000i q^{63} -64.0000 q^{64} +12.0000 q^{66} -448.000i q^{67} -24.0000 q^{68} +60.0000 q^{69} -32.0000i q^{70} -926.000i q^{71} +72.0000i q^{72} +254.000i q^{73} -516.000 q^{74} +327.000 q^{75} -144.000i q^{76} -8.00000 q^{77} +1328.00 q^{79} -64.0000i q^{80} +81.0000 q^{81} -168.000 q^{82} -186.000i q^{83} -48.0000i q^{84} -24.0000i q^{85} -376.000i q^{86} -42.0000 q^{87} -16.0000 q^{88} -336.000i q^{89} -72.0000 q^{90} -80.0000 q^{92} +456.000i q^{93} +508.000 q^{94} +144.000 q^{95} -96.0000i q^{96} -614.000i q^{97} -654.000i q^{98} +18.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} - 8 q^{4} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} - 8 q^{4} + 18 q^{9} - 16 q^{10} - 24 q^{12} + 16 q^{14} + 32 q^{16} + 12 q^{17} + 8 q^{22} + 40 q^{23} + 218 q^{25} + 54 q^{27} - 28 q^{29} - 48 q^{30} + 32 q^{35} - 72 q^{36} + 144 q^{38} + 64 q^{40} + 48 q^{42} + 376 q^{43} + 96 q^{48} + 654 q^{49} + 36 q^{51} + 732 q^{53} + 16 q^{55} - 64 q^{56} - 28 q^{61} + 608 q^{62} - 128 q^{64} + 24 q^{66} - 48 q^{68} + 120 q^{69} - 1032 q^{74} + 654 q^{75} - 16 q^{77} + 2656 q^{79} + 162 q^{81} - 336 q^{82} - 84 q^{87} - 32 q^{88} - 144 q^{90} - 160 q^{92} + 1016 q^{94} + 288 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.00000i − 0.707107i
\(3\) 3.00000 0.577350
\(4\) −4.00000 −0.500000
\(5\) − 4.00000i − 0.357771i −0.983870 0.178885i \(-0.942751\pi\)
0.983870 0.178885i \(-0.0572491\pi\)
\(6\) − 6.00000i − 0.408248i
\(7\) 4.00000i 0.215980i 0.994152 + 0.107990i \(0.0344414\pi\)
−0.994152 + 0.107990i \(0.965559\pi\)
\(8\) 8.00000i 0.353553i
\(9\) 9.00000 0.333333
\(10\) −8.00000 −0.252982
\(11\) 2.00000i 0.0548202i 0.999624 + 0.0274101i \(0.00872601\pi\)
−0.999624 + 0.0274101i \(0.991274\pi\)
\(12\) −12.0000 −0.288675
\(13\) 0 0
\(14\) 8.00000 0.152721
\(15\) − 12.0000i − 0.206559i
\(16\) 16.0000 0.250000
\(17\) 6.00000 0.0856008 0.0428004 0.999084i \(-0.486372\pi\)
0.0428004 + 0.999084i \(0.486372\pi\)
\(18\) − 18.0000i − 0.235702i
\(19\) 36.0000i 0.434682i 0.976096 + 0.217341i \(0.0697384\pi\)
−0.976096 + 0.217341i \(0.930262\pi\)
\(20\) 16.0000i 0.178885i
\(21\) 12.0000i 0.124696i
\(22\) 4.00000 0.0387638
\(23\) 20.0000 0.181317 0.0906584 0.995882i \(-0.471103\pi\)
0.0906584 + 0.995882i \(0.471103\pi\)
\(24\) 24.0000i 0.204124i
\(25\) 109.000 0.872000
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) − 16.0000i − 0.107990i
\(29\) −14.0000 −0.0896460 −0.0448230 0.998995i \(-0.514272\pi\)
−0.0448230 + 0.998995i \(0.514272\pi\)
\(30\) −24.0000 −0.146059
\(31\) 152.000i 0.880645i 0.897840 + 0.440323i \(0.145136\pi\)
−0.897840 + 0.440323i \(0.854864\pi\)
\(32\) − 32.0000i − 0.176777i
\(33\) 6.00000i 0.0316505i
\(34\) − 12.0000i − 0.0605289i
\(35\) 16.0000 0.0772712
\(36\) −36.0000 −0.166667
\(37\) − 258.000i − 1.14635i −0.819433 0.573175i \(-0.805712\pi\)
0.819433 0.573175i \(-0.194288\pi\)
\(38\) 72.0000 0.307367
\(39\) 0 0
\(40\) 32.0000 0.126491
\(41\) − 84.0000i − 0.319966i −0.987120 0.159983i \(-0.948856\pi\)
0.987120 0.159983i \(-0.0511439\pi\)
\(42\) 24.0000 0.0881733
\(43\) 188.000 0.666738 0.333369 0.942796i \(-0.391815\pi\)
0.333369 + 0.942796i \(0.391815\pi\)
\(44\) − 8.00000i − 0.0274101i
\(45\) − 36.0000i − 0.119257i
\(46\) − 40.0000i − 0.128210i
\(47\) 254.000i 0.788292i 0.919048 + 0.394146i \(0.128960\pi\)
−0.919048 + 0.394146i \(0.871040\pi\)
\(48\) 48.0000 0.144338
\(49\) 327.000 0.953353
\(50\) − 218.000i − 0.616597i
\(51\) 18.0000 0.0494217
\(52\) 0 0
\(53\) 366.000 0.948565 0.474283 0.880373i \(-0.342707\pi\)
0.474283 + 0.880373i \(0.342707\pi\)
\(54\) − 54.0000i − 0.136083i
\(55\) 8.00000 0.0196131
\(56\) −32.0000 −0.0763604
\(57\) 108.000i 0.250964i
\(58\) 28.0000i 0.0633893i
\(59\) 550.000i 1.21363i 0.794845 + 0.606813i \(0.207552\pi\)
−0.794845 + 0.606813i \(0.792448\pi\)
\(60\) 48.0000i 0.103280i
\(61\) −14.0000 −0.0293855 −0.0146928 0.999892i \(-0.504677\pi\)
−0.0146928 + 0.999892i \(0.504677\pi\)
\(62\) 304.000 0.622710
\(63\) 36.0000i 0.0719932i
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) 12.0000 0.0223803
\(67\) − 448.000i − 0.816894i −0.912782 0.408447i \(-0.866070\pi\)
0.912782 0.408447i \(-0.133930\pi\)
\(68\) −24.0000 −0.0428004
\(69\) 60.0000 0.104683
\(70\) − 32.0000i − 0.0546390i
\(71\) − 926.000i − 1.54783i −0.633289 0.773915i \(-0.718296\pi\)
0.633289 0.773915i \(-0.281704\pi\)
\(72\) 72.0000i 0.117851i
\(73\) 254.000i 0.407239i 0.979050 + 0.203620i \(0.0652706\pi\)
−0.979050 + 0.203620i \(0.934729\pi\)
\(74\) −516.000 −0.810592
\(75\) 327.000 0.503449
\(76\) − 144.000i − 0.217341i
\(77\) −8.00000 −0.0118401
\(78\) 0 0
\(79\) 1328.00 1.89129 0.945644 0.325205i \(-0.105433\pi\)
0.945644 + 0.325205i \(0.105433\pi\)
\(80\) − 64.0000i − 0.0894427i
\(81\) 81.0000 0.111111
\(82\) −168.000 −0.226250
\(83\) − 186.000i − 0.245978i −0.992408 0.122989i \(-0.960752\pi\)
0.992408 0.122989i \(-0.0392479\pi\)
\(84\) − 48.0000i − 0.0623480i
\(85\) − 24.0000i − 0.0306255i
\(86\) − 376.000i − 0.471455i
\(87\) −42.0000 −0.0517572
\(88\) −16.0000 −0.0193819
\(89\) − 336.000i − 0.400179i −0.979778 0.200089i \(-0.935877\pi\)
0.979778 0.200089i \(-0.0641233\pi\)
\(90\) −72.0000 −0.0843274
\(91\) 0 0
\(92\) −80.0000 −0.0906584
\(93\) 456.000i 0.508441i
\(94\) 508.000 0.557406
\(95\) 144.000 0.155517
\(96\) − 96.0000i − 0.102062i
\(97\) − 614.000i − 0.642704i −0.946960 0.321352i \(-0.895863\pi\)
0.946960 0.321352i \(-0.104137\pi\)
\(98\) − 654.000i − 0.674122i
\(99\) 18.0000i 0.0182734i
\(100\) −436.000 −0.436000
\(101\) 1606.00 1.58221 0.791104 0.611682i \(-0.209507\pi\)
0.791104 + 0.611682i \(0.209507\pi\)
\(102\) − 36.0000i − 0.0349464i
\(103\) −208.000 −0.198979 −0.0994896 0.995039i \(-0.531721\pi\)
−0.0994896 + 0.995039i \(0.531721\pi\)
\(104\) 0 0
\(105\) 48.0000 0.0446126
\(106\) − 732.000i − 0.670737i
\(107\) −248.000 −0.224066 −0.112033 0.993704i \(-0.535736\pi\)
−0.112033 + 0.993704i \(0.535736\pi\)
\(108\) −108.000 −0.0962250
\(109\) 542.000i 0.476277i 0.971231 + 0.238138i \(0.0765372\pi\)
−0.971231 + 0.238138i \(0.923463\pi\)
\(110\) − 16.0000i − 0.0138685i
\(111\) − 774.000i − 0.661845i
\(112\) 64.0000i 0.0539949i
\(113\) −2042.00 −1.69996 −0.849979 0.526817i \(-0.823385\pi\)
−0.849979 + 0.526817i \(0.823385\pi\)
\(114\) 216.000 0.177458
\(115\) − 80.0000i − 0.0648699i
\(116\) 56.0000 0.0448230
\(117\) 0 0
\(118\) 1100.00 0.858163
\(119\) 24.0000i 0.0184880i
\(120\) 96.0000 0.0730297
\(121\) 1327.00 0.996995
\(122\) 28.0000i 0.0207787i
\(123\) − 252.000i − 0.184732i
\(124\) − 608.000i − 0.440323i
\(125\) − 936.000i − 0.669747i
\(126\) 72.0000 0.0509069
\(127\) 488.000 0.340968 0.170484 0.985360i \(-0.445467\pi\)
0.170484 + 0.985360i \(0.445467\pi\)
\(128\) 128.000i 0.0883883i
\(129\) 564.000 0.384941
\(130\) 0 0
\(131\) 1744.00 1.16316 0.581580 0.813489i \(-0.302435\pi\)
0.581580 + 0.813489i \(0.302435\pi\)
\(132\) − 24.0000i − 0.0158252i
\(133\) −144.000 −0.0938826
\(134\) −896.000 −0.577631
\(135\) − 108.000i − 0.0688530i
\(136\) 48.0000i 0.0302645i
\(137\) − 828.000i − 0.516356i −0.966097 0.258178i \(-0.916878\pi\)
0.966097 0.258178i \(-0.0831222\pi\)
\(138\) − 120.000i − 0.0740223i
\(139\) −404.000 −0.246524 −0.123262 0.992374i \(-0.539336\pi\)
−0.123262 + 0.992374i \(0.539336\pi\)
\(140\) −64.0000 −0.0386356
\(141\) 762.000i 0.455120i
\(142\) −1852.00 −1.09448
\(143\) 0 0
\(144\) 144.000 0.0833333
\(145\) 56.0000i 0.0320727i
\(146\) 508.000 0.287962
\(147\) 981.000 0.550418
\(148\) 1032.00i 0.573175i
\(149\) − 2928.00i − 1.60987i −0.593361 0.804937i \(-0.702199\pi\)
0.593361 0.804937i \(-0.297801\pi\)
\(150\) − 654.000i − 0.355993i
\(151\) 1944.00i 1.04769i 0.851815 + 0.523843i \(0.175502\pi\)
−0.851815 + 0.523843i \(0.824498\pi\)
\(152\) −288.000 −0.153683
\(153\) 54.0000 0.0285336
\(154\) 16.0000i 0.00837219i
\(155\) 608.000 0.315069
\(156\) 0 0
\(157\) 3590.00 1.82492 0.912462 0.409161i \(-0.134178\pi\)
0.912462 + 0.409161i \(0.134178\pi\)
\(158\) − 2656.00i − 1.33734i
\(159\) 1098.00 0.547654
\(160\) −128.000 −0.0632456
\(161\) 80.0000i 0.0391608i
\(162\) − 162.000i − 0.0785674i
\(163\) − 2284.00i − 1.09753i −0.835978 0.548763i \(-0.815099\pi\)
0.835978 0.548763i \(-0.184901\pi\)
\(164\) 336.000i 0.159983i
\(165\) 24.0000 0.0113236
\(166\) −372.000 −0.173933
\(167\) 3174.00i 1.47073i 0.677673 + 0.735364i \(0.262989\pi\)
−0.677673 + 0.735364i \(0.737011\pi\)
\(168\) −96.0000 −0.0440867
\(169\) 0 0
\(170\) −48.0000 −0.0216555
\(171\) 324.000i 0.144894i
\(172\) −752.000 −0.333369
\(173\) 1358.00 0.596802 0.298401 0.954441i \(-0.403547\pi\)
0.298401 + 0.954441i \(0.403547\pi\)
\(174\) 84.0000i 0.0365978i
\(175\) 436.000i 0.188334i
\(176\) 32.0000i 0.0137051i
\(177\) 1650.00i 0.700687i
\(178\) −672.000 −0.282969
\(179\) −708.000 −0.295634 −0.147817 0.989015i \(-0.547225\pi\)
−0.147817 + 0.989015i \(0.547225\pi\)
\(180\) 144.000i 0.0596285i
\(181\) 546.000 0.224220 0.112110 0.993696i \(-0.464239\pi\)
0.112110 + 0.993696i \(0.464239\pi\)
\(182\) 0 0
\(183\) −42.0000 −0.0169657
\(184\) 160.000i 0.0641052i
\(185\) −1032.00 −0.410131
\(186\) 912.000 0.359522
\(187\) 12.0000i 0.00469266i
\(188\) − 1016.00i − 0.394146i
\(189\) 108.000i 0.0415653i
\(190\) − 288.000i − 0.109967i
\(191\) −3472.00 −1.31531 −0.657657 0.753317i \(-0.728453\pi\)
−0.657657 + 0.753317i \(0.728453\pi\)
\(192\) −192.000 −0.0721688
\(193\) − 310.000i − 0.115618i −0.998328 0.0578090i \(-0.981589\pi\)
0.998328 0.0578090i \(-0.0184115\pi\)
\(194\) −1228.00 −0.454460
\(195\) 0 0
\(196\) −1308.00 −0.476676
\(197\) − 1020.00i − 0.368893i −0.982843 0.184447i \(-0.940951\pi\)
0.982843 0.184447i \(-0.0590493\pi\)
\(198\) 36.0000 0.0129213
\(199\) 3256.00 1.15986 0.579929 0.814667i \(-0.303080\pi\)
0.579929 + 0.814667i \(0.303080\pi\)
\(200\) 872.000i 0.308299i
\(201\) − 1344.00i − 0.471634i
\(202\) − 3212.00i − 1.11879i
\(203\) − 56.0000i − 0.0193617i
\(204\) −72.0000 −0.0247108
\(205\) −336.000 −0.114474
\(206\) 416.000i 0.140699i
\(207\) 180.000 0.0604390
\(208\) 0 0
\(209\) −72.0000 −0.0238294
\(210\) − 96.0000i − 0.0315459i
\(211\) −4564.00 −1.48909 −0.744547 0.667570i \(-0.767334\pi\)
−0.744547 + 0.667570i \(0.767334\pi\)
\(212\) −1464.00 −0.474283
\(213\) − 2778.00i − 0.893640i
\(214\) 496.000i 0.158439i
\(215\) − 752.000i − 0.238539i
\(216\) 216.000i 0.0680414i
\(217\) −608.000 −0.190202
\(218\) 1084.00 0.336779
\(219\) 762.000i 0.235120i
\(220\) −32.0000 −0.00980654
\(221\) 0 0
\(222\) −1548.00 −0.467995
\(223\) 72.0000i 0.0216210i 0.999942 + 0.0108105i \(0.00344115\pi\)
−0.999942 + 0.0108105i \(0.996559\pi\)
\(224\) 128.000 0.0381802
\(225\) 981.000 0.290667
\(226\) 4084.00i 1.20205i
\(227\) − 2694.00i − 0.787696i −0.919176 0.393848i \(-0.871144\pi\)
0.919176 0.393848i \(-0.128856\pi\)
\(228\) − 432.000i − 0.125482i
\(229\) 5922.00i 1.70889i 0.519538 + 0.854447i \(0.326104\pi\)
−0.519538 + 0.854447i \(0.673896\pi\)
\(230\) −160.000 −0.0458699
\(231\) −24.0000 −0.00683586
\(232\) − 112.000i − 0.0316947i
\(233\) 5122.00 1.44014 0.720072 0.693900i \(-0.244109\pi\)
0.720072 + 0.693900i \(0.244109\pi\)
\(234\) 0 0
\(235\) 1016.00 0.282028
\(236\) − 2200.00i − 0.606813i
\(237\) 3984.00 1.09194
\(238\) 48.0000 0.0130730
\(239\) − 5022.00i − 1.35919i −0.733588 0.679595i \(-0.762156\pi\)
0.733588 0.679595i \(-0.237844\pi\)
\(240\) − 192.000i − 0.0516398i
\(241\) − 1218.00i − 0.325553i −0.986663 0.162777i \(-0.947955\pi\)
0.986663 0.162777i \(-0.0520450\pi\)
\(242\) − 2654.00i − 0.704982i
\(243\) 243.000 0.0641500
\(244\) 56.0000 0.0146928
\(245\) − 1308.00i − 0.341082i
\(246\) −504.000 −0.130625
\(247\) 0 0
\(248\) −1216.00 −0.311355
\(249\) − 558.000i − 0.142015i
\(250\) −1872.00 −0.473583
\(251\) 2112.00 0.531109 0.265554 0.964096i \(-0.414445\pi\)
0.265554 + 0.964096i \(0.414445\pi\)
\(252\) − 144.000i − 0.0359966i
\(253\) 40.0000i 0.00993984i
\(254\) − 976.000i − 0.241101i
\(255\) − 72.0000i − 0.0176816i
\(256\) 256.000 0.0625000
\(257\) −2814.00 −0.683006 −0.341503 0.939881i \(-0.610936\pi\)
−0.341503 + 0.939881i \(0.610936\pi\)
\(258\) − 1128.00i − 0.272195i
\(259\) 1032.00 0.247588
\(260\) 0 0
\(261\) −126.000 −0.0298820
\(262\) − 3488.00i − 0.822478i
\(263\) −4044.00 −0.948151 −0.474076 0.880484i \(-0.657218\pi\)
−0.474076 + 0.880484i \(0.657218\pi\)
\(264\) −48.0000 −0.0111901
\(265\) − 1464.00i − 0.339369i
\(266\) 288.000i 0.0663850i
\(267\) − 1008.00i − 0.231043i
\(268\) 1792.00i 0.408447i
\(269\) −1470.00 −0.333188 −0.166594 0.986026i \(-0.553277\pi\)
−0.166594 + 0.986026i \(0.553277\pi\)
\(270\) −216.000 −0.0486864
\(271\) − 1844.00i − 0.413340i −0.978411 0.206670i \(-0.933737\pi\)
0.978411 0.206670i \(-0.0662626\pi\)
\(272\) 96.0000 0.0214002
\(273\) 0 0
\(274\) −1656.00 −0.365119
\(275\) 218.000i 0.0478033i
\(276\) −240.000 −0.0523417
\(277\) −5766.00 −1.25071 −0.625353 0.780342i \(-0.715045\pi\)
−0.625353 + 0.780342i \(0.715045\pi\)
\(278\) 808.000i 0.174319i
\(279\) 1368.00i 0.293548i
\(280\) 128.000i 0.0273195i
\(281\) − 7468.00i − 1.58542i −0.609598 0.792711i \(-0.708669\pi\)
0.609598 0.792711i \(-0.291331\pi\)
\(282\) 1524.00 0.321819
\(283\) −1228.00 −0.257940 −0.128970 0.991648i \(-0.541167\pi\)
−0.128970 + 0.991648i \(0.541167\pi\)
\(284\) 3704.00i 0.773915i
\(285\) 432.000 0.0897876
\(286\) 0 0
\(287\) 336.000 0.0691061
\(288\) − 288.000i − 0.0589256i
\(289\) −4877.00 −0.992673
\(290\) 112.000 0.0226788
\(291\) − 1842.00i − 0.371065i
\(292\) − 1016.00i − 0.203620i
\(293\) 6608.00i 1.31755i 0.752338 + 0.658777i \(0.228926\pi\)
−0.752338 + 0.658777i \(0.771074\pi\)
\(294\) − 1962.00i − 0.389205i
\(295\) 2200.00 0.434200
\(296\) 2064.00 0.405296
\(297\) 54.0000i 0.0105502i
\(298\) −5856.00 −1.13835
\(299\) 0 0
\(300\) −1308.00 −0.251725
\(301\) 752.000i 0.144002i
\(302\) 3888.00 0.740825
\(303\) 4818.00 0.913488
\(304\) 576.000i 0.108671i
\(305\) 56.0000i 0.0105133i
\(306\) − 108.000i − 0.0201763i
\(307\) 7664.00i 1.42478i 0.701784 + 0.712390i \(0.252387\pi\)
−0.701784 + 0.712390i \(0.747613\pi\)
\(308\) 32.0000 0.00592003
\(309\) −624.000 −0.114881
\(310\) − 1216.00i − 0.222788i
\(311\) 2340.00 0.426653 0.213327 0.976981i \(-0.431570\pi\)
0.213327 + 0.976981i \(0.431570\pi\)
\(312\) 0 0
\(313\) 6710.00 1.21173 0.605865 0.795567i \(-0.292827\pi\)
0.605865 + 0.795567i \(0.292827\pi\)
\(314\) − 7180.00i − 1.29042i
\(315\) 144.000 0.0257571
\(316\) −5312.00 −0.945644
\(317\) − 4164.00i − 0.737771i −0.929475 0.368886i \(-0.879739\pi\)
0.929475 0.368886i \(-0.120261\pi\)
\(318\) − 2196.00i − 0.387250i
\(319\) − 28.0000i − 0.00491442i
\(320\) 256.000i 0.0447214i
\(321\) −744.000 −0.129365
\(322\) 160.000 0.0276908
\(323\) 216.000i 0.0372092i
\(324\) −324.000 −0.0555556
\(325\) 0 0
\(326\) −4568.00 −0.776068
\(327\) 1626.00i 0.274979i
\(328\) 672.000 0.113125
\(329\) −1016.00 −0.170255
\(330\) − 48.0000i − 0.00800701i
\(331\) 10072.0i 1.67253i 0.548326 + 0.836265i \(0.315265\pi\)
−0.548326 + 0.836265i \(0.684735\pi\)
\(332\) 744.000i 0.122989i
\(333\) − 2322.00i − 0.382117i
\(334\) 6348.00 1.03996
\(335\) −1792.00 −0.292261
\(336\) 192.000i 0.0311740i
\(337\) −2990.00 −0.483311 −0.241655 0.970362i \(-0.577690\pi\)
−0.241655 + 0.970362i \(0.577690\pi\)
\(338\) 0 0
\(339\) −6126.00 −0.981471
\(340\) 96.0000i 0.0153127i
\(341\) −304.000 −0.0482772
\(342\) 648.000 0.102456
\(343\) 2680.00i 0.421885i
\(344\) 1504.00i 0.235727i
\(345\) − 240.000i − 0.0374527i
\(346\) − 2716.00i − 0.422003i
\(347\) 6564.00 1.01549 0.507743 0.861508i \(-0.330480\pi\)
0.507743 + 0.861508i \(0.330480\pi\)
\(348\) 168.000 0.0258786
\(349\) − 674.000i − 0.103376i −0.998663 0.0516882i \(-0.983540\pi\)
0.998663 0.0516882i \(-0.0164602\pi\)
\(350\) 872.000 0.133172
\(351\) 0 0
\(352\) 64.0000 0.00969094
\(353\) 10732.0i 1.61815i 0.587706 + 0.809075i \(0.300031\pi\)
−0.587706 + 0.809075i \(0.699969\pi\)
\(354\) 3300.00 0.495461
\(355\) −3704.00 −0.553769
\(356\) 1344.00i 0.200089i
\(357\) 72.0000i 0.0106741i
\(358\) 1416.00i 0.209044i
\(359\) − 4842.00i − 0.711841i −0.934516 0.355921i \(-0.884167\pi\)
0.934516 0.355921i \(-0.115833\pi\)
\(360\) 288.000 0.0421637
\(361\) 5563.00 0.811051
\(362\) − 1092.00i − 0.158548i
\(363\) 3981.00 0.575615
\(364\) 0 0
\(365\) 1016.00 0.145698
\(366\) 84.0000i 0.0119966i
\(367\) −6280.00 −0.893224 −0.446612 0.894728i \(-0.647370\pi\)
−0.446612 + 0.894728i \(0.647370\pi\)
\(368\) 320.000 0.0453292
\(369\) − 756.000i − 0.106655i
\(370\) 2064.00i 0.290006i
\(371\) 1464.00i 0.204871i
\(372\) − 1824.00i − 0.254220i
\(373\) 6434.00 0.893136 0.446568 0.894750i \(-0.352646\pi\)
0.446568 + 0.894750i \(0.352646\pi\)
\(374\) 24.0000 0.00331821
\(375\) − 2808.00i − 0.386679i
\(376\) −2032.00 −0.278703
\(377\) 0 0
\(378\) 216.000 0.0293911
\(379\) 9068.00i 1.22900i 0.788916 + 0.614501i \(0.210643\pi\)
−0.788916 + 0.614501i \(0.789357\pi\)
\(380\) −576.000 −0.0777584
\(381\) 1464.00 0.196858
\(382\) 6944.00i 0.930068i
\(383\) − 3162.00i − 0.421855i −0.977502 0.210928i \(-0.932352\pi\)
0.977502 0.210928i \(-0.0676485\pi\)
\(384\) 384.000i 0.0510310i
\(385\) 32.0000i 0.00423603i
\(386\) −620.000 −0.0817543
\(387\) 1692.00 0.222246
\(388\) 2456.00i 0.321352i
\(389\) 3666.00 0.477824 0.238912 0.971041i \(-0.423209\pi\)
0.238912 + 0.971041i \(0.423209\pi\)
\(390\) 0 0
\(391\) 120.000 0.0155209
\(392\) 2616.00i 0.337061i
\(393\) 5232.00 0.671551
\(394\) −2040.00 −0.260847
\(395\) − 5312.00i − 0.676647i
\(396\) − 72.0000i − 0.00913671i
\(397\) 11054.0i 1.39744i 0.715394 + 0.698721i \(0.246247\pi\)
−0.715394 + 0.698721i \(0.753753\pi\)
\(398\) − 6512.00i − 0.820143i
\(399\) −432.000 −0.0542031
\(400\) 1744.00 0.218000
\(401\) − 5328.00i − 0.663510i −0.943366 0.331755i \(-0.892359\pi\)
0.943366 0.331755i \(-0.107641\pi\)
\(402\) −2688.00 −0.333496
\(403\) 0 0
\(404\) −6424.00 −0.791104
\(405\) − 324.000i − 0.0397523i
\(406\) −112.000 −0.0136908
\(407\) 516.000 0.0628432
\(408\) 144.000i 0.0174732i
\(409\) 12074.0i 1.45971i 0.683603 + 0.729854i \(0.260412\pi\)
−0.683603 + 0.729854i \(0.739588\pi\)
\(410\) 672.000i 0.0809456i
\(411\) − 2484.00i − 0.298118i
\(412\) 832.000 0.0994896
\(413\) −2200.00 −0.262118
\(414\) − 360.000i − 0.0427368i
\(415\) −744.000 −0.0880037
\(416\) 0 0
\(417\) −1212.00 −0.142331
\(418\) 144.000i 0.0168499i
\(419\) 13584.0 1.58382 0.791911 0.610636i \(-0.209086\pi\)
0.791911 + 0.610636i \(0.209086\pi\)
\(420\) −192.000 −0.0223063
\(421\) 7406.00i 0.857355i 0.903458 + 0.428677i \(0.141020\pi\)
−0.903458 + 0.428677i \(0.858980\pi\)
\(422\) 9128.00i 1.05295i
\(423\) 2286.00i 0.262764i
\(424\) 2928.00i 0.335369i
\(425\) 654.000 0.0746439
\(426\) −5556.00 −0.631899
\(427\) − 56.0000i − 0.00634667i
\(428\) 992.000 0.112033
\(429\) 0 0
\(430\) −1504.00 −0.168673
\(431\) 10134.0i 1.13257i 0.824210 + 0.566285i \(0.191620\pi\)
−0.824210 + 0.566285i \(0.808380\pi\)
\(432\) 432.000 0.0481125
\(433\) −9406.00 −1.04393 −0.521967 0.852966i \(-0.674802\pi\)
−0.521967 + 0.852966i \(0.674802\pi\)
\(434\) 1216.00i 0.134493i
\(435\) 168.000i 0.0185172i
\(436\) − 2168.00i − 0.238138i
\(437\) 720.000i 0.0788153i
\(438\) 1524.00 0.166255
\(439\) −4088.00 −0.444441 −0.222220 0.974996i \(-0.571330\pi\)
−0.222220 + 0.974996i \(0.571330\pi\)
\(440\) 64.0000i 0.00693427i
\(441\) 2943.00 0.317784
\(442\) 0 0
\(443\) −5328.00 −0.571424 −0.285712 0.958315i \(-0.592230\pi\)
−0.285712 + 0.958315i \(0.592230\pi\)
\(444\) 3096.00i 0.330923i
\(445\) −1344.00 −0.143172
\(446\) 144.000 0.0152883
\(447\) − 8784.00i − 0.929461i
\(448\) − 256.000i − 0.0269975i
\(449\) 13160.0i 1.38320i 0.722279 + 0.691602i \(0.243095\pi\)
−0.722279 + 0.691602i \(0.756905\pi\)
\(450\) − 1962.00i − 0.205532i
\(451\) 168.000 0.0175406
\(452\) 8168.00 0.849979
\(453\) 5832.00i 0.604881i
\(454\) −5388.00 −0.556985
\(455\) 0 0
\(456\) −864.000 −0.0887292
\(457\) 9146.00i 0.936175i 0.883682 + 0.468087i \(0.155057\pi\)
−0.883682 + 0.468087i \(0.844943\pi\)
\(458\) 11844.0 1.20837
\(459\) 162.000 0.0164739
\(460\) 320.000i 0.0324349i
\(461\) − 5580.00i − 0.563745i −0.959452 0.281873i \(-0.909044\pi\)
0.959452 0.281873i \(-0.0909555\pi\)
\(462\) 48.0000i 0.00483368i
\(463\) 14788.0i 1.48436i 0.670203 + 0.742178i \(0.266207\pi\)
−0.670203 + 0.742178i \(0.733793\pi\)
\(464\) −224.000 −0.0224115
\(465\) 1824.00 0.181905
\(466\) − 10244.0i − 1.01834i
\(467\) −12376.0 −1.22632 −0.613162 0.789957i \(-0.710103\pi\)
−0.613162 + 0.789957i \(0.710103\pi\)
\(468\) 0 0
\(469\) 1792.00 0.176433
\(470\) − 2032.00i − 0.199424i
\(471\) 10770.0 1.05362
\(472\) −4400.00 −0.429081
\(473\) 376.000i 0.0365507i
\(474\) − 7968.00i − 0.772115i
\(475\) 3924.00i 0.379043i
\(476\) − 96.0000i − 0.00924402i
\(477\) 3294.00 0.316188
\(478\) −10044.0 −0.961092
\(479\) 834.000i 0.0795541i 0.999209 + 0.0397771i \(0.0126648\pi\)
−0.999209 + 0.0397771i \(0.987335\pi\)
\(480\) −384.000 −0.0365148
\(481\) 0 0
\(482\) −2436.00 −0.230201
\(483\) 240.000i 0.0226095i
\(484\) −5308.00 −0.498497
\(485\) −2456.00 −0.229941
\(486\) − 486.000i − 0.0453609i
\(487\) 13192.0i 1.22749i 0.789505 + 0.613744i \(0.210337\pi\)
−0.789505 + 0.613744i \(0.789663\pi\)
\(488\) − 112.000i − 0.0103893i
\(489\) − 6852.00i − 0.633657i
\(490\) −2616.00 −0.241181
\(491\) −16568.0 −1.52282 −0.761409 0.648272i \(-0.775492\pi\)
−0.761409 + 0.648272i \(0.775492\pi\)
\(492\) 1008.00i 0.0923662i
\(493\) −84.0000 −0.00767377
\(494\) 0 0
\(495\) 72.0000 0.00653770
\(496\) 2432.00i 0.220161i
\(497\) 3704.00 0.334300
\(498\) −1116.00 −0.100420
\(499\) 10136.0i 0.909318i 0.890666 + 0.454659i \(0.150239\pi\)
−0.890666 + 0.454659i \(0.849761\pi\)
\(500\) 3744.00i 0.334874i
\(501\) 9522.00i 0.849125i
\(502\) − 4224.00i − 0.375550i
\(503\) 10412.0 0.922959 0.461479 0.887151i \(-0.347319\pi\)
0.461479 + 0.887151i \(0.347319\pi\)
\(504\) −288.000 −0.0254535
\(505\) − 6424.00i − 0.566068i
\(506\) 80.0000 0.00702853
\(507\) 0 0
\(508\) −1952.00 −0.170484
\(509\) 4180.00i 0.363999i 0.983299 + 0.181999i \(0.0582568\pi\)
−0.983299 + 0.181999i \(0.941743\pi\)
\(510\) −144.000 −0.0125028
\(511\) −1016.00 −0.0879554
\(512\) − 512.000i − 0.0441942i
\(513\) 972.000i 0.0836547i
\(514\) 5628.00i 0.482958i
\(515\) 832.000i 0.0711889i
\(516\) −2256.00 −0.192471
\(517\) −508.000 −0.0432143
\(518\) − 2064.00i − 0.175071i
\(519\) 4074.00 0.344564
\(520\) 0 0
\(521\) −14610.0 −1.22855 −0.614276 0.789091i \(-0.710552\pi\)
−0.614276 + 0.789091i \(0.710552\pi\)
\(522\) 252.000i 0.0211298i
\(523\) −2172.00 −0.181596 −0.0907982 0.995869i \(-0.528942\pi\)
−0.0907982 + 0.995869i \(0.528942\pi\)
\(524\) −6976.00 −0.581580
\(525\) 1308.00i 0.108735i
\(526\) 8088.00i 0.670444i
\(527\) 912.000i 0.0753840i
\(528\) 96.0000i 0.00791262i
\(529\) −11767.0 −0.967124
\(530\) −2928.00 −0.239970
\(531\) 4950.00i 0.404542i
\(532\) 576.000 0.0469413
\(533\) 0 0
\(534\) −2016.00 −0.163372
\(535\) 992.000i 0.0801643i
\(536\) 3584.00 0.288816
\(537\) −2124.00 −0.170684
\(538\) 2940.00i 0.235599i
\(539\) 654.000i 0.0522630i
\(540\) 432.000i 0.0344265i
\(541\) − 11758.0i − 0.934410i −0.884149 0.467205i \(-0.845261\pi\)
0.884149 0.467205i \(-0.154739\pi\)
\(542\) −3688.00 −0.292275
\(543\) 1638.00 0.129454
\(544\) − 192.000i − 0.0151322i
\(545\) 2168.00 0.170398
\(546\) 0 0
\(547\) 340.000 0.0265765 0.0132883 0.999912i \(-0.495770\pi\)
0.0132883 + 0.999912i \(0.495770\pi\)
\(548\) 3312.00i 0.258178i
\(549\) −126.000 −0.00979517
\(550\) 436.000 0.0338020
\(551\) − 504.000i − 0.0389676i
\(552\) 480.000i 0.0370112i
\(553\) 5312.00i 0.408480i
\(554\) 11532.0i 0.884382i
\(555\) −3096.00 −0.236789
\(556\) 1616.00 0.123262
\(557\) − 3768.00i − 0.286634i −0.989677 0.143317i \(-0.954223\pi\)
0.989677 0.143317i \(-0.0457769\pi\)
\(558\) 2736.00 0.207570
\(559\) 0 0
\(560\) 256.000 0.0193178
\(561\) 36.0000i 0.00270931i
\(562\) −14936.0 −1.12106
\(563\) 10172.0 0.761454 0.380727 0.924687i \(-0.375674\pi\)
0.380727 + 0.924687i \(0.375674\pi\)
\(564\) − 3048.00i − 0.227560i
\(565\) 8168.00i 0.608195i
\(566\) 2456.00i 0.182391i
\(567\) 324.000i 0.0239977i
\(568\) 7408.00 0.547241
\(569\) 5506.00 0.405665 0.202833 0.979213i \(-0.434985\pi\)
0.202833 + 0.979213i \(0.434985\pi\)
\(570\) − 864.000i − 0.0634894i
\(571\) −2340.00 −0.171499 −0.0857495 0.996317i \(-0.527328\pi\)
−0.0857495 + 0.996317i \(0.527328\pi\)
\(572\) 0 0
\(573\) −10416.0 −0.759397
\(574\) − 672.000i − 0.0488654i
\(575\) 2180.00 0.158108
\(576\) −576.000 −0.0416667
\(577\) 20094.0i 1.44978i 0.688864 + 0.724891i \(0.258110\pi\)
−0.688864 + 0.724891i \(0.741890\pi\)
\(578\) 9754.00i 0.701925i
\(579\) − 930.000i − 0.0667521i
\(580\) − 224.000i − 0.0160364i
\(581\) 744.000 0.0531262
\(582\) −3684.00 −0.262383
\(583\) 732.000i 0.0520006i
\(584\) −2032.00 −0.143981
\(585\) 0 0
\(586\) 13216.0 0.931652
\(587\) 7118.00i 0.500496i 0.968182 + 0.250248i \(0.0805122\pi\)
−0.968182 + 0.250248i \(0.919488\pi\)
\(588\) −3924.00 −0.275209
\(589\) −5472.00 −0.382801
\(590\) − 4400.00i − 0.307026i
\(591\) − 3060.00i − 0.212981i
\(592\) − 4128.00i − 0.286587i
\(593\) − 10328.0i − 0.715211i −0.933873 0.357606i \(-0.883593\pi\)
0.933873 0.357606i \(-0.116407\pi\)
\(594\) 108.000 0.00746009
\(595\) 96.0000 0.00661448
\(596\) 11712.0i 0.804937i
\(597\) 9768.00 0.669644
\(598\) 0 0
\(599\) −19732.0 −1.34596 −0.672978 0.739662i \(-0.734985\pi\)
−0.672978 + 0.739662i \(0.734985\pi\)
\(600\) 2616.00i 0.177996i
\(601\) −12026.0 −0.816224 −0.408112 0.912932i \(-0.633813\pi\)
−0.408112 + 0.912932i \(0.633813\pi\)
\(602\) 1504.00 0.101825
\(603\) − 4032.00i − 0.272298i
\(604\) − 7776.00i − 0.523843i
\(605\) − 5308.00i − 0.356696i
\(606\) − 9636.00i − 0.645934i
\(607\) 17016.0 1.13782 0.568911 0.822399i \(-0.307365\pi\)
0.568911 + 0.822399i \(0.307365\pi\)
\(608\) 1152.00 0.0768417
\(609\) − 168.000i − 0.0111785i
\(610\) 112.000 0.00743401
\(611\) 0 0
\(612\) −216.000 −0.0142668
\(613\) − 11654.0i − 0.767864i −0.923361 0.383932i \(-0.874570\pi\)
0.923361 0.383932i \(-0.125430\pi\)
\(614\) 15328.0 1.00747
\(615\) −1008.00 −0.0660918
\(616\) − 64.0000i − 0.00418609i
\(617\) − 11612.0i − 0.757669i −0.925465 0.378834i \(-0.876325\pi\)
0.925465 0.378834i \(-0.123675\pi\)
\(618\) 1248.00i 0.0812329i
\(619\) 4024.00i 0.261290i 0.991429 + 0.130645i \(0.0417047\pi\)
−0.991429 + 0.130645i \(0.958295\pi\)
\(620\) −2432.00 −0.157535
\(621\) 540.000 0.0348945
\(622\) − 4680.00i − 0.301690i
\(623\) 1344.00 0.0864305
\(624\) 0 0
\(625\) 9881.00 0.632384
\(626\) − 13420.0i − 0.856823i
\(627\) −216.000 −0.0137579
\(628\) −14360.0 −0.912462
\(629\) − 1548.00i − 0.0981285i
\(630\) − 288.000i − 0.0182130i
\(631\) − 1088.00i − 0.0686412i −0.999411 0.0343206i \(-0.989073\pi\)
0.999411 0.0343206i \(-0.0109267\pi\)
\(632\) 10624.0i 0.668671i
\(633\) −13692.0 −0.859729
\(634\) −8328.00 −0.521683
\(635\) − 1952.00i − 0.121989i
\(636\) −4392.00 −0.273827
\(637\) 0 0
\(638\) −56.0000 −0.00347502
\(639\) − 8334.00i − 0.515944i
\(640\) 512.000 0.0316228
\(641\) 7078.00 0.436138 0.218069 0.975933i \(-0.430024\pi\)
0.218069 + 0.975933i \(0.430024\pi\)
\(642\) 1488.00i 0.0914746i
\(643\) − 8336.00i − 0.511259i −0.966775 0.255630i \(-0.917717\pi\)
0.966775 0.255630i \(-0.0822828\pi\)
\(644\) − 320.000i − 0.0195804i
\(645\) − 2256.00i − 0.137721i
\(646\) 432.000 0.0263109
\(647\) −32.0000 −0.00194444 −0.000972218 1.00000i \(-0.500309\pi\)
−0.000972218 1.00000i \(0.500309\pi\)
\(648\) 648.000i 0.0392837i
\(649\) −1100.00 −0.0665312
\(650\) 0 0
\(651\) −1824.00 −0.109813
\(652\) 9136.00i 0.548763i
\(653\) −15822.0 −0.948182 −0.474091 0.880476i \(-0.657223\pi\)
−0.474091 + 0.880476i \(0.657223\pi\)
\(654\) 3252.00 0.194439
\(655\) − 6976.00i − 0.416145i
\(656\) − 1344.00i − 0.0799914i
\(657\) 2286.00i 0.135746i
\(658\) 2032.00i 0.120388i
\(659\) 21540.0 1.27326 0.636631 0.771169i \(-0.280328\pi\)
0.636631 + 0.771169i \(0.280328\pi\)
\(660\) −96.0000 −0.00566181
\(661\) 8270.00i 0.486635i 0.969947 + 0.243317i \(0.0782357\pi\)
−0.969947 + 0.243317i \(0.921764\pi\)
\(662\) 20144.0 1.18266
\(663\) 0 0
\(664\) 1488.00 0.0869663
\(665\) 576.000i 0.0335885i
\(666\) −4644.00 −0.270197
\(667\) −280.000 −0.0162543
\(668\) − 12696.0i − 0.735364i
\(669\) 216.000i 0.0124829i
\(670\) 3584.00i 0.206660i
\(671\) − 28.0000i − 0.00161092i
\(672\) 384.000 0.0220433
\(673\) −8482.00 −0.485820 −0.242910 0.970049i \(-0.578102\pi\)
−0.242910 + 0.970049i \(0.578102\pi\)
\(674\) 5980.00i 0.341752i
\(675\) 2943.00 0.167816
\(676\) 0 0
\(677\) 2550.00 0.144763 0.0723814 0.997377i \(-0.476940\pi\)
0.0723814 + 0.997377i \(0.476940\pi\)
\(678\) 12252.0i 0.694005i
\(679\) 2456.00 0.138811
\(680\) 192.000 0.0108277
\(681\) − 8082.00i − 0.454777i
\(682\) 608.000i 0.0341371i
\(683\) − 31534.0i − 1.76664i −0.468771 0.883320i \(-0.655303\pi\)
0.468771 0.883320i \(-0.344697\pi\)
\(684\) − 1296.00i − 0.0724471i
\(685\) −3312.00 −0.184737
\(686\) 5360.00 0.298317
\(687\) 17766.0i 0.986631i
\(688\) 3008.00 0.166684
\(689\) 0 0
\(690\) −480.000 −0.0264830
\(691\) − 33832.0i − 1.86256i −0.364302 0.931281i \(-0.618693\pi\)
0.364302 0.931281i \(-0.381307\pi\)
\(692\) −5432.00 −0.298401
\(693\) −72.0000 −0.00394669
\(694\) − 13128.0i − 0.718058i
\(695\) 1616.00i 0.0881991i
\(696\) − 336.000i − 0.0182989i
\(697\) − 504.000i − 0.0273893i
\(698\) −1348.00 −0.0730982
\(699\) 15366.0 0.831467
\(700\) − 1744.00i − 0.0941671i
\(701\) −19422.0 −1.04645 −0.523223 0.852196i \(-0.675271\pi\)
−0.523223 + 0.852196i \(0.675271\pi\)
\(702\) 0 0
\(703\) 9288.00 0.498298
\(704\) − 128.000i − 0.00685253i
\(705\) 3048.00 0.162829
\(706\) 21464.0 1.14420
\(707\) 6424.00i 0.341725i
\(708\) − 6600.00i − 0.350343i
\(709\) − 1894.00i − 0.100325i −0.998741 0.0501627i \(-0.984026\pi\)
0.998741 0.0501627i \(-0.0159740\pi\)
\(710\) 7408.00i 0.391574i
\(711\) 11952.0 0.630429
\(712\) 2688.00 0.141485
\(713\) 3040.00i 0.159676i
\(714\) 144.000 0.00754771
\(715\) 0 0
\(716\) 2832.00 0.147817
\(717\) − 15066.0i − 0.784728i
\(718\) −9684.00 −0.503348
\(719\) 20156.0 1.04547 0.522734 0.852496i \(-0.324912\pi\)
0.522734 + 0.852496i \(0.324912\pi\)
\(720\) − 576.000i − 0.0298142i
\(721\) − 832.000i − 0.0429754i
\(722\) − 11126.0i − 0.573500i
\(723\) − 3654.00i − 0.187958i
\(724\) −2184.00 −0.112110
\(725\) −1526.00 −0.0781713
\(726\) − 7962.00i − 0.407021i
\(727\) −11128.0 −0.567696 −0.283848 0.958869i \(-0.591611\pi\)
−0.283848 + 0.958869i \(0.591611\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) − 2032.00i − 0.103024i
\(731\) 1128.00 0.0570733
\(732\) 168.000 0.00848287
\(733\) − 16202.0i − 0.816418i −0.912888 0.408209i \(-0.866153\pi\)
0.912888 0.408209i \(-0.133847\pi\)
\(734\) 12560.0i 0.631605i
\(735\) − 3924.00i − 0.196924i
\(736\) − 640.000i − 0.0320526i
\(737\) 896.000 0.0447823
\(738\) −1512.00 −0.0754167
\(739\) − 5328.00i − 0.265215i −0.991169 0.132607i \(-0.957665\pi\)
0.991169 0.132607i \(-0.0423349\pi\)
\(740\) 4128.00 0.205065
\(741\) 0 0
\(742\) 2928.00 0.144866
\(743\) 20482.0i 1.01132i 0.862732 + 0.505661i \(0.168751\pi\)
−0.862732 + 0.505661i \(0.831249\pi\)
\(744\) −3648.00 −0.179761
\(745\) −11712.0 −0.575966
\(746\) − 12868.0i − 0.631543i
\(747\) − 1674.00i − 0.0819926i
\(748\) − 48.0000i − 0.00234633i
\(749\) − 992.000i − 0.0483937i
\(750\) −5616.00 −0.273423
\(751\) −8040.00 −0.390657 −0.195329 0.980738i \(-0.562577\pi\)
−0.195329 + 0.980738i \(0.562577\pi\)
\(752\) 4064.00i 0.197073i
\(753\) 6336.00 0.306636
\(754\) 0 0
\(755\) 7776.00 0.374831
\(756\) − 432.000i − 0.0207827i
\(757\) −15822.0 −0.759657 −0.379829 0.925057i \(-0.624017\pi\)
−0.379829 + 0.925057i \(0.624017\pi\)
\(758\) 18136.0 0.869036
\(759\) 120.000i 0.00573877i
\(760\) 1152.00i 0.0549835i
\(761\) − 1452.00i − 0.0691655i −0.999402 0.0345828i \(-0.988990\pi\)
0.999402 0.0345828i \(-0.0110102\pi\)
\(762\) − 2928.00i − 0.139200i
\(763\) −2168.00 −0.102866
\(764\) 13888.0 0.657657
\(765\) − 216.000i − 0.0102085i
\(766\) −6324.00 −0.298297
\(767\) 0 0
\(768\) 768.000 0.0360844
\(769\) − 32298.0i − 1.51456i −0.653091 0.757279i \(-0.726528\pi\)
0.653091 0.757279i \(-0.273472\pi\)
\(770\) 64.0000 0.00299532
\(771\) −8442.00 −0.394334
\(772\) 1240.00i 0.0578090i
\(773\) − 18736.0i − 0.871781i −0.900000 0.435891i \(-0.856433\pi\)
0.900000 0.435891i \(-0.143567\pi\)
\(774\) − 3384.00i − 0.157152i
\(775\) 16568.0i 0.767923i
\(776\) 4912.00 0.227230
\(777\) 3096.00 0.142945
\(778\) − 7332.00i − 0.337873i
\(779\) 3024.00 0.139083
\(780\) 0 0
\(781\) 1852.00 0.0848525
\(782\) − 240.000i − 0.0109749i
\(783\) −378.000 −0.0172524
\(784\) 5232.00 0.238338
\(785\) − 14360.0i − 0.652905i
\(786\) − 10464.0i − 0.474858i
\(787\) − 40816.0i − 1.84871i −0.381536 0.924354i \(-0.624605\pi\)
0.381536 0.924354i \(-0.375395\pi\)
\(788\) 4080.00i 0.184447i
\(789\) −12132.0 −0.547415
\(790\) −10624.0 −0.478462
\(791\) − 8168.00i − 0.367156i
\(792\) −144.000 −0.00646063
\(793\) 0 0
\(794\) 22108.0 0.988141
\(795\) − 4392.00i − 0.195935i
\(796\) −13024.0 −0.579929
\(797\) −4518.00 −0.200798 −0.100399 0.994947i \(-0.532012\pi\)
−0.100399 + 0.994947i \(0.532012\pi\)
\(798\) 864.000i 0.0383274i
\(799\) 1524.00i 0.0674784i
\(800\) − 3488.00i − 0.154149i
\(801\) − 3024.00i − 0.133393i
\(802\) −10656.0 −0.469173
\(803\) −508.000 −0.0223249
\(804\) 5376.00i 0.235817i
\(805\) 320.000 0.0140106
\(806\) 0 0
\(807\) −4410.00 −0.192366
\(808\) 12848.0i 0.559395i
\(809\) −5058.00 −0.219814 −0.109907 0.993942i \(-0.535055\pi\)
−0.109907 + 0.993942i \(0.535055\pi\)
\(810\) −648.000 −0.0281091
\(811\) 22564.0i 0.976978i 0.872570 + 0.488489i \(0.162452\pi\)
−0.872570 + 0.488489i \(0.837548\pi\)
\(812\) 224.000i 0.00968086i
\(813\) − 5532.00i − 0.238642i
\(814\) − 1032.00i − 0.0444368i
\(815\) −9136.00 −0.392663
\(816\) 288.000 0.0123554
\(817\) 6768.00i 0.289819i
\(818\) 24148.0 1.03217
\(819\) 0 0
\(820\) 1344.00 0.0572372
\(821\) − 32584.0i − 1.38513i −0.721357 0.692564i \(-0.756481\pi\)
0.721357 0.692564i \(-0.243519\pi\)
\(822\) −4968.00 −0.210802
\(823\) 9288.00 0.393389 0.196695 0.980465i \(-0.436979\pi\)
0.196695 + 0.980465i \(0.436979\pi\)
\(824\) − 1664.00i − 0.0703497i
\(825\) 654.000i 0.0275992i
\(826\) 4400.00i 0.185346i
\(827\) 20586.0i 0.865593i 0.901492 + 0.432796i \(0.142473\pi\)
−0.901492 + 0.432796i \(0.857527\pi\)
\(828\) −720.000 −0.0302195
\(829\) 46118.0 1.93214 0.966070 0.258280i \(-0.0831556\pi\)
0.966070 + 0.258280i \(0.0831556\pi\)
\(830\) 1488.00i 0.0622280i
\(831\) −17298.0 −0.722095
\(832\) 0 0
\(833\) 1962.00 0.0816078
\(834\) 2424.00i 0.100643i
\(835\) 12696.0 0.526183
\(836\) 288.000 0.0119147
\(837\) 4104.00i 0.169480i
\(838\) − 27168.0i − 1.11993i
\(839\) − 39230.0i − 1.61427i −0.590369 0.807133i \(-0.701018\pi\)
0.590369 0.807133i \(-0.298982\pi\)
\(840\) 384.000i 0.0157729i
\(841\) −24193.0 −0.991964
\(842\) 14812.0 0.606241
\(843\) − 22404.0i − 0.915344i
\(844\) 18256.0 0.744547
\(845\) 0 0
\(846\) 4572.00 0.185802
\(847\) 5308.00i 0.215331i
\(848\) 5856.00 0.237141
\(849\) −3684.00 −0.148922
\(850\) − 1308.00i − 0.0527812i
\(851\) − 5160.00i − 0.207853i
\(852\) 11112.0i 0.446820i
\(853\) − 18674.0i − 0.749573i −0.927111 0.374786i \(-0.877716\pi\)
0.927111 0.374786i \(-0.122284\pi\)
\(854\) −112.000 −0.00448778
\(855\) 1296.00 0.0518389
\(856\) − 1984.00i − 0.0792193i
\(857\) −41678.0 −1.66125 −0.830626 0.556830i \(-0.812017\pi\)
−0.830626 + 0.556830i \(0.812017\pi\)
\(858\) 0 0
\(859\) −14740.0 −0.585474 −0.292737 0.956193i \(-0.594566\pi\)
−0.292737 + 0.956193i \(0.594566\pi\)
\(860\) 3008.00i 0.119270i
\(861\) 1008.00 0.0398984
\(862\) 20268.0 0.800848
\(863\) 24982.0i 0.985396i 0.870200 + 0.492698i \(0.163989\pi\)
−0.870200 + 0.492698i \(0.836011\pi\)
\(864\) − 864.000i − 0.0340207i
\(865\) − 5432.00i − 0.213519i
\(866\) 18812.0i 0.738173i
\(867\) −14631.0 −0.573120
\(868\) 2432.00 0.0951008
\(869\) 2656.00i 0.103681i
\(870\) 336.000 0.0130936
\(871\) 0 0
\(872\) −4336.00 −0.168389
\(873\) − 5526.00i − 0.214235i
\(874\) 1440.00 0.0557308
\(875\) 3744.00 0.144652
\(876\) − 3048.00i − 0.117560i
\(877\) − 1134.00i − 0.0436630i −0.999762 0.0218315i \(-0.993050\pi\)
0.999762 0.0218315i \(-0.00694974\pi\)
\(878\) 8176.00i 0.314267i
\(879\) 19824.0i 0.760690i
\(880\) 128.000 0.00490327
\(881\) −34950.0 −1.33654 −0.668272 0.743917i \(-0.732966\pi\)
−0.668272 + 0.743917i \(0.732966\pi\)
\(882\) − 5886.00i − 0.224707i
\(883\) 3068.00 0.116927 0.0584634 0.998290i \(-0.481380\pi\)
0.0584634 + 0.998290i \(0.481380\pi\)
\(884\) 0 0
\(885\) 6600.00 0.250685
\(886\) 10656.0i 0.404058i
\(887\) −14080.0 −0.532988 −0.266494 0.963837i \(-0.585865\pi\)
−0.266494 + 0.963837i \(0.585865\pi\)
\(888\) 6192.00 0.233998
\(889\) 1952.00i 0.0736423i
\(890\) 2688.00i 0.101238i
\(891\) 162.000i 0.00609114i
\(892\) − 288.000i − 0.0108105i
\(893\) −9144.00 −0.342657
\(894\) −17568.0 −0.657228
\(895\) 2832.00i 0.105769i
\(896\) −512.000 −0.0190901
\(897\) 0 0
\(898\) 26320.0 0.978073
\(899\) − 2128.00i − 0.0789464i
\(900\) −3924.00 −0.145333
\(901\) 2196.00 0.0811980
\(902\) − 336.000i − 0.0124031i
\(903\) 2256.00i 0.0831395i
\(904\) − 16336.0i − 0.601026i
\(905\) − 2184.00i − 0.0802195i
\(906\) 11664.0 0.427716
\(907\) 24876.0 0.910688 0.455344 0.890316i \(-0.349516\pi\)
0.455344 + 0.890316i \(0.349516\pi\)
\(908\) 10776.0i 0.393848i
\(909\) 14454.0 0.527403
\(910\) 0 0
\(911\) 51456.0 1.87136 0.935682 0.352843i \(-0.114785\pi\)
0.935682 + 0.352843i \(0.114785\pi\)
\(912\) 1728.00i 0.0627410i
\(913\) 372.000 0.0134846
\(914\) 18292.0 0.661975
\(915\) 168.000i 0.00606985i
\(916\) − 23688.0i − 0.854447i
\(917\) 6976.00i 0.251219i
\(918\) − 324.000i − 0.0116488i
\(919\) −31032.0 −1.11388 −0.556938 0.830554i \(-0.688024\pi\)
−0.556938 + 0.830554i \(0.688024\pi\)
\(920\) 640.000 0.0229350
\(921\) 22992.0i 0.822597i
\(922\) −11160.0 −0.398628
\(923\) 0 0
\(924\) 96.0000 0.00341793
\(925\) − 28122.0i − 0.999617i
\(926\) 29576.0 1.04960
\(927\) −1872.00 −0.0663264
\(928\) 448.000i 0.0158473i
\(929\) − 50820.0i − 1.79478i −0.441239 0.897390i \(-0.645461\pi\)
0.441239 0.897390i \(-0.354539\pi\)
\(930\) − 3648.00i − 0.128626i
\(931\) 11772.0i 0.414406i
\(932\) −20488.0 −0.720072
\(933\) 7020.00 0.246328
\(934\) 24752.0i 0.867142i
\(935\) 48.0000 0.00167890
\(936\) 0 0
\(937\) 5982.00 0.208563 0.104281 0.994548i \(-0.466746\pi\)
0.104281 + 0.994548i \(0.466746\pi\)
\(938\) − 3584.00i − 0.124757i
\(939\) 20130.0 0.699593
\(940\) −4064.00 −0.141014
\(941\) 20224.0i 0.700620i 0.936634 + 0.350310i \(0.113924\pi\)
−0.936634 + 0.350310i \(0.886076\pi\)
\(942\) − 21540.0i − 0.745022i
\(943\) − 1680.00i − 0.0580152i
\(944\) 8800.00i 0.303406i
\(945\) 432.000 0.0148709
\(946\) 752.000 0.0258453
\(947\) 8478.00i 0.290917i 0.989364 + 0.145458i \(0.0464657\pi\)
−0.989364 + 0.145458i \(0.953534\pi\)
\(948\) −15936.0 −0.545968
\(949\) 0 0
\(950\) 7848.00 0.268024
\(951\) − 12492.0i − 0.425953i
\(952\) −192.000 −0.00653651
\(953\) −40918.0 −1.39083 −0.695417 0.718607i \(-0.744780\pi\)
−0.695417 + 0.718607i \(0.744780\pi\)
\(954\) − 6588.00i − 0.223579i
\(955\) 13888.0i 0.470581i
\(956\) 20088.0i 0.679595i
\(957\) − 84.0000i − 0.00283734i
\(958\) 1668.00 0.0562533
\(959\) 3312.00 0.111522
\(960\) 768.000i 0.0258199i
\(961\) 6687.00 0.224464
\(962\) 0 0
\(963\) −2232.00 −0.0746887
\(964\) 4872.00i 0.162777i
\(965\) −1240.00 −0.0413648
\(966\) 480.000 0.0159873
\(967\) 4624.00i 0.153772i 0.997040 + 0.0768862i \(0.0244978\pi\)
−0.997040 + 0.0768862i \(0.975502\pi\)
\(968\) 10616.0i 0.352491i
\(969\) 648.000i 0.0214827i
\(970\) 4912.00i 0.162593i
\(971\) 15300.0 0.505665 0.252832 0.967510i \(-0.418638\pi\)
0.252832 + 0.967510i \(0.418638\pi\)
\(972\) −972.000 −0.0320750
\(973\) − 1616.00i − 0.0532442i
\(974\) 26384.0 0.867965
\(975\) 0 0
\(976\) −224.000 −0.00734638
\(977\) − 19584.0i − 0.641298i −0.947198 0.320649i \(-0.896099\pi\)
0.947198 0.320649i \(-0.103901\pi\)
\(978\) −13704.0 −0.448063
\(979\) 672.000 0.0219379
\(980\) 5232.00i 0.170541i
\(981\) 4878.00i 0.158759i
\(982\) 33136.0i 1.07679i
\(983\) − 17582.0i − 0.570477i −0.958457 0.285238i \(-0.907927\pi\)
0.958457 0.285238i \(-0.0920728\pi\)
\(984\) 2016.00 0.0653127
\(985\) −4080.00 −0.131979
\(986\) 168.000i 0.00542618i
\(987\) −3048.00 −0.0982968
\(988\) 0 0
\(989\) 3760.00 0.120891
\(990\) − 144.000i − 0.00462285i
\(991\) 47904.0 1.53554 0.767770 0.640725i \(-0.221366\pi\)
0.767770 + 0.640725i \(0.221366\pi\)
\(992\) 4864.00 0.155678
\(993\) 30216.0i 0.965635i
\(994\) − 7408.00i − 0.236386i
\(995\) − 13024.0i − 0.414963i
\(996\) 2232.00i 0.0710077i
\(997\) −44578.0 −1.41605 −0.708024 0.706189i \(-0.750413\pi\)
−0.708024 + 0.706189i \(0.750413\pi\)
\(998\) 20272.0 0.642985
\(999\) − 6966.00i − 0.220615i
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 1014.4.b.g.337.1 2
13.5 odd 4 1014.4.a.e.1.1 1
13.8 odd 4 78.4.a.f.1.1 1
13.12 even 2 inner 1014.4.b.g.337.2 2
39.8 even 4 234.4.a.c.1.1 1
52.47 even 4 624.4.a.c.1.1 1
65.34 odd 4 1950.4.a.a.1.1 1
104.21 odd 4 2496.4.a.c.1.1 1
104.99 even 4 2496.4.a.l.1.1 1
156.47 odd 4 1872.4.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.4.a.f.1.1 1 13.8 odd 4
234.4.a.c.1.1 1 39.8 even 4
624.4.a.c.1.1 1 52.47 even 4
1014.4.a.e.1.1 1 13.5 odd 4
1014.4.b.g.337.1 2 1.1 even 1 trivial
1014.4.b.g.337.2 2 13.12 even 2 inner
1872.4.a.f.1.1 1 156.47 odd 4
1950.4.a.a.1.1 1 65.34 odd 4
2496.4.a.c.1.1 1 104.21 odd 4
2496.4.a.l.1.1 1 104.99 even 4