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Properties

Label 2310.2.a.c.1.1
Level $2310$
Weight $2$
Character 2310.1
Self dual yes
Analytic conductor $18.445$
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] = [2310,2,Mod(1,2310)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2310, 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("2310.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2310.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: \(18.4454428669\)
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 \(\chi\) \(=\) 2310.1

$q$-expansion

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

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 1.00000 0.301511
\(12\) −1.00000 −0.288675
\(13\) −6.00000 −1.66410 −0.832050 − 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) −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
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −1.00000 −0.223607
\(21\) −1.00000 −0.218218
\(22\) −1.00000 −0.213201
\(23\) −4.00000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 6.00000 1.17670
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) −1.00000 −0.182574
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.00000 −0.174078
\(34\) −6.00000 −1.02899
\(35\) −1.00000 −0.169031
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 6.00000 0.960769
\(40\) 1.00000 0.158114
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 1.00000 0.154303
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 1.00000 0.150756
\(45\) −1.00000 −0.149071
\(46\) 4.00000 0.589768
\(47\) 12.0000 1.75038 0.875190 − 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −6.00000 −0.840168
\(52\) −6.00000 −0.832050
\(53\) −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
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 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
\(64\) 1.00000 0.125000
\(65\) 6.00000 0.744208
\(66\) 1.00000 0.123091
\(67\) −12.0000 −1.46603 −0.733017 − 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 6.00000 0.727607
\(69\) 4.00000 0.481543
\(70\) 1.00000 0.119523
\(71\) −4.00000 −0.474713 −0.237356 − 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) −1.00000 −0.117851
\(73\) −6.00000 −0.702247 −0.351123 − 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 2.00000 0.232495
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) −6.00000 −0.679366
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) 12.0000 1.31717 0.658586 − 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) −1.00000 −0.109109
\(85\) −6.00000 −0.650791
\(86\) 8.00000 0.862662
\(87\) −2.00000 −0.214423
\(88\) −1.00000 −0.106600
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 1.00000 0.105409
\(91\) −6.00000 −0.628971
\(92\) −4.00000 −0.417029
\(93\) 0 0
\(94\) −12.0000 −1.23771
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −1.00000 −0.101015
\(99\) 1.00000 0.100504
\(100\) 1.00000 0.100000
\(101\) 10.0000 0.995037 0.497519 − 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 6.00000 0.594089
\(103\) −8.00000 −0.788263 −0.394132 − 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 6.00000 0.588348
\(105\) 1.00000 0.0975900
\(106\) 6.00000 0.582772
\(107\) −12.0000 −1.16008 −0.580042 − 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −14.0000 −1.34096 −0.670478 − 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 1.00000 0.0953463
\(111\) 2.00000 0.189832
\(112\) 1.00000 0.0944911
\(113\) 2.00000 0.188144 0.0940721 − 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) 2.00000 0.185695
\(117\) −6.00000 −0.554700
\(118\) −4.00000 −0.368230
\(119\) 6.00000 0.550019
\(120\) −1.00000 −0.0912871
\(121\) 1.00000 0.0909091
\(122\) −2.00000 −0.181071
\(123\) 2.00000 0.180334
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) −1.00000 −0.0890871
\(127\) −8.00000 −0.709885 −0.354943 − 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 8.00000 0.704361
\(130\) −6.00000 −0.526235
\(131\) 4.00000 0.349482 0.174741 − 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 0 0
\(134\) 12.0000 1.03664
\(135\) 1.00000 0.0860663
\(136\) −6.00000 −0.514496
\(137\) −14.0000 −1.19610 −0.598050 − 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) −4.00000 −0.340503
\(139\) −8.00000 −0.678551 −0.339276 − 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −12.0000 −1.01058
\(142\) 4.00000 0.335673
\(143\) −6.00000 −0.501745
\(144\) 1.00000 0.0833333
\(145\) −2.00000 −0.166091
\(146\) 6.00000 0.496564
\(147\) −1.00000 −0.0824786
\(148\) −2.00000 −0.164399
\(149\) −6.00000 −0.491539 −0.245770 − 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 1.00000 0.0816497
\(151\) −16.0000 −1.30206 −0.651031 − 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) −1.00000 −0.0805823
\(155\) 0 0
\(156\) 6.00000 0.480384
\(157\) −10.0000 −0.798087 −0.399043 − 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) −8.00000 −0.636446
\(159\) 6.00000 0.475831
\(160\) 1.00000 0.0790569
\(161\) −4.00000 −0.315244
\(162\) −1.00000 −0.0785674
\(163\) −12.0000 −0.939913 −0.469956 − 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) −2.00000 −0.156174
\(165\) 1.00000 0.0778499
\(166\) −12.0000 −0.931381
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 1.00000 0.0771517
\(169\) 23.0000 1.76923
\(170\) 6.00000 0.460179
\(171\) 0 0
\(172\) −8.00000 −0.609994
\(173\) −14.0000 −1.06440 −0.532200 − 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 2.00000 0.151620
\(175\) 1.00000 0.0755929
\(176\) 1.00000 0.0753778
\(177\) −4.00000 −0.300658
\(178\) 6.00000 0.449719
\(179\) −12.0000 −0.896922 −0.448461 − 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −26.0000 −1.93256 −0.966282 − 0.257485i \(-0.917106\pi\)
−0.966282 + 0.257485i \(0.917106\pi\)
\(182\) 6.00000 0.444750
\(183\) −2.00000 −0.147844
\(184\) 4.00000 0.294884
\(185\) 2.00000 0.147043
\(186\) 0 0
\(187\) 6.00000 0.438763
\(188\) 12.0000 0.875190
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −4.00000 −0.289430 −0.144715 − 0.989473i \(-0.546227\pi\)
−0.144715 + 0.989473i \(0.546227\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −6.00000 −0.431889 −0.215945 − 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) −2.00000 −0.143592
\(195\) −6.00000 −0.429669
\(196\) 1.00000 0.0714286
\(197\) −6.00000 −0.427482 −0.213741 − 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) −1.00000 −0.0710669
\(199\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 12.0000 0.846415
\(202\) −10.0000 −0.703598
\(203\) 2.00000 0.140372
\(204\) −6.00000 −0.420084
\(205\) 2.00000 0.139686
\(206\) 8.00000 0.557386
\(207\) −4.00000 −0.278019
\(208\) −6.00000 −0.416025
\(209\) 0 0
\(210\) −1.00000 −0.0690066
\(211\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) −6.00000 −0.412082
\(213\) 4.00000 0.274075
\(214\) 12.0000 0.820303
\(215\) 8.00000 0.545595
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 14.0000 0.948200
\(219\) 6.00000 0.405442
\(220\) −1.00000 −0.0674200
\(221\) −36.0000 −2.42162
\(222\) −2.00000 −0.134231
\(223\) −8.00000 −0.535720 −0.267860 − 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 1.00000 0.0666667
\(226\) −2.00000 −0.133038
\(227\) −12.0000 −0.796468 −0.398234 − 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 0 0
\(229\) 22.0000 1.45380 0.726900 − 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) −4.00000 −0.263752
\(231\) −1.00000 −0.0657952
\(232\) −2.00000 −0.131306
\(233\) 14.0000 0.917170 0.458585 − 0.888650i \(-0.348356\pi\)
0.458585 + 0.888650i \(0.348356\pi\)
\(234\) 6.00000 0.392232
\(235\) −12.0000 −0.782794
\(236\) 4.00000 0.260378
\(237\) −8.00000 −0.519656
\(238\) −6.00000 −0.388922
\(239\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 1.00000 0.0645497
\(241\) 10.0000 0.644157 0.322078 − 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) −1.00000 −0.0642824
\(243\) −1.00000 −0.0641500
\(244\) 2.00000 0.128037
\(245\) −1.00000 −0.0638877
\(246\) −2.00000 −0.127515
\(247\) 0 0
\(248\) 0 0
\(249\) −12.0000 −0.760469
\(250\) 1.00000 0.0632456
\(251\) 12.0000 0.757433 0.378717 − 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 1.00000 0.0629941
\(253\) −4.00000 −0.251478
\(254\) 8.00000 0.501965
\(255\) 6.00000 0.375735
\(256\) 1.00000 0.0625000
\(257\) −22.0000 −1.37232 −0.686161 − 0.727450i \(-0.740706\pi\)
−0.686161 + 0.727450i \(0.740706\pi\)
\(258\) −8.00000 −0.498058
\(259\) −2.00000 −0.124274
\(260\) 6.00000 0.372104
\(261\) 2.00000 0.123797
\(262\) −4.00000 −0.247121
\(263\) −8.00000 −0.493301 −0.246651 − 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) 1.00000 0.0615457
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) −12.0000 −0.733017
\(269\) −22.0000 −1.34136 −0.670682 − 0.741745i \(-0.733998\pi\)
−0.670682 + 0.741745i \(0.733998\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 8.00000 0.485965 0.242983 − 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 6.00000 0.363803
\(273\) 6.00000 0.363137
\(274\) 14.0000 0.845771
\(275\) 1.00000 0.0603023
\(276\) 4.00000 0.240772
\(277\) −22.0000 −1.32185 −0.660926 − 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) 8.00000 0.479808
\(279\) 0 0
\(280\) 1.00000 0.0597614
\(281\) −10.0000 −0.596550 −0.298275 − 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 12.0000 0.714590
\(283\) 16.0000 0.951101 0.475551 − 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) −4.00000 −0.237356
\(285\) 0 0
\(286\) 6.00000 0.354787
\(287\) −2.00000 −0.118056
\(288\) −1.00000 −0.0589256
\(289\) 19.0000 1.11765
\(290\) 2.00000 0.117444
\(291\) −2.00000 −0.117242
\(292\) −6.00000 −0.351123
\(293\) 10.0000 0.584206 0.292103 − 0.956387i \(-0.405645\pi\)
0.292103 + 0.956387i \(0.405645\pi\)
\(294\) 1.00000 0.0583212
\(295\) −4.00000 −0.232889
\(296\) 2.00000 0.116248
\(297\) −1.00000 −0.0580259
\(298\) 6.00000 0.347571
\(299\) 24.0000 1.38796
\(300\) −1.00000 −0.0577350
\(301\) −8.00000 −0.461112
\(302\) 16.0000 0.920697
\(303\) −10.0000 −0.574485
\(304\) 0 0
\(305\) −2.00000 −0.114520
\(306\) −6.00000 −0.342997
\(307\) 8.00000 0.456584 0.228292 − 0.973593i \(-0.426686\pi\)
0.228292 + 0.973593i \(0.426686\pi\)
\(308\) 1.00000 0.0569803
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −20.0000 −1.13410 −0.567048 − 0.823685i \(-0.691915\pi\)
−0.567048 + 0.823685i \(0.691915\pi\)
\(312\) −6.00000 −0.339683
\(313\) −6.00000 −0.339140 −0.169570 − 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 10.0000 0.564333
\(315\) −1.00000 −0.0563436
\(316\) 8.00000 0.450035
\(317\) 2.00000 0.112331 0.0561656 − 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) −6.00000 −0.336463
\(319\) 2.00000 0.111979
\(320\) −1.00000 −0.0559017
\(321\) 12.0000 0.669775
\(322\) 4.00000 0.222911
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) −6.00000 −0.332820
\(326\) 12.0000 0.664619
\(327\) 14.0000 0.774202
\(328\) 2.00000 0.110432
\(329\) 12.0000 0.661581
\(330\) −1.00000 −0.0550482
\(331\) 20.0000 1.09930 0.549650 − 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 12.0000 0.658586
\(333\) −2.00000 −0.109599
\(334\) 0 0
\(335\) 12.0000 0.655630
\(336\) −1.00000 −0.0545545
\(337\) 26.0000 1.41631 0.708155 − 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) −23.0000 −1.25104
\(339\) −2.00000 −0.108625
\(340\) −6.00000 −0.325396
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 8.00000 0.431331
\(345\) −4.00000 −0.215353
\(346\) 14.0000 0.752645
\(347\) 28.0000 1.50312 0.751559 − 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) −2.00000 −0.107211
\(349\) −22.0000 −1.17763 −0.588817 − 0.808267i \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 6.00000 0.320256
\(352\) −1.00000 −0.0533002
\(353\) −22.0000 −1.17094 −0.585471 − 0.810693i \(-0.699090\pi\)
−0.585471 + 0.810693i \(0.699090\pi\)
\(354\) 4.00000 0.212598
\(355\) 4.00000 0.212298
\(356\) −6.00000 −0.317999
\(357\) −6.00000 −0.317554
\(358\) 12.0000 0.634220
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 1.00000 0.0527046
\(361\) −19.0000 −1.00000
\(362\) 26.0000 1.36653
\(363\) −1.00000 −0.0524864
\(364\) −6.00000 −0.314485
\(365\) 6.00000 0.314054
\(366\) 2.00000 0.104542
\(367\) −8.00000 −0.417597 −0.208798 − 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) −4.00000 −0.208514
\(369\) −2.00000 −0.104116
\(370\) −2.00000 −0.103975
\(371\) −6.00000 −0.311504
\(372\) 0 0
\(373\) −22.0000 −1.13912 −0.569558 − 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) −6.00000 −0.310253
\(375\) 1.00000 0.0516398
\(376\) −12.0000 −0.618853
\(377\) −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 \(0.328287\pi\)
\(380\) 0 0
\(381\) 8.00000 0.409852
\(382\) 4.00000 0.204658
\(383\) −4.00000 −0.204390 −0.102195 − 0.994764i \(-0.532587\pi\)
−0.102195 + 0.994764i \(0.532587\pi\)
\(384\) 1.00000 0.0510310
\(385\) −1.00000 −0.0509647
\(386\) 6.00000 0.305392
\(387\) −8.00000 −0.406663
\(388\) 2.00000 0.101535
\(389\) −30.0000 −1.52106 −0.760530 − 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 6.00000 0.303822
\(391\) −24.0000 −1.21373
\(392\) −1.00000 −0.0505076
\(393\) −4.00000 −0.201773
\(394\) 6.00000 0.302276
\(395\) −8.00000 −0.402524
\(396\) 1.00000 0.0502519
\(397\) −10.0000 −0.501886 −0.250943 − 0.968002i \(-0.580741\pi\)
−0.250943 + 0.968002i \(0.580741\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 2.00000 0.0998752 0.0499376 − 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) −12.0000 −0.598506
\(403\) 0 0
\(404\) 10.0000 0.497519
\(405\) −1.00000 −0.0496904
\(406\) −2.00000 −0.0992583
\(407\) −2.00000 −0.0991363
\(408\) 6.00000 0.297044
\(409\) −30.0000 −1.48340 −0.741702 − 0.670729i \(-0.765981\pi\)
−0.741702 + 0.670729i \(0.765981\pi\)
\(410\) −2.00000 −0.0987730
\(411\) 14.0000 0.690569
\(412\) −8.00000 −0.394132
\(413\) 4.00000 0.196827
\(414\) 4.00000 0.196589
\(415\) −12.0000 −0.589057
\(416\) 6.00000 0.294174
\(417\) 8.00000 0.391762
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 − 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 1.00000 0.0487950
\(421\) −26.0000 −1.26716 −0.633581 − 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) 0 0
\(423\) 12.0000 0.583460
\(424\) 6.00000 0.291386
\(425\) 6.00000 0.291043
\(426\) −4.00000 −0.193801
\(427\) 2.00000 0.0967868
\(428\) −12.0000 −0.580042
\(429\) 6.00000 0.289683
\(430\) −8.00000 −0.385794
\(431\) 24.0000 1.15604 0.578020 − 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −14.0000 −0.672797 −0.336399 − 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) 2.00000 0.0958927
\(436\) −14.0000 −0.670478
\(437\) 0 0
\(438\) −6.00000 −0.286691
\(439\) 40.0000 1.90910 0.954548 − 0.298057i \(-0.0963387\pi\)
0.954548 + 0.298057i \(0.0963387\pi\)
\(440\) 1.00000 0.0476731
\(441\) 1.00000 0.0476190
\(442\) 36.0000 1.71235
\(443\) 20.0000 0.950229 0.475114 − 0.879924i \(-0.342407\pi\)
0.475114 + 0.879924i \(0.342407\pi\)
\(444\) 2.00000 0.0949158
\(445\) 6.00000 0.284427
\(446\) 8.00000 0.378811
\(447\) 6.00000 0.283790
\(448\) 1.00000 0.0472456
\(449\) 26.0000 1.22702 0.613508 − 0.789689i \(-0.289758\pi\)
0.613508 + 0.789689i \(0.289758\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −2.00000 −0.0941763
\(452\) 2.00000 0.0940721
\(453\) 16.0000 0.751746
\(454\) 12.0000 0.563188
\(455\) 6.00000 0.281284
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 − 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) −22.0000 −1.02799
\(459\) −6.00000 −0.280056
\(460\) 4.00000 0.186501
\(461\) 18.0000 0.838344 0.419172 − 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 1.00000 0.0465242
\(463\) 24.0000 1.11537 0.557687 − 0.830051i \(-0.311689\pi\)
0.557687 + 0.830051i \(0.311689\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) −14.0000 −0.648537
\(467\) 36.0000 1.66588 0.832941 − 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) −6.00000 −0.277350
\(469\) −12.0000 −0.554109
\(470\) 12.0000 0.553519
\(471\) 10.0000 0.460776
\(472\) −4.00000 −0.184115
\(473\) −8.00000 −0.367840
\(474\) 8.00000 0.367452
\(475\) 0 0
\(476\) 6.00000 0.275010
\(477\) −6.00000 −0.274721
\(478\) 0 0
\(479\) 24.0000 1.09659 0.548294 − 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 12.0000 0.547153
\(482\) −10.0000 −0.455488
\(483\) 4.00000 0.182006
\(484\) 1.00000 0.0454545
\(485\) −2.00000 −0.0908153
\(486\) 1.00000 0.0453609
\(487\) 8.00000 0.362515 0.181257 − 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 12.0000 0.542659
\(490\) 1.00000 0.0451754
\(491\) −12.0000 −0.541552 −0.270776 − 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 2.00000 0.0901670
\(493\) 12.0000 0.540453
\(494\) 0 0
\(495\) −1.00000 −0.0449467
\(496\) 0 0
\(497\) −4.00000 −0.179425
\(498\) 12.0000 0.537733
\(499\) 36.0000 1.61158 0.805791 − 0.592200i \(-0.201741\pi\)
0.805791 + 0.592200i \(0.201741\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −12.0000 −0.535586
\(503\) −16.0000 −0.713405 −0.356702 − 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) −1.00000 −0.0445435
\(505\) −10.0000 −0.444994
\(506\) 4.00000 0.177822
\(507\) −23.0000 −1.02147
\(508\) −8.00000 −0.354943
\(509\) 10.0000 0.443242 0.221621 − 0.975133i \(-0.428865\pi\)
0.221621 + 0.975133i \(0.428865\pi\)
\(510\) −6.00000 −0.265684
\(511\) −6.00000 −0.265424
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 22.0000 0.970378
\(515\) 8.00000 0.352522
\(516\) 8.00000 0.352180
\(517\) 12.0000 0.527759
\(518\) 2.00000 0.0878750
\(519\) 14.0000 0.614532
\(520\) −6.00000 −0.263117
\(521\) 42.0000 1.84005 0.920027 − 0.391856i \(-0.128167\pi\)
0.920027 + 0.391856i \(0.128167\pi\)
\(522\) −2.00000 −0.0875376
\(523\) 16.0000 0.699631 0.349816 − 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 4.00000 0.174741
\(525\) −1.00000 −0.0436436
\(526\) 8.00000 0.348817
\(527\) 0 0
\(528\) −1.00000 −0.0435194
\(529\) −7.00000 −0.304348
\(530\) −6.00000 −0.260623
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) 12.0000 0.519778
\(534\) −6.00000 −0.259645
\(535\) 12.0000 0.518805
\(536\) 12.0000 0.518321
\(537\) 12.0000 0.517838
\(538\) 22.0000 0.948487
\(539\) 1.00000 0.0430730
\(540\) 1.00000 0.0430331
\(541\) −14.0000 −0.601907 −0.300954 − 0.953639i \(-0.597305\pi\)
−0.300954 + 0.953639i \(0.597305\pi\)
\(542\) −8.00000 −0.343629
\(543\) 26.0000 1.11577
\(544\) −6.00000 −0.257248
\(545\) 14.0000 0.599694
\(546\) −6.00000 −0.256776
\(547\) −8.00000 −0.342055 −0.171028 − 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) −14.0000 −0.598050
\(549\) 2.00000 0.0853579
\(550\) −1.00000 −0.0426401
\(551\) 0 0
\(552\) −4.00000 −0.170251
\(553\) 8.00000 0.340195
\(554\) 22.0000 0.934690
\(555\) −2.00000 −0.0848953
\(556\) −8.00000 −0.339276
\(557\) 18.0000 0.762684 0.381342 − 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 0 0
\(559\) 48.0000 2.03018
\(560\) −1.00000 −0.0422577
\(561\) −6.00000 −0.253320
\(562\) 10.0000 0.421825
\(563\) 44.0000 1.85438 0.927189 − 0.374593i \(-0.122217\pi\)
0.927189 + 0.374593i \(0.122217\pi\)
\(564\) −12.0000 −0.505291
\(565\) −2.00000 −0.0841406
\(566\) −16.0000 −0.672530
\(567\) 1.00000 0.0419961
\(568\) 4.00000 0.167836
\(569\) 46.0000 1.92842 0.964210 − 0.265139i \(-0.0854179\pi\)
0.964210 + 0.265139i \(0.0854179\pi\)
\(570\) 0 0
\(571\) 24.0000 1.00437 0.502184 − 0.864761i \(-0.332530\pi\)
0.502184 + 0.864761i \(0.332530\pi\)
\(572\) −6.00000 −0.250873
\(573\) 4.00000 0.167102
\(574\) 2.00000 0.0834784
\(575\) −4.00000 −0.166812
\(576\) 1.00000 0.0416667
\(577\) 18.0000 0.749350 0.374675 − 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(578\) −19.0000 −0.790296
\(579\) 6.00000 0.249351
\(580\) −2.00000 −0.0830455
\(581\) 12.0000 0.497844
\(582\) 2.00000 0.0829027
\(583\) −6.00000 −0.248495
\(584\) 6.00000 0.248282
\(585\) 6.00000 0.248069
\(586\) −10.0000 −0.413096
\(587\) −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
\(590\) 4.00000 0.164677
\(591\) 6.00000 0.246807
\(592\) −2.00000 −0.0821995
\(593\) 14.0000 0.574911 0.287456 − 0.957794i \(-0.407191\pi\)
0.287456 + 0.957794i \(0.407191\pi\)
\(594\) 1.00000 0.0410305
\(595\) −6.00000 −0.245976
\(596\) −6.00000 −0.245770
\(597\) 0 0
\(598\) −24.0000 −0.981433
\(599\) 36.0000 1.47092 0.735460 − 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) 1.00000 0.0408248
\(601\) −30.0000 −1.22373 −0.611863 − 0.790964i \(-0.709580\pi\)
−0.611863 + 0.790964i \(0.709580\pi\)
\(602\) 8.00000 0.326056
\(603\) −12.0000 −0.488678
\(604\) −16.0000 −0.651031
\(605\) −1.00000 −0.0406558
\(606\) 10.0000 0.406222
\(607\) 40.0000 1.62355 0.811775 − 0.583970i \(-0.198502\pi\)
0.811775 + 0.583970i \(0.198502\pi\)
\(608\) 0 0
\(609\) −2.00000 −0.0810441
\(610\) 2.00000 0.0809776
\(611\) −72.0000 −2.91281
\(612\) 6.00000 0.242536
\(613\) −14.0000 −0.565455 −0.282727 − 0.959200i \(-0.591239\pi\)
−0.282727 + 0.959200i \(0.591239\pi\)
\(614\) −8.00000 −0.322854
\(615\) −2.00000 −0.0806478
\(616\) −1.00000 −0.0402911
\(617\) −22.0000 −0.885687 −0.442843 − 0.896599i \(-0.646030\pi\)
−0.442843 + 0.896599i \(0.646030\pi\)
\(618\) −8.00000 −0.321807
\(619\) 28.0000 1.12542 0.562708 − 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 20.0000 0.801927
\(623\) −6.00000 −0.240385
\(624\) 6.00000 0.240192
\(625\) 1.00000 0.0400000
\(626\) 6.00000 0.239808
\(627\) 0 0
\(628\) −10.0000 −0.399043
\(629\) −12.0000 −0.478471
\(630\) 1.00000 0.0398410
\(631\) −16.0000 −0.636950 −0.318475 − 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) −8.00000 −0.318223
\(633\) 0 0
\(634\) −2.00000 −0.0794301
\(635\) 8.00000 0.317470
\(636\) 6.00000 0.237915
\(637\) −6.00000 −0.237729
\(638\) −2.00000 −0.0791808
\(639\) −4.00000 −0.158238
\(640\) 1.00000 0.0395285
\(641\) −6.00000 −0.236986 −0.118493 − 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) −12.0000 −0.473602
\(643\) −20.0000 −0.788723 −0.394362 − 0.918955i \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(644\) −4.00000 −0.157622
\(645\) −8.00000 −0.315000
\(646\) 0 0
\(647\) −28.0000 −1.10079 −0.550397 − 0.834903i \(-0.685524\pi\)
−0.550397 + 0.834903i \(0.685524\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 4.00000 0.157014
\(650\) 6.00000 0.235339
\(651\) 0 0
\(652\) −12.0000 −0.469956
\(653\) −6.00000 −0.234798 −0.117399 − 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) −14.0000 −0.547443
\(655\) −4.00000 −0.156293
\(656\) −2.00000 −0.0780869
\(657\) −6.00000 −0.234082
\(658\) −12.0000 −0.467809
\(659\) −12.0000 −0.467454 −0.233727 − 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 1.00000 0.0389249
\(661\) −42.0000 −1.63361 −0.816805 − 0.576913i \(-0.804257\pi\)
−0.816805 + 0.576913i \(0.804257\pi\)
\(662\) −20.0000 −0.777322
\(663\) 36.0000 1.39812
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) 2.00000 0.0774984
\(667\) −8.00000 −0.309761
\(668\) 0 0
\(669\) 8.00000 0.309298
\(670\) −12.0000 −0.463600
\(671\) 2.00000 0.0772091
\(672\) 1.00000 0.0385758
\(673\) 26.0000 1.00223 0.501113 − 0.865382i \(-0.332924\pi\)
0.501113 + 0.865382i \(0.332924\pi\)
\(674\) −26.0000 −1.00148
\(675\) −1.00000 −0.0384900
\(676\) 23.0000 0.884615
\(677\) 26.0000 0.999261 0.499631 − 0.866239i \(-0.333469\pi\)
0.499631 + 0.866239i \(0.333469\pi\)
\(678\) 2.00000 0.0768095
\(679\) 2.00000 0.0767530
\(680\) 6.00000 0.230089
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) −52.0000 −1.98972 −0.994862 − 0.101237i \(-0.967720\pi\)
−0.994862 + 0.101237i \(0.967720\pi\)
\(684\) 0 0
\(685\) 14.0000 0.534913
\(686\) −1.00000 −0.0381802
\(687\) −22.0000 −0.839352
\(688\) −8.00000 −0.304997
\(689\) 36.0000 1.37149
\(690\) 4.00000 0.152277
\(691\) 20.0000 0.760836 0.380418 − 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) −14.0000 −0.532200
\(693\) 1.00000 0.0379869
\(694\) −28.0000 −1.06287
\(695\) 8.00000 0.303457
\(696\) 2.00000 0.0758098
\(697\) −12.0000 −0.454532
\(698\) 22.0000 0.832712
\(699\) −14.0000 −0.529529
\(700\) 1.00000 0.0377964
\(701\) 2.00000 0.0755390 0.0377695 − 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) −6.00000 −0.226455
\(703\) 0 0
\(704\) 1.00000 0.0376889
\(705\) 12.0000 0.451946
\(706\) 22.0000 0.827981
\(707\) 10.0000 0.376089
\(708\) −4.00000 −0.150329
\(709\) −18.0000 −0.676004 −0.338002 − 0.941145i \(-0.609751\pi\)
−0.338002 + 0.941145i \(0.609751\pi\)
\(710\) −4.00000 −0.150117
\(711\) 8.00000 0.300023
\(712\) 6.00000 0.224860
\(713\) 0 0
\(714\) 6.00000 0.224544
\(715\) 6.00000 0.224387
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) 0 0
\(719\) −36.0000 −1.34257 −0.671287 − 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −8.00000 −0.297936
\(722\) 19.0000 0.707107
\(723\) −10.0000 −0.371904
\(724\) −26.0000 −0.966282
\(725\) 2.00000 0.0742781
\(726\) 1.00000 0.0371135
\(727\) −16.0000 −0.593407 −0.296704 − 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 6.00000 0.222375
\(729\) 1.00000 0.0370370
\(730\) −6.00000 −0.222070
\(731\) −48.0000 −1.77534
\(732\) −2.00000 −0.0739221
\(733\) 42.0000 1.55131 0.775653 − 0.631160i \(-0.217421\pi\)
0.775653 + 0.631160i \(0.217421\pi\)
\(734\) 8.00000 0.295285
\(735\) 1.00000 0.0368856
\(736\) 4.00000 0.147442
\(737\) −12.0000 −0.442026
\(738\) 2.00000 0.0736210
\(739\) 40.0000 1.47142 0.735712 − 0.677295i \(-0.236848\pi\)
0.735712 + 0.677295i \(0.236848\pi\)
\(740\) 2.00000 0.0735215
\(741\) 0 0
\(742\) 6.00000 0.220267
\(743\) −24.0000 −0.880475 −0.440237 − 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) 22.0000 0.805477
\(747\) 12.0000 0.439057
\(748\) 6.00000 0.219382
\(749\) −12.0000 −0.438470
\(750\) −1.00000 −0.0365148
\(751\) 8.00000 0.291924 0.145962 − 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 12.0000 0.437595
\(753\) −12.0000 −0.437304
\(754\) 12.0000 0.437014
\(755\) 16.0000 0.582300
\(756\) −1.00000 −0.0363696
\(757\) 30.0000 1.09037 0.545184 − 0.838316i \(-0.316460\pi\)
0.545184 + 0.838316i \(0.316460\pi\)
\(758\) −20.0000 −0.726433
\(759\) 4.00000 0.145191
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 − 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) −8.00000 −0.289809
\(763\) −14.0000 −0.506834
\(764\) −4.00000 −0.144715
\(765\) −6.00000 −0.216930
\(766\) 4.00000 0.144526
\(767\) −24.0000 −0.866590
\(768\) −1.00000 −0.0360844
\(769\) 18.0000 0.649097 0.324548 − 0.945869i \(-0.394788\pi\)
0.324548 + 0.945869i \(0.394788\pi\)
\(770\) 1.00000 0.0360375
\(771\) 22.0000 0.792311
\(772\) −6.00000 −0.215945
\(773\) −46.0000 −1.65451 −0.827253 − 0.561830i \(-0.810097\pi\)
−0.827253 + 0.561830i \(0.810097\pi\)
\(774\) 8.00000 0.287554
\(775\) 0 0
\(776\) −2.00000 −0.0717958
\(777\) 2.00000 0.0717496
\(778\) 30.0000 1.07555
\(779\) 0 0
\(780\) −6.00000 −0.214834
\(781\) −4.00000 −0.143131
\(782\) 24.0000 0.858238
\(783\) −2.00000 −0.0714742
\(784\) 1.00000 0.0357143
\(785\) 10.0000 0.356915
\(786\) 4.00000 0.142675
\(787\) 8.00000 0.285169 0.142585 − 0.989783i \(-0.454459\pi\)
0.142585 + 0.989783i \(0.454459\pi\)
\(788\) −6.00000 −0.213741
\(789\) 8.00000 0.284808
\(790\) 8.00000 0.284627
\(791\) 2.00000 0.0711118
\(792\) −1.00000 −0.0355335
\(793\) −12.0000 −0.426132
\(794\) 10.0000 0.354887
\(795\) −6.00000 −0.212798
\(796\) 0 0
\(797\) 18.0000 0.637593 0.318796 − 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 0 0
\(799\) 72.0000 2.54718
\(800\) −1.00000 −0.0353553
\(801\) −6.00000 −0.212000
\(802\) −2.00000 −0.0706225
\(803\) −6.00000 −0.211735
\(804\) 12.0000 0.423207
\(805\) 4.00000 0.140981
\(806\) 0 0
\(807\) 22.0000 0.774437
\(808\) −10.0000 −0.351799
\(809\) −50.0000 −1.75791 −0.878953 − 0.476908i \(-0.841757\pi\)
−0.878953 + 0.476908i \(0.841757\pi\)
\(810\) 1.00000 0.0351364
\(811\) −48.0000 −1.68551 −0.842754 − 0.538299i \(-0.819067\pi\)
−0.842754 + 0.538299i \(0.819067\pi\)
\(812\) 2.00000 0.0701862
\(813\) −8.00000 −0.280572
\(814\) 2.00000 0.0701000
\(815\) 12.0000 0.420342
\(816\) −6.00000 −0.210042
\(817\) 0 0
\(818\) 30.0000 1.04893
\(819\) −6.00000 −0.209657
\(820\) 2.00000 0.0698430
\(821\) −22.0000 −0.767805 −0.383903 − 0.923374i \(-0.625420\pi\)
−0.383903 + 0.923374i \(0.625420\pi\)
\(822\) −14.0000 −0.488306
\(823\) 24.0000 0.836587 0.418294 − 0.908312i \(-0.362628\pi\)
0.418294 + 0.908312i \(0.362628\pi\)
\(824\) 8.00000 0.278693
\(825\) −1.00000 −0.0348155
\(826\) −4.00000 −0.139178
\(827\) 12.0000 0.417281 0.208640 − 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) −4.00000 −0.139010
\(829\) −10.0000 −0.347314 −0.173657 − 0.984806i \(-0.555558\pi\)
−0.173657 + 0.984806i \(0.555558\pi\)
\(830\) 12.0000 0.416526
\(831\) 22.0000 0.763172
\(832\) −6.00000 −0.208013
\(833\) 6.00000 0.207888
\(834\) −8.00000 −0.277017
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 12.0000 0.414533
\(839\) 4.00000 0.138095 0.0690477 − 0.997613i \(-0.478004\pi\)
0.0690477 + 0.997613i \(0.478004\pi\)
\(840\) −1.00000 −0.0345033
\(841\) −25.0000 −0.862069
\(842\) 26.0000 0.896019
\(843\) 10.0000 0.344418
\(844\) 0 0
\(845\) −23.0000 −0.791224
\(846\) −12.0000 −0.412568
\(847\) 1.00000 0.0343604
\(848\) −6.00000 −0.206041
\(849\) −16.0000 −0.549119
\(850\) −6.00000 −0.205798
\(851\) 8.00000 0.274236
\(852\) 4.00000 0.137038
\(853\) 2.00000 0.0684787 0.0342393 − 0.999414i \(-0.489099\pi\)
0.0342393 + 0.999414i \(0.489099\pi\)
\(854\) −2.00000 −0.0684386
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 14.0000 0.478231 0.239115 − 0.970991i \(-0.423143\pi\)
0.239115 + 0.970991i \(0.423143\pi\)
\(858\) −6.00000 −0.204837
\(859\) −44.0000 −1.50126 −0.750630 − 0.660722i \(-0.770250\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) 8.00000 0.272798
\(861\) 2.00000 0.0681598
\(862\) −24.0000 −0.817443
\(863\) 12.0000 0.408485 0.204242 − 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) 1.00000 0.0340207
\(865\) 14.0000 0.476014
\(866\) 14.0000 0.475739
\(867\) −19.0000 −0.645274
\(868\) 0 0
\(869\) 8.00000 0.271381
\(870\) −2.00000 −0.0678064
\(871\) 72.0000 2.43963
\(872\) 14.0000 0.474100
\(873\) 2.00000 0.0676897
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 6.00000 0.202721
\(877\) 18.0000 0.607817 0.303908 − 0.952701i \(-0.401708\pi\)
0.303908 + 0.952701i \(0.401708\pi\)
\(878\) −40.0000 −1.34993
\(879\) −10.0000 −0.337292
\(880\) −1.00000 −0.0337100
\(881\) −54.0000 −1.81931 −0.909653 − 0.415369i \(-0.863653\pi\)
−0.909653 + 0.415369i \(0.863653\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 20.0000 0.673054 0.336527 − 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) −36.0000 −1.21081
\(885\) 4.00000 0.134459
\(886\) −20.0000 −0.671913
\(887\) 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
\(890\) −6.00000 −0.201120
\(891\) 1.00000 0.0335013
\(892\) −8.00000 −0.267860
\(893\) 0 0
\(894\) −6.00000 −0.200670
\(895\) 12.0000 0.401116
\(896\) −1.00000 −0.0334077
\(897\) −24.0000 −0.801337
\(898\) −26.0000 −0.867631
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) −36.0000 −1.19933
\(902\) 2.00000 0.0665927
\(903\) 8.00000 0.266223
\(904\) −2.00000 −0.0665190
\(905\) 26.0000 0.864269
\(906\) −16.0000 −0.531564
\(907\) −44.0000 −1.46100 −0.730498 − 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) −12.0000 −0.398234
\(909\) 10.0000 0.331679
\(910\) −6.00000 −0.198898
\(911\) −28.0000 −0.927681 −0.463841 − 0.885919i \(-0.653529\pi\)
−0.463841 + 0.885919i \(0.653529\pi\)
\(912\) 0 0
\(913\) 12.0000 0.397142
\(914\) 22.0000 0.727695
\(915\) 2.00000 0.0661180
\(916\) 22.0000 0.726900
\(917\) 4.00000 0.132092
\(918\) 6.00000 0.198030
\(919\) −32.0000 −1.05558 −0.527791 − 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) −4.00000 −0.131876
\(921\) −8.00000 −0.263609
\(922\) −18.0000 −0.592798
\(923\) 24.0000 0.789970
\(924\) −1.00000 −0.0328976
\(925\) −2.00000 −0.0657596
\(926\) −24.0000 −0.788689
\(927\) −8.00000 −0.262754
\(928\) −2.00000 −0.0656532
\(929\) −30.0000 −0.984268 −0.492134 − 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 14.0000 0.458585
\(933\) 20.0000 0.654771
\(934\) −36.0000 −1.17796
\(935\) −6.00000 −0.196221
\(936\) 6.00000 0.196116
\(937\) 34.0000 1.11073 0.555366 − 0.831606i \(-0.312578\pi\)
0.555366 + 0.831606i \(0.312578\pi\)
\(938\) 12.0000 0.391814
\(939\) 6.00000 0.195803
\(940\) −12.0000 −0.391397
\(941\) −14.0000 −0.456387 −0.228193 − 0.973616i \(-0.573282\pi\)
−0.228193 + 0.973616i \(0.573282\pi\)
\(942\) −10.0000 −0.325818
\(943\) 8.00000 0.260516
\(944\) 4.00000 0.130189
\(945\) 1.00000 0.0325300
\(946\) 8.00000 0.260102
\(947\) 12.0000 0.389948 0.194974 − 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) −8.00000 −0.259828
\(949\) 36.0000 1.16861
\(950\) 0 0
\(951\) −2.00000 −0.0648544
\(952\) −6.00000 −0.194461
\(953\) 54.0000 1.74923 0.874616 − 0.484817i \(-0.161114\pi\)
0.874616 + 0.484817i \(0.161114\pi\)
\(954\) 6.00000 0.194257
\(955\) 4.00000 0.129437
\(956\) 0 0
\(957\) −2.00000 −0.0646508
\(958\) −24.0000 −0.775405
\(959\) −14.0000 −0.452084
\(960\) 1.00000 0.0322749
\(961\) −31.0000 −1.00000
\(962\) −12.0000 −0.386896
\(963\) −12.0000 −0.386695
\(964\) 10.0000 0.322078
\(965\) 6.00000 0.193147
\(966\) −4.00000 −0.128698
\(967\) 16.0000 0.514525 0.257263 − 0.966342i \(-0.417179\pi\)
0.257263 + 0.966342i \(0.417179\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) 2.00000 0.0642161
\(971\) 20.0000 0.641831 0.320915 − 0.947108i \(-0.396010\pi\)
0.320915 + 0.947108i \(0.396010\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −8.00000 −0.256468
\(974\) −8.00000 −0.256337
\(975\) 6.00000 0.192154
\(976\) 2.00000 0.0640184
\(977\) 10.0000 0.319928 0.159964 − 0.987123i \(-0.448862\pi\)
0.159964 + 0.987123i \(0.448862\pi\)
\(978\) −12.0000 −0.383718
\(979\) −6.00000 −0.191761
\(980\) −1.00000 −0.0319438
\(981\) −14.0000 −0.446986
\(982\) 12.0000 0.382935
\(983\) 36.0000 1.14822 0.574111 − 0.818778i \(-0.305348\pi\)
0.574111 + 0.818778i \(0.305348\pi\)
\(984\) −2.00000 −0.0637577
\(985\) 6.00000 0.191176
\(986\) −12.0000 −0.382158
\(987\) −12.0000 −0.381964
\(988\) 0 0
\(989\) 32.0000 1.01754
\(990\) 1.00000 0.0317821
\(991\) 40.0000 1.27064 0.635321 − 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) 0 0
\(993\) −20.0000 −0.634681
\(994\) 4.00000 0.126872
\(995\) 0 0
\(996\) −12.0000 −0.380235
\(997\) 42.0000 1.33015 0.665077 − 0.746775i \(-0.268399\pi\)
0.665077 + 0.746775i \(0.268399\pi\)
\(998\) −36.0000 −1.13956
\(999\) 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 2310.2.a.c.1.1 ✓ 1
3.2 odd 2 6930.2.a.bf.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2310.2.a.c.1.1 ✓ 1 1.1 even 1 trivial
6930.2.a.bf.1.1 1 3.2 odd 2