Properties

Label 1503.2.c.a.1502.16
Level $1503$
Weight $2$
Character 1503.1502
Analytic conductor $12.002$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1503,2,Mod(1502,1503)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1503, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1503.1502");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1503 = 3^{2} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1503.c (of order \(2\), degree \(1\), 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: \(12.0015154238\)
Analytic rank: \(0\)
Dimension: \(56\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1502.16
Character \(\chi\) \(=\) 1503.1502
Dual form 1503.2.c.a.1502.15

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.16730i q^{2} -2.69717 q^{4} -1.88079 q^{5} -3.73517 q^{7} -1.51098i q^{8} +O(q^{10})\) \(q+2.16730i q^{2} -2.69717 q^{4} -1.88079 q^{5} -3.73517 q^{7} -1.51098i q^{8} -4.07624i q^{10} +1.85446i q^{11} +4.16402i q^{13} -8.09521i q^{14} -2.11961 q^{16} -2.49228 q^{17} -2.11128 q^{19} +5.07283 q^{20} -4.01915 q^{22} +8.93205 q^{23} -1.46261 q^{25} -9.02466 q^{26} +10.0744 q^{28} -6.40845i q^{29} -1.50442 q^{31} -7.61577i q^{32} -5.40150i q^{34} +7.02508 q^{35} -2.34193i q^{37} -4.57577i q^{38} +2.84184i q^{40} +8.37955 q^{41} -9.29965i q^{43} -5.00178i q^{44} +19.3584i q^{46} -2.45945i q^{47} +6.95147 q^{49} -3.16991i q^{50} -11.2311i q^{52} -6.33808 q^{53} -3.48785i q^{55} +5.64376i q^{56} +13.8890 q^{58} +9.45469 q^{59} +0.660137 q^{61} -3.26051i q^{62} +12.2664 q^{64} -7.83167i q^{65} +4.29499i q^{67} +6.72210 q^{68} +15.2254i q^{70} -1.12478 q^{71} +6.87565i q^{73} +5.07567 q^{74} +5.69449 q^{76} -6.92670i q^{77} +2.70341i q^{79} +3.98655 q^{80} +18.1610i q^{82} -7.93347 q^{83} +4.68746 q^{85} +20.1551 q^{86} +2.80204 q^{88} -15.3639i q^{89} -15.5533i q^{91} -24.0913 q^{92} +5.33036 q^{94} +3.97089 q^{95} -12.9800 q^{97} +15.0659i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q - 48 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 56 q - 48 q^{4} + 32 q^{16} - 8 q^{19} + 16 q^{22} + 64 q^{25} - 32 q^{28} + 40 q^{31} + 56 q^{49} - 32 q^{58} + 24 q^{61} - 56 q^{76} + 72 q^{88} - 56 q^{94} - 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1503\mathbb{Z}\right)^\times\).

\(n\) \(172\) \(335\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.16730i 1.53251i 0.642537 + 0.766255i \(0.277882\pi\)
−0.642537 + 0.766255i \(0.722118\pi\)
\(3\) 0 0
\(4\) −2.69717 −1.34859
\(5\) −1.88079 −0.841117 −0.420558 0.907265i \(-0.638166\pi\)
−0.420558 + 0.907265i \(0.638166\pi\)
\(6\) 0 0
\(7\) −3.73517 −1.41176 −0.705880 0.708331i \(-0.749448\pi\)
−0.705880 + 0.708331i \(0.749448\pi\)
\(8\) 1.51098i 0.534211i
\(9\) 0 0
\(10\) 4.07624i 1.28902i
\(11\) 1.85446i 0.559139i 0.960125 + 0.279570i \(0.0901918\pi\)
−0.960125 + 0.279570i \(0.909808\pi\)
\(12\) 0 0
\(13\) 4.16402i 1.15489i 0.816429 + 0.577446i \(0.195950\pi\)
−0.816429 + 0.577446i \(0.804050\pi\)
\(14\) 8.09521i 2.16354i
\(15\) 0 0
\(16\) −2.11961 −0.529902
\(17\) −2.49228 −0.604466 −0.302233 0.953234i \(-0.597732\pi\)
−0.302233 + 0.953234i \(0.597732\pi\)
\(18\) 0 0
\(19\) −2.11128 −0.484361 −0.242181 0.970231i \(-0.577863\pi\)
−0.242181 + 0.970231i \(0.577863\pi\)
\(20\) 5.07283 1.13432
\(21\) 0 0
\(22\) −4.01915 −0.856886
\(23\) 8.93205 1.86246 0.931230 0.364431i \(-0.118736\pi\)
0.931230 + 0.364431i \(0.118736\pi\)
\(24\) 0 0
\(25\) −1.46261 −0.292522
\(26\) −9.02466 −1.76988
\(27\) 0 0
\(28\) 10.0744 1.90388
\(29\) 6.40845i 1.19002i −0.803718 0.595010i \(-0.797148\pi\)
0.803718 0.595010i \(-0.202852\pi\)
\(30\) 0 0
\(31\) −1.50442 −0.270201 −0.135100 0.990832i \(-0.543136\pi\)
−0.135100 + 0.990832i \(0.543136\pi\)
\(32\) 7.61577i 1.34629i
\(33\) 0 0
\(34\) 5.40150i 0.926350i
\(35\) 7.02508 1.18746
\(36\) 0 0
\(37\) 2.34193i 0.385012i −0.981296 0.192506i \(-0.938339\pi\)
0.981296 0.192506i \(-0.0616614\pi\)
\(38\) 4.57577i 0.742289i
\(39\) 0 0
\(40\) 2.84184i 0.449334i
\(41\) 8.37955 1.30867 0.654333 0.756207i \(-0.272950\pi\)
0.654333 + 0.756207i \(0.272950\pi\)
\(42\) 0 0
\(43\) 9.29965i 1.41818i −0.705116 0.709092i \(-0.749105\pi\)
0.705116 0.709092i \(-0.250895\pi\)
\(44\) 5.00178i 0.754047i
\(45\) 0 0
\(46\) 19.3584i 2.85424i
\(47\) 2.45945i 0.358748i −0.983781 0.179374i \(-0.942593\pi\)
0.983781 0.179374i \(-0.0574071\pi\)
\(48\) 0 0
\(49\) 6.95147 0.993068
\(50\) 3.16991i 0.448293i
\(51\) 0 0
\(52\) 11.2311i 1.55747i
\(53\) −6.33808 −0.870602 −0.435301 0.900285i \(-0.643358\pi\)
−0.435301 + 0.900285i \(0.643358\pi\)
\(54\) 0 0
\(55\) 3.48785i 0.470301i
\(56\) 5.64376i 0.754179i
\(57\) 0 0
\(58\) 13.8890 1.82372
\(59\) 9.45469 1.23090 0.615448 0.788178i \(-0.288975\pi\)
0.615448 + 0.788178i \(0.288975\pi\)
\(60\) 0 0
\(61\) 0.660137 0.0845219 0.0422610 0.999107i \(-0.486544\pi\)
0.0422610 + 0.999107i \(0.486544\pi\)
\(62\) 3.26051i 0.414086i
\(63\) 0 0
\(64\) 12.2664 1.53330
\(65\) 7.83167i 0.971399i
\(66\) 0 0
\(67\) 4.29499i 0.524717i 0.964970 + 0.262359i \(0.0845003\pi\)
−0.964970 + 0.262359i \(0.915500\pi\)
\(68\) 6.72210 0.815174
\(69\) 0 0
\(70\) 15.2254i 1.81979i
\(71\) −1.12478 −0.133487 −0.0667436 0.997770i \(-0.521261\pi\)
−0.0667436 + 0.997770i \(0.521261\pi\)
\(72\) 0 0
\(73\) 6.87565i 0.804734i 0.915478 + 0.402367i \(0.131812\pi\)
−0.915478 + 0.402367i \(0.868188\pi\)
\(74\) 5.07567 0.590034
\(75\) 0 0
\(76\) 5.69449 0.653203
\(77\) 6.92670i 0.789371i
\(78\) 0 0
\(79\) 2.70341i 0.304158i 0.988368 + 0.152079i \(0.0485968\pi\)
−0.988368 + 0.152079i \(0.951403\pi\)
\(80\) 3.98655 0.445709
\(81\) 0 0
\(82\) 18.1610i 2.00554i
\(83\) −7.93347 −0.870811 −0.435406 0.900234i \(-0.643395\pi\)
−0.435406 + 0.900234i \(0.643395\pi\)
\(84\) 0 0
\(85\) 4.68746 0.508427
\(86\) 20.1551 2.17338
\(87\) 0 0
\(88\) 2.80204 0.298699
\(89\) 15.3639i 1.62857i −0.580462 0.814287i \(-0.697128\pi\)
0.580462 0.814287i \(-0.302872\pi\)
\(90\) 0 0
\(91\) 15.5533i 1.63043i
\(92\) −24.0913 −2.51169
\(93\) 0 0
\(94\) 5.33036 0.549784
\(95\) 3.97089 0.407405
\(96\) 0 0
\(97\) −12.9800 −1.31792 −0.658961 0.752177i \(-0.729004\pi\)
−0.658961 + 0.752177i \(0.729004\pi\)
\(98\) 15.0659i 1.52189i
\(99\) 0 0
\(100\) 3.94492 0.394492
\(101\) 10.4273 1.03755 0.518776 0.854910i \(-0.326388\pi\)
0.518776 + 0.854910i \(0.326388\pi\)
\(102\) 0 0
\(103\) 0.181534i 0.0178871i −0.999960 0.00894354i \(-0.997153\pi\)
0.999960 0.00894354i \(-0.00284686\pi\)
\(104\) 6.29174 0.616956
\(105\) 0 0
\(106\) 13.7365i 1.33421i
\(107\) 3.16099i 0.305584i 0.988258 + 0.152792i \(0.0488265\pi\)
−0.988258 + 0.152792i \(0.951174\pi\)
\(108\) 0 0
\(109\) 10.4737i 1.00320i −0.865099 0.501601i \(-0.832745\pi\)
0.865099 0.501601i \(-0.167255\pi\)
\(110\) 7.55920 0.720742
\(111\) 0 0
\(112\) 7.91709 0.748094
\(113\) −18.1883 −1.71102 −0.855508 0.517790i \(-0.826755\pi\)
−0.855508 + 0.517790i \(0.826755\pi\)
\(114\) 0 0
\(115\) −16.7993 −1.56655
\(116\) 17.2847i 1.60484i
\(117\) 0 0
\(118\) 20.4911i 1.88636i
\(119\) 9.30907 0.853361
\(120\) 0 0
\(121\) 7.56100 0.687363
\(122\) 1.43071i 0.129531i
\(123\) 0 0
\(124\) 4.05767 0.364389
\(125\) 12.1548 1.08716
\(126\) 0 0
\(127\) −21.5671 −1.91377 −0.956885 0.290469i \(-0.906189\pi\)
−0.956885 + 0.290469i \(0.906189\pi\)
\(128\) 11.3534i 1.00351i
\(129\) 0 0
\(130\) 16.9735 1.48868
\(131\) −17.2904 −1.51067 −0.755335 0.655339i \(-0.772526\pi\)
−0.755335 + 0.655339i \(0.772526\pi\)
\(132\) 0 0
\(133\) 7.88599 0.683802
\(134\) −9.30852 −0.804134
\(135\) 0 0
\(136\) 3.76578i 0.322913i
\(137\) 9.75406i 0.833346i 0.909057 + 0.416673i \(0.136804\pi\)
−0.909057 + 0.416673i \(0.863196\pi\)
\(138\) 0 0
\(139\) 8.50529i 0.721410i 0.932680 + 0.360705i \(0.117464\pi\)
−0.932680 + 0.360705i \(0.882536\pi\)
\(140\) −18.9479 −1.60139
\(141\) 0 0
\(142\) 2.43774i 0.204571i
\(143\) −7.72199 −0.645745
\(144\) 0 0
\(145\) 12.0530i 1.00095i
\(146\) −14.9016 −1.23326
\(147\) 0 0
\(148\) 6.31660i 0.519221i
\(149\) 3.84755 0.315203 0.157602 0.987503i \(-0.449624\pi\)
0.157602 + 0.987503i \(0.449624\pi\)
\(150\) 0 0
\(151\) 11.0238i 0.897100i −0.893758 0.448550i \(-0.851941\pi\)
0.893758 0.448550i \(-0.148059\pi\)
\(152\) 3.19010i 0.258751i
\(153\) 0 0
\(154\) 15.0122 1.20972
\(155\) 2.82950 0.227271
\(156\) 0 0
\(157\) 10.9707 0.875559 0.437779 0.899082i \(-0.355765\pi\)
0.437779 + 0.899082i \(0.355765\pi\)
\(158\) −5.85910 −0.466125
\(159\) 0 0
\(160\) 14.3237i 1.13239i
\(161\) −33.3627 −2.62935
\(162\) 0 0
\(163\) 13.3073i 1.04231i −0.853463 0.521154i \(-0.825502\pi\)
0.853463 0.521154i \(-0.174498\pi\)
\(164\) −22.6011 −1.76485
\(165\) 0 0
\(166\) 17.1942i 1.33453i
\(167\) −5.93764 11.4780i −0.459468 0.888194i
\(168\) 0 0
\(169\) −4.33906 −0.333774
\(170\) 10.1591i 0.779169i
\(171\) 0 0
\(172\) 25.0828i 1.91254i
\(173\) 11.8056i 0.897561i 0.893642 + 0.448781i \(0.148142\pi\)
−0.893642 + 0.448781i \(0.851858\pi\)
\(174\) 0 0
\(175\) 5.46310 0.412972
\(176\) 3.93072i 0.296289i
\(177\) 0 0
\(178\) 33.2982 2.49581
\(179\) 7.65053i 0.571828i 0.958255 + 0.285914i \(0.0922971\pi\)
−0.958255 + 0.285914i \(0.907703\pi\)
\(180\) 0 0
\(181\) −23.8712 −1.77433 −0.887167 0.461449i \(-0.847330\pi\)
−0.887167 + 0.461449i \(0.847330\pi\)
\(182\) 33.7086 2.49865
\(183\) 0 0
\(184\) 13.4961i 0.994948i
\(185\) 4.40470i 0.323840i
\(186\) 0 0
\(187\) 4.62182i 0.337981i
\(188\) 6.63356i 0.483802i
\(189\) 0 0
\(190\) 8.60609i 0.624351i
\(191\) 9.08004i 0.657009i 0.944502 + 0.328504i \(0.106545\pi\)
−0.944502 + 0.328504i \(0.893455\pi\)
\(192\) 0 0
\(193\) 26.0841i 1.87757i −0.344498 0.938787i \(-0.611951\pi\)
0.344498 0.938787i \(-0.388049\pi\)
\(194\) 28.1316i 2.01973i
\(195\) 0 0
\(196\) −18.7493 −1.33924
\(197\) 16.2096 1.15488 0.577442 0.816432i \(-0.304051\pi\)
0.577442 + 0.816432i \(0.304051\pi\)
\(198\) 0 0
\(199\) −4.94229 −0.350350 −0.175175 0.984537i \(-0.556049\pi\)
−0.175175 + 0.984537i \(0.556049\pi\)
\(200\) 2.20998i 0.156269i
\(201\) 0 0
\(202\) 22.5990i 1.59006i
\(203\) 23.9366i 1.68002i
\(204\) 0 0
\(205\) −15.7602 −1.10074
\(206\) 0.393438 0.0274121
\(207\) 0 0
\(208\) 8.82609i 0.611979i
\(209\) 3.91528i 0.270826i
\(210\) 0 0
\(211\) −22.7402 −1.56550 −0.782749 0.622338i \(-0.786183\pi\)
−0.782749 + 0.622338i \(0.786183\pi\)
\(212\) 17.0949 1.17408
\(213\) 0 0
\(214\) −6.85080 −0.468311
\(215\) 17.4907i 1.19286i
\(216\) 0 0
\(217\) 5.61924 0.381459
\(218\) 22.6997 1.53742
\(219\) 0 0
\(220\) 9.40733i 0.634242i
\(221\) 10.3779i 0.698093i
\(222\) 0 0
\(223\) −7.06331 −0.472994 −0.236497 0.971632i \(-0.575999\pi\)
−0.236497 + 0.971632i \(0.575999\pi\)
\(224\) 28.4462i 1.90064i
\(225\) 0 0
\(226\) 39.4195i 2.62215i
\(227\) −3.92199 −0.260312 −0.130156 0.991494i \(-0.541548\pi\)
−0.130156 + 0.991494i \(0.541548\pi\)
\(228\) 0 0
\(229\) 11.9412 0.789094 0.394547 0.918876i \(-0.370902\pi\)
0.394547 + 0.918876i \(0.370902\pi\)
\(230\) 36.4092i 2.40075i
\(231\) 0 0
\(232\) −9.68303 −0.635722
\(233\) 17.0336i 1.11591i 0.829873 + 0.557953i \(0.188413\pi\)
−0.829873 + 0.557953i \(0.811587\pi\)
\(234\) 0 0
\(235\) 4.62572i 0.301749i
\(236\) −25.5009 −1.65997
\(237\) 0 0
\(238\) 20.1755i 1.30778i
\(239\) 11.9142i 0.770668i 0.922777 + 0.385334i \(0.125914\pi\)
−0.922777 + 0.385334i \(0.874086\pi\)
\(240\) 0 0
\(241\) 6.49274i 0.418234i −0.977891 0.209117i \(-0.932941\pi\)
0.977891 0.209117i \(-0.0670589\pi\)
\(242\) 16.3869i 1.05339i
\(243\) 0 0
\(244\) −1.78050 −0.113985
\(245\) −13.0743 −0.835286
\(246\) 0 0
\(247\) 8.79142i 0.559385i
\(248\) 2.27314i 0.144344i
\(249\) 0 0
\(250\) 26.3431i 1.66609i
\(251\) 0.0745657i 0.00470655i 0.999997 + 0.00235327i \(0.000749071\pi\)
−0.999997 + 0.00235327i \(0.999251\pi\)
\(252\) 0 0
\(253\) 16.5641i 1.04138i
\(254\) 46.7423i 2.93287i
\(255\) 0 0
\(256\) −0.0733770 −0.00458606
\(257\) 13.8692 0.865140 0.432570 0.901600i \(-0.357607\pi\)
0.432570 + 0.901600i \(0.357607\pi\)
\(258\) 0 0
\(259\) 8.74752i 0.543544i
\(260\) 21.1234i 1.31001i
\(261\) 0 0
\(262\) 37.4734i 2.31512i
\(263\) 5.00657i 0.308719i 0.988015 + 0.154359i \(0.0493313\pi\)
−0.988015 + 0.154359i \(0.950669\pi\)
\(264\) 0 0
\(265\) 11.9206 0.732278
\(266\) 17.0913i 1.04793i
\(267\) 0 0
\(268\) 11.5843i 0.707626i
\(269\) 26.1035 1.59156 0.795778 0.605588i \(-0.207062\pi\)
0.795778 + 0.605588i \(0.207062\pi\)
\(270\) 0 0
\(271\) 13.2610i 0.805550i 0.915299 + 0.402775i \(0.131954\pi\)
−0.915299 + 0.402775i \(0.868046\pi\)
\(272\) 5.28265 0.320308
\(273\) 0 0
\(274\) −21.1399 −1.27711
\(275\) 2.71235i 0.163561i
\(276\) 0 0
\(277\) 18.0561i 1.08489i 0.840092 + 0.542443i \(0.182501\pi\)
−0.840092 + 0.542443i \(0.817499\pi\)
\(278\) −18.4335 −1.10557
\(279\) 0 0
\(280\) 10.6147i 0.634352i
\(281\) 21.4639i 1.28043i −0.768195 0.640215i \(-0.778845\pi\)
0.768195 0.640215i \(-0.221155\pi\)
\(282\) 0 0
\(283\) −22.1803 −1.31848 −0.659242 0.751930i \(-0.729123\pi\)
−0.659242 + 0.751930i \(0.729123\pi\)
\(284\) 3.03374 0.180019
\(285\) 0 0
\(286\) 16.7358i 0.989611i
\(287\) −31.2990 −1.84752
\(288\) 0 0
\(289\) −10.7886 −0.634621
\(290\) −26.1224 −1.53396
\(291\) 0 0
\(292\) 18.5448i 1.08525i
\(293\) 20.2245i 1.18153i −0.806845 0.590763i \(-0.798827\pi\)
0.806845 0.590763i \(-0.201173\pi\)
\(294\) 0 0
\(295\) −17.7823 −1.03533
\(296\) −3.53861 −0.205678
\(297\) 0 0
\(298\) 8.33877i 0.483052i
\(299\) 37.1932i 2.15094i
\(300\) 0 0
\(301\) 34.7357i 2.00214i
\(302\) 23.8917 1.37481
\(303\) 0 0
\(304\) 4.47509 0.256664
\(305\) −1.24158 −0.0710928
\(306\) 0 0
\(307\) 4.32662i 0.246933i 0.992349 + 0.123467i \(0.0394012\pi\)
−0.992349 + 0.123467i \(0.960599\pi\)
\(308\) 18.6825i 1.06453i
\(309\) 0 0
\(310\) 6.13235i 0.348294i
\(311\) 15.5067i 0.879303i −0.898169 0.439651i \(-0.855102\pi\)
0.898169 0.439651i \(-0.144898\pi\)
\(312\) 0 0
\(313\) 0.663701i 0.0375146i 0.999824 + 0.0187573i \(0.00597098\pi\)
−0.999824 + 0.0187573i \(0.994029\pi\)
\(314\) 23.7768i 1.34180i
\(315\) 0 0
\(316\) 7.29157i 0.410183i
\(317\) 2.78217i 0.156262i 0.996943 + 0.0781310i \(0.0248952\pi\)
−0.996943 + 0.0781310i \(0.975105\pi\)
\(318\) 0 0
\(319\) 11.8842 0.665387
\(320\) −23.0706 −1.28969
\(321\) 0 0
\(322\) 72.3068i 4.02950i
\(323\) 5.26190 0.292780
\(324\) 0 0
\(325\) 6.09035i 0.337832i
\(326\) 28.8409 1.59735
\(327\) 0 0
\(328\) 12.6613i 0.699104i
\(329\) 9.18645i 0.506466i
\(330\) 0 0
\(331\) 34.2735i 1.88384i −0.335838 0.941920i \(-0.609019\pi\)
0.335838 0.941920i \(-0.390981\pi\)
\(332\) 21.3979 1.17436
\(333\) 0 0
\(334\) 24.8762 12.8686i 1.36117 0.704140i
\(335\) 8.07800i 0.441348i
\(336\) 0 0
\(337\) 9.64893 0.525611 0.262805 0.964849i \(-0.415352\pi\)
0.262805 + 0.964849i \(0.415352\pi\)
\(338\) 9.40404i 0.511512i
\(339\) 0 0
\(340\) −12.6429 −0.685657
\(341\) 2.78987i 0.151080i
\(342\) 0 0
\(343\) 0.181258 0.00978701
\(344\) −14.0516 −0.757610
\(345\) 0 0
\(346\) −25.5862 −1.37552
\(347\) −16.0340 −0.860750 −0.430375 0.902650i \(-0.641619\pi\)
−0.430375 + 0.902650i \(0.641619\pi\)
\(348\) 0 0
\(349\) 20.5841i 1.10184i −0.834558 0.550921i \(-0.814277\pi\)
0.834558 0.550921i \(-0.185723\pi\)
\(350\) 11.8402i 0.632883i
\(351\) 0 0
\(352\) 14.1231 0.752764
\(353\) 9.22419i 0.490954i 0.969402 + 0.245477i \(0.0789446\pi\)
−0.969402 + 0.245477i \(0.921055\pi\)
\(354\) 0 0
\(355\) 2.11549 0.112278
\(356\) 41.4392i 2.19627i
\(357\) 0 0
\(358\) −16.5810 −0.876331
\(359\) 30.5458i 1.61215i −0.591816 0.806073i \(-0.701589\pi\)
0.591816 0.806073i \(-0.298411\pi\)
\(360\) 0 0
\(361\) −14.5425 −0.765394
\(362\) 51.7360i 2.71918i
\(363\) 0 0
\(364\) 41.9500i 2.19878i
\(365\) 12.9317i 0.676875i
\(366\) 0 0
\(367\) 4.54381 0.237185 0.118592 0.992943i \(-0.462162\pi\)
0.118592 + 0.992943i \(0.462162\pi\)
\(368\) −18.9324 −0.986921
\(369\) 0 0
\(370\) −9.54628 −0.496288
\(371\) 23.6738 1.22908
\(372\) 0 0
\(373\) 18.6283i 0.964538i −0.876023 0.482269i \(-0.839813\pi\)
0.876023 0.482269i \(-0.160187\pi\)
\(374\) 10.0168 0.517959
\(375\) 0 0
\(376\) −3.71617 −0.191647
\(377\) 26.6849 1.37434
\(378\) 0 0
\(379\) 18.1414i 0.931861i −0.884821 0.465931i \(-0.845720\pi\)
0.884821 0.465931i \(-0.154280\pi\)
\(380\) −10.7102 −0.549420
\(381\) 0 0
\(382\) −19.6791 −1.00687
\(383\) 25.6015i 1.30817i −0.756419 0.654087i \(-0.773053\pi\)
0.756419 0.654087i \(-0.226947\pi\)
\(384\) 0 0
\(385\) 13.0277i 0.663953i
\(386\) 56.5320 2.87740
\(387\) 0 0
\(388\) 35.0094 1.77733
\(389\) −8.60355 −0.436217 −0.218109 0.975924i \(-0.569989\pi\)
−0.218109 + 0.975924i \(0.569989\pi\)
\(390\) 0 0
\(391\) −22.2611 −1.12579
\(392\) 10.5035i 0.530508i
\(393\) 0 0
\(394\) 35.1309i 1.76987i
\(395\) 5.08457i 0.255832i
\(396\) 0 0
\(397\) −3.94816 −0.198152 −0.0990761 0.995080i \(-0.531589\pi\)
−0.0990761 + 0.995080i \(0.531589\pi\)
\(398\) 10.7114i 0.536914i
\(399\) 0 0
\(400\) 3.10016 0.155008
\(401\) −3.37506 −0.168542 −0.0842712 0.996443i \(-0.526856\pi\)
−0.0842712 + 0.996443i \(0.526856\pi\)
\(402\) 0 0
\(403\) 6.26441i 0.312053i
\(404\) −28.1241 −1.39923
\(405\) 0 0
\(406\) −51.8778 −2.57465
\(407\) 4.34301 0.215275
\(408\) 0 0
\(409\) 2.65075 0.131071 0.0655356 0.997850i \(-0.479124\pi\)
0.0655356 + 0.997850i \(0.479124\pi\)
\(410\) 34.1570i 1.68690i
\(411\) 0 0
\(412\) 0.489629i 0.0241223i
\(413\) −35.3148 −1.73773
\(414\) 0 0
\(415\) 14.9212 0.732454
\(416\) 31.7122 1.55482
\(417\) 0 0
\(418\) 8.48557 0.415043
\(419\) 23.3304i 1.13977i −0.821726 0.569883i \(-0.806988\pi\)
0.821726 0.569883i \(-0.193012\pi\)
\(420\) 0 0
\(421\) −29.5887 −1.44207 −0.721034 0.692900i \(-0.756333\pi\)
−0.721034 + 0.692900i \(0.756333\pi\)
\(422\) 49.2847i 2.39914i
\(423\) 0 0
\(424\) 9.57670i 0.465085i
\(425\) 3.64524 0.176820
\(426\) 0 0
\(427\) −2.46572 −0.119325
\(428\) 8.52573i 0.412107i
\(429\) 0 0
\(430\) −37.9076 −1.82807
\(431\) 25.5345i 1.22995i −0.788546 0.614976i \(-0.789166\pi\)
0.788546 0.614976i \(-0.210834\pi\)
\(432\) 0 0
\(433\) 6.62340 0.318300 0.159150 0.987254i \(-0.449125\pi\)
0.159150 + 0.987254i \(0.449125\pi\)
\(434\) 12.1786i 0.584590i
\(435\) 0 0
\(436\) 28.2495i 1.35290i
\(437\) −18.8581 −0.902104
\(438\) 0 0
\(439\) 21.5668i 1.02933i 0.857392 + 0.514664i \(0.172083\pi\)
−0.857392 + 0.514664i \(0.827917\pi\)
\(440\) −5.27006 −0.251240
\(441\) 0 0
\(442\) 22.4920 1.06983
\(443\) −0.831700 −0.0395153 −0.0197576 0.999805i \(-0.506289\pi\)
−0.0197576 + 0.999805i \(0.506289\pi\)
\(444\) 0 0
\(445\) 28.8964i 1.36982i
\(446\) 15.3083i 0.724868i
\(447\) 0 0
\(448\) −45.8171 −2.16466
\(449\) 29.3604i 1.38560i −0.721127 0.692802i \(-0.756376\pi\)
0.721127 0.692802i \(-0.243624\pi\)
\(450\) 0 0
\(451\) 15.5395i 0.731727i
\(452\) 49.0571 2.30745
\(453\) 0 0
\(454\) 8.50012i 0.398930i
\(455\) 29.2526i 1.37138i
\(456\) 0 0
\(457\) 10.8560i 0.507822i −0.967228 0.253911i \(-0.918283\pi\)
0.967228 0.253911i \(-0.0817171\pi\)
\(458\) 25.8800i 1.20929i
\(459\) 0 0
\(460\) 45.3107 2.11262
\(461\) 0.0680390i 0.00316889i −0.999999 0.00158445i \(-0.999496\pi\)
0.999999 0.00158445i \(-0.000504345\pi\)
\(462\) 0 0
\(463\) 12.2704i 0.570254i −0.958490 0.285127i \(-0.907964\pi\)
0.958490 0.285127i \(-0.0920358\pi\)
\(464\) 13.5834i 0.630594i
\(465\) 0 0
\(466\) −36.9168 −1.71014
\(467\) 9.30810i 0.430727i 0.976534 + 0.215364i \(0.0690937\pi\)
−0.976534 + 0.215364i \(0.930906\pi\)
\(468\) 0 0
\(469\) 16.0425i 0.740775i
\(470\) −10.0253 −0.462433
\(471\) 0 0
\(472\) 14.2858i 0.657559i
\(473\) 17.2458 0.792962
\(474\) 0 0
\(475\) 3.08799 0.141687
\(476\) −25.1082 −1.15083
\(477\) 0 0
\(478\) −25.8217 −1.18106
\(479\) 2.11731 0.0967423 0.0483711 0.998829i \(-0.484597\pi\)
0.0483711 + 0.998829i \(0.484597\pi\)
\(480\) 0 0
\(481\) 9.75186 0.444647
\(482\) 14.0717 0.640948
\(483\) 0 0
\(484\) −20.3933 −0.926968
\(485\) 24.4128 1.10853
\(486\) 0 0
\(487\) 6.68431i 0.302895i 0.988465 + 0.151447i \(0.0483934\pi\)
−0.988465 + 0.151447i \(0.951607\pi\)
\(488\) 0.997453i 0.0451526i
\(489\) 0 0
\(490\) 28.3359i 1.28008i
\(491\) 33.2018i 1.49838i 0.662357 + 0.749188i \(0.269556\pi\)
−0.662357 + 0.749188i \(0.730444\pi\)
\(492\) 0 0
\(493\) 15.9716i 0.719327i
\(494\) 19.0536 0.857263
\(495\) 0 0
\(496\) 3.18877 0.143180
\(497\) 4.20126 0.188452
\(498\) 0 0
\(499\) 10.2781i 0.460112i −0.973177 0.230056i \(-0.926109\pi\)
0.973177 0.230056i \(-0.0738909\pi\)
\(500\) −32.7837 −1.46613
\(501\) 0 0
\(502\) −0.161606 −0.00721283
\(503\) 10.2410i 0.456622i 0.973588 + 0.228311i \(0.0733203\pi\)
−0.973588 + 0.228311i \(0.926680\pi\)
\(504\) 0 0
\(505\) −19.6115 −0.872703
\(506\) −35.8993 −1.59592
\(507\) 0 0
\(508\) 58.1701 2.58088
\(509\) 3.47827i 0.154171i 0.997024 + 0.0770857i \(0.0245615\pi\)
−0.997024 + 0.0770857i \(0.975438\pi\)
\(510\) 0 0
\(511\) 25.6817i 1.13609i
\(512\) 22.5478i 0.996482i
\(513\) 0 0
\(514\) 30.0588i 1.32584i
\(515\) 0.341428i 0.0150451i
\(516\) 0 0
\(517\) 4.56094 0.200590
\(518\) −18.9585 −0.832987
\(519\) 0 0
\(520\) −11.8335 −0.518932
\(521\) −19.8435 −0.869359 −0.434679 0.900585i \(-0.643138\pi\)
−0.434679 + 0.900585i \(0.643138\pi\)
\(522\) 0 0
\(523\) 8.54400 0.373603 0.186801 0.982398i \(-0.440188\pi\)
0.186801 + 0.982398i \(0.440188\pi\)
\(524\) 46.6352 2.03727
\(525\) 0 0
\(526\) −10.8507 −0.473114
\(527\) 3.74942 0.163327
\(528\) 0 0
\(529\) 56.7815 2.46876
\(530\) 25.8355i 1.12222i
\(531\) 0 0
\(532\) −21.2699 −0.922166
\(533\) 34.8926i 1.51137i
\(534\) 0 0
\(535\) 5.94517i 0.257032i
\(536\) 6.48964 0.280310
\(537\) 0 0
\(538\) 56.5739i 2.43908i
\(539\) 12.8912i 0.555263i
\(540\) 0 0
\(541\) 40.7283i 1.75105i 0.483175 + 0.875524i \(0.339484\pi\)
−0.483175 + 0.875524i \(0.660516\pi\)
\(542\) −28.7406 −1.23451
\(543\) 0 0
\(544\) 18.9806i 0.813787i
\(545\) 19.6989i 0.843810i
\(546\) 0 0
\(547\) 32.4836i 1.38890i −0.719543 0.694448i \(-0.755648\pi\)
0.719543 0.694448i \(-0.244352\pi\)
\(548\) 26.3084i 1.12384i
\(549\) 0 0
\(550\) 5.87846 0.250658
\(551\) 13.5301i 0.576400i
\(552\) 0 0
\(553\) 10.0977i 0.429398i
\(554\) −39.1329 −1.66260
\(555\) 0 0
\(556\) 22.9402i 0.972883i
\(557\) 20.9341i 0.887004i 0.896273 + 0.443502i \(0.146264\pi\)
−0.896273 + 0.443502i \(0.853736\pi\)
\(558\) 0 0
\(559\) 38.7239 1.63785
\(560\) −14.8904 −0.629235
\(561\) 0 0
\(562\) 46.5187 1.96227
\(563\) 31.1313i 1.31203i 0.754749 + 0.656014i \(0.227759\pi\)
−0.754749 + 0.656014i \(0.772241\pi\)
\(564\) 0 0
\(565\) 34.2085 1.43916
\(566\) 48.0714i 2.02059i
\(567\) 0 0
\(568\) 1.69952i 0.0713104i
\(569\) −20.2604 −0.849361 −0.424681 0.905343i \(-0.639614\pi\)
−0.424681 + 0.905343i \(0.639614\pi\)
\(570\) 0 0
\(571\) 42.8322i 1.79247i −0.443580 0.896235i \(-0.646292\pi\)
0.443580 0.896235i \(-0.353708\pi\)
\(572\) 20.8275 0.870843
\(573\) 0 0
\(574\) 67.8342i 2.83135i
\(575\) −13.0641 −0.544812
\(576\) 0 0
\(577\) 6.83395 0.284501 0.142251 0.989831i \(-0.454566\pi\)
0.142251 + 0.989831i \(0.454566\pi\)
\(578\) 23.3820i 0.972563i
\(579\) 0 0
\(580\) 32.5090i 1.34986i
\(581\) 29.6328 1.22938
\(582\) 0 0
\(583\) 11.7537i 0.486788i
\(584\) 10.3890 0.429898
\(585\) 0 0
\(586\) 43.8324 1.81070
\(587\) −13.4295 −0.554295 −0.277148 0.960827i \(-0.589389\pi\)
−0.277148 + 0.960827i \(0.589389\pi\)
\(588\) 0 0
\(589\) 3.17625 0.130875
\(590\) 38.5396i 1.58665i
\(591\) 0 0
\(592\) 4.96398i 0.204018i
\(593\) 12.6609 0.519922 0.259961 0.965619i \(-0.416290\pi\)
0.259961 + 0.965619i \(0.416290\pi\)
\(594\) 0 0
\(595\) −17.5085 −0.717777
\(596\) −10.3775 −0.425079
\(597\) 0 0
\(598\) −80.6087 −3.29634
\(599\) 4.69675i 0.191904i −0.995386 0.0959521i \(-0.969410\pi\)
0.995386 0.0959521i \(-0.0305896\pi\)
\(600\) 0 0
\(601\) −44.4167 −1.81180 −0.905898 0.423496i \(-0.860803\pi\)
−0.905898 + 0.423496i \(0.860803\pi\)
\(602\) −75.2826 −3.06829
\(603\) 0 0
\(604\) 29.7330i 1.20982i
\(605\) −14.2207 −0.578153
\(606\) 0 0
\(607\) 6.23920i 0.253241i −0.991951 0.126621i \(-0.959587\pi\)
0.991951 0.126621i \(-0.0404131\pi\)
\(608\) 16.0790i 0.652091i
\(609\) 0 0
\(610\) 2.69088i 0.108950i
\(611\) 10.2412 0.414314
\(612\) 0 0
\(613\) 22.8817 0.924184 0.462092 0.886832i \(-0.347099\pi\)
0.462092 + 0.886832i \(0.347099\pi\)
\(614\) −9.37707 −0.378428
\(615\) 0 0
\(616\) −10.4661 −0.421691
\(617\) 7.89901i 0.318002i −0.987278 0.159001i \(-0.949173\pi\)
0.987278 0.159001i \(-0.0508274\pi\)
\(618\) 0 0
\(619\) 40.5098i 1.62823i 0.580705 + 0.814114i \(0.302777\pi\)
−0.580705 + 0.814114i \(0.697223\pi\)
\(620\) −7.63164 −0.306494
\(621\) 0 0
\(622\) 33.6076 1.34754
\(623\) 57.3869i 2.29916i
\(624\) 0 0
\(625\) −15.5477 −0.621908
\(626\) −1.43844 −0.0574915
\(627\) 0 0
\(628\) −29.5899 −1.18077
\(629\) 5.83675i 0.232726i
\(630\) 0 0
\(631\) −27.2662 −1.08545 −0.542725 0.839910i \(-0.682607\pi\)
−0.542725 + 0.839910i \(0.682607\pi\)
\(632\) 4.08480 0.162485
\(633\) 0 0
\(634\) −6.02978 −0.239473
\(635\) 40.5632 1.60970
\(636\) 0 0
\(637\) 28.9461i 1.14689i
\(638\) 25.7566i 1.01971i
\(639\) 0 0
\(640\) 21.3534i 0.844069i
\(641\) −36.8500 −1.45549 −0.727744 0.685848i \(-0.759431\pi\)
−0.727744 + 0.685848i \(0.759431\pi\)
\(642\) 0 0
\(643\) 33.6664i 1.32767i 0.747878 + 0.663837i \(0.231073\pi\)
−0.747878 + 0.663837i \(0.768927\pi\)
\(644\) 89.9849 3.54590
\(645\) 0 0
\(646\) 11.4041i 0.448688i
\(647\) −33.9298 −1.33392 −0.666958 0.745095i \(-0.732404\pi\)
−0.666958 + 0.745095i \(0.732404\pi\)
\(648\) 0 0
\(649\) 17.5333i 0.688242i
\(650\) 13.1996 0.517730
\(651\) 0 0
\(652\) 35.8921i 1.40564i
\(653\) 19.8232i 0.775743i −0.921713 0.387871i \(-0.873210\pi\)
0.921713 0.387871i \(-0.126790\pi\)
\(654\) 0 0
\(655\) 32.5197 1.27065
\(656\) −17.7614 −0.693464
\(657\) 0 0
\(658\) −19.9098 −0.776164
\(659\) −10.4088 −0.405469 −0.202734 0.979234i \(-0.564983\pi\)
−0.202734 + 0.979234i \(0.564983\pi\)
\(660\) 0 0
\(661\) 37.1265i 1.44405i −0.691865 0.722027i \(-0.743211\pi\)
0.691865 0.722027i \(-0.256789\pi\)
\(662\) 74.2807 2.88700
\(663\) 0 0
\(664\) 11.9873i 0.465197i
\(665\) −14.8319 −0.575158
\(666\) 0 0
\(667\) 57.2406i 2.21637i
\(668\) 16.0148 + 30.9581i 0.619633 + 1.19781i
\(669\) 0 0
\(670\) 17.5074 0.676371
\(671\) 1.22420i 0.0472595i
\(672\) 0 0
\(673\) 3.49554i 0.134743i −0.997728 0.0673715i \(-0.978539\pi\)
0.997728 0.0673715i \(-0.0214613\pi\)
\(674\) 20.9121i 0.805504i
\(675\) 0 0
\(676\) 11.7032 0.450123
\(677\) 47.1091i 1.81055i 0.424828 + 0.905274i \(0.360335\pi\)
−0.424828 + 0.905274i \(0.639665\pi\)
\(678\) 0 0
\(679\) 48.4826 1.86059
\(680\) 7.08265i 0.271607i
\(681\) 0 0
\(682\) 6.04647 0.231531
\(683\) −14.0600 −0.537990 −0.268995 0.963142i \(-0.586692\pi\)
−0.268995 + 0.963142i \(0.586692\pi\)
\(684\) 0 0
\(685\) 18.3454i 0.700941i
\(686\) 0.392840i 0.0149987i
\(687\) 0 0
\(688\) 19.7116i 0.751498i
\(689\) 26.3919i 1.00545i
\(690\) 0 0
\(691\) 10.7832i 0.410212i −0.978740 0.205106i \(-0.934246\pi\)
0.978740 0.205106i \(-0.0657539\pi\)
\(692\) 31.8417i 1.21044i
\(693\) 0 0
\(694\) 34.7504i 1.31911i
\(695\) 15.9967i 0.606790i
\(696\) 0 0
\(697\) −20.8842 −0.791044
\(698\) 44.6118 1.68858
\(699\) 0 0
\(700\) −14.7349 −0.556928
\(701\) 20.4488i 0.772339i −0.922428 0.386170i \(-0.873798\pi\)
0.922428 0.386170i \(-0.126202\pi\)
\(702\) 0 0
\(703\) 4.94449i 0.186485i
\(704\) 22.7475i 0.857330i
\(705\) 0 0
\(706\) −19.9915 −0.752392
\(707\) −38.9476 −1.46477
\(708\) 0 0
\(709\) 14.6246i 0.549237i 0.961553 + 0.274619i \(0.0885516\pi\)
−0.961553 + 0.274619i \(0.911448\pi\)
\(710\) 4.58489i 0.172068i
\(711\) 0 0
\(712\) −23.2146 −0.870003
\(713\) −13.4375 −0.503239
\(714\) 0 0
\(715\) 14.5235 0.543147
\(716\) 20.6348i 0.771159i
\(717\) 0 0
\(718\) 66.2018 2.47063
\(719\) 44.1959 1.64823 0.824115 0.566422i \(-0.191673\pi\)
0.824115 + 0.566422i \(0.191673\pi\)
\(720\) 0 0
\(721\) 0.678060i 0.0252523i
\(722\) 31.5179i 1.17297i
\(723\) 0 0
\(724\) 64.3848 2.39284
\(725\) 9.37308i 0.348108i
\(726\) 0 0
\(727\) 44.6592i 1.65632i 0.560493 + 0.828159i \(0.310611\pi\)
−0.560493 + 0.828159i \(0.689389\pi\)
\(728\) −23.5007 −0.870994
\(729\) 0 0
\(730\) 28.0268 1.03732
\(731\) 23.1773i 0.857244i
\(732\) 0 0
\(733\) 44.8148 1.65527 0.827637 0.561264i \(-0.189685\pi\)
0.827637 + 0.561264i \(0.189685\pi\)
\(734\) 9.84778i 0.363488i
\(735\) 0 0
\(736\) 68.0244i 2.50741i
\(737\) −7.96487 −0.293390
\(738\) 0 0
\(739\) 40.9767i 1.50735i 0.657246 + 0.753676i \(0.271721\pi\)
−0.657246 + 0.753676i \(0.728279\pi\)
\(740\) 11.8802i 0.436726i
\(741\) 0 0
\(742\) 51.3081i 1.88358i
\(743\) 9.15050i 0.335699i 0.985813 + 0.167850i \(0.0536823\pi\)
−0.985813 + 0.167850i \(0.946318\pi\)
\(744\) 0 0
\(745\) −7.23644 −0.265123
\(746\) 40.3731 1.47816
\(747\) 0 0
\(748\) 12.4658i 0.455796i
\(749\) 11.8068i 0.431412i
\(750\) 0 0
\(751\) 11.4203i 0.416733i 0.978051 + 0.208367i \(0.0668147\pi\)
−0.978051 + 0.208367i \(0.933185\pi\)
\(752\) 5.21307i 0.190101i
\(753\) 0 0
\(754\) 57.8341i 2.10620i
\(755\) 20.7334i 0.754566i
\(756\) 0 0
\(757\) 9.33170 0.339166 0.169583 0.985516i \(-0.445758\pi\)
0.169583 + 0.985516i \(0.445758\pi\)
\(758\) 39.3178 1.42809
\(759\) 0 0
\(760\) 5.99993i 0.217640i
\(761\) 1.08239i 0.0392366i −0.999808 0.0196183i \(-0.993755\pi\)
0.999808 0.0196183i \(-0.00624510\pi\)
\(762\) 0 0
\(763\) 39.1212i 1.41628i
\(764\) 24.4904i 0.886033i
\(765\) 0 0
\(766\) 55.4860 2.00479
\(767\) 39.3695i 1.42155i
\(768\) 0 0
\(769\) 10.5572i 0.380701i 0.981716 + 0.190351i \(0.0609625\pi\)
−0.981716 + 0.190351i \(0.939038\pi\)
\(770\) −28.2349 −1.01751
\(771\) 0 0
\(772\) 70.3533i 2.53207i
\(773\) 38.0265 1.36772 0.683859 0.729614i \(-0.260300\pi\)
0.683859 + 0.729614i \(0.260300\pi\)
\(774\) 0 0
\(775\) 2.20038 0.0790398
\(776\) 19.6125i 0.704049i
\(777\) 0 0
\(778\) 18.6464i 0.668507i
\(779\) −17.6916 −0.633867
\(780\) 0 0
\(781\) 2.08586i 0.0746380i
\(782\) 48.2465i 1.72529i
\(783\) 0 0
\(784\) −14.7344 −0.526228
\(785\) −20.6337 −0.736447
\(786\) 0 0
\(787\) 54.3549i 1.93754i 0.247957 + 0.968771i \(0.420241\pi\)
−0.247957 + 0.968771i \(0.579759\pi\)
\(788\) −43.7200 −1.55746
\(789\) 0 0
\(790\) 11.0198 0.392066
\(791\) 67.9365 2.41554
\(792\) 0 0
\(793\) 2.74883i 0.0976136i
\(794\) 8.55682i 0.303670i
\(795\) 0 0
\(796\) 13.3302 0.472477
\(797\) 15.2054 0.538605 0.269302 0.963056i \(-0.413207\pi\)
0.269302 + 0.963056i \(0.413207\pi\)
\(798\) 0 0
\(799\) 6.12963i 0.216851i
\(800\) 11.1389i 0.393820i
\(801\) 0 0
\(802\) 7.31475i 0.258293i
\(803\) −12.7506 −0.449958
\(804\) 0 0
\(805\) 62.7484 2.21159
\(806\) 13.5768 0.478224
\(807\) 0 0
\(808\) 15.7554i 0.554272i
\(809\) 6.44710i 0.226668i 0.993557 + 0.113334i \(0.0361530\pi\)
−0.993557 + 0.113334i \(0.963847\pi\)
\(810\) 0 0
\(811\) 8.70976i 0.305841i 0.988238 + 0.152920i \(0.0488678\pi\)
−0.988238 + 0.152920i \(0.951132\pi\)
\(812\) 64.5612i 2.26566i
\(813\) 0 0
\(814\) 9.41259i 0.329911i
\(815\) 25.0283i 0.876703i
\(816\) 0 0
\(817\) 19.6342i 0.686913i
\(818\) 5.74496i 0.200868i
\(819\) 0 0
\(820\) 42.5080 1.48444
\(821\) −24.5958 −0.858401 −0.429200 0.903209i \(-0.641205\pi\)
−0.429200 + 0.903209i \(0.641205\pi\)
\(822\) 0 0
\(823\) 11.6078i 0.404624i 0.979321 + 0.202312i \(0.0648455\pi\)
−0.979321 + 0.202312i \(0.935155\pi\)
\(824\) −0.274294 −0.00955549
\(825\) 0 0
\(826\) 76.5377i 2.66309i
\(827\) −41.5678 −1.44545 −0.722727 0.691134i \(-0.757112\pi\)
−0.722727 + 0.691134i \(0.757112\pi\)
\(828\) 0 0
\(829\) 23.8514i 0.828394i −0.910187 0.414197i \(-0.864062\pi\)
0.910187 0.414197i \(-0.135938\pi\)
\(830\) 32.3387i 1.12249i
\(831\) 0 0
\(832\) 51.0776i 1.77080i
\(833\) −17.3250 −0.600276
\(834\) 0 0
\(835\) 11.1675 + 21.5878i 0.386467 + 0.747075i
\(836\) 10.5602i 0.365231i
\(837\) 0 0
\(838\) 50.5640 1.74670
\(839\) 28.2628i 0.975741i −0.872916 0.487871i \(-0.837774\pi\)
0.872916 0.487871i \(-0.162226\pi\)
\(840\) 0 0
\(841\) −12.0683 −0.416147
\(842\) 64.1276i 2.20998i
\(843\) 0 0
\(844\) 61.3341 2.11121
\(845\) 8.16089 0.280743
\(846\) 0 0
\(847\) −28.2416 −0.970392
\(848\) 13.4342 0.461333
\(849\) 0 0
\(850\) 7.90030i 0.270978i
\(851\) 20.9183i 0.717069i
\(852\) 0 0
\(853\) 4.65148 0.159264 0.0796319 0.996824i \(-0.474626\pi\)
0.0796319 + 0.996824i \(0.474626\pi\)
\(854\) 5.34395i 0.182866i
\(855\) 0 0
\(856\) 4.77618 0.163247
\(857\) 40.0158i 1.36691i 0.729992 + 0.683456i \(0.239524\pi\)
−0.729992 + 0.683456i \(0.760476\pi\)
\(858\) 0 0
\(859\) −22.6909 −0.774203 −0.387101 0.922037i \(-0.626524\pi\)
−0.387101 + 0.922037i \(0.626524\pi\)
\(860\) 47.1755i 1.60867i
\(861\) 0 0
\(862\) 55.3407 1.88491
\(863\) 31.0931i 1.05842i −0.848490 0.529212i \(-0.822488\pi\)
0.848490 0.529212i \(-0.177512\pi\)
\(864\) 0 0
\(865\) 22.2039i 0.754954i
\(866\) 14.3549i 0.487798i
\(867\) 0 0
\(868\) −15.1561 −0.514430
\(869\) −5.01336 −0.170067
\(870\) 0 0
\(871\) −17.8844 −0.605991
\(872\) −15.8256 −0.535922
\(873\) 0 0
\(874\) 40.8710i 1.38248i
\(875\) −45.4004 −1.53481
\(876\) 0 0
\(877\) 47.6084 1.60762 0.803810 0.594886i \(-0.202803\pi\)
0.803810 + 0.594886i \(0.202803\pi\)
\(878\) −46.7417 −1.57745
\(879\) 0 0
\(880\) 7.39287i 0.249214i
\(881\) 58.5074 1.97116 0.985582 0.169201i \(-0.0541187\pi\)
0.985582 + 0.169201i \(0.0541187\pi\)
\(882\) 0 0
\(883\) −47.8561 −1.61049 −0.805243 0.592945i \(-0.797965\pi\)
−0.805243 + 0.592945i \(0.797965\pi\)
\(884\) 27.9910i 0.941438i
\(885\) 0 0
\(886\) 1.80254i 0.0605575i
\(887\) −44.9871 −1.51052 −0.755259 0.655426i \(-0.772489\pi\)
−0.755259 + 0.655426i \(0.772489\pi\)
\(888\) 0 0
\(889\) 80.5567 2.70178
\(890\) −62.6271 −2.09927
\(891\) 0 0
\(892\) 19.0510 0.637873
\(893\) 5.19259i 0.173763i
\(894\) 0 0
\(895\) 14.3891i 0.480974i
\(896\) 42.4069i 1.41672i
\(897\) 0 0
\(898\) 63.6328 2.12345
\(899\) 9.64097i 0.321544i
\(900\) 0 0
\(901\) 15.7962 0.526249
\(902\) −33.6787 −1.12138
\(903\) 0 0
\(904\) 27.4822i 0.914044i
\(905\) 44.8969 1.49242
\(906\) 0 0
\(907\) −22.9092 −0.760688 −0.380344 0.924845i \(-0.624194\pi\)
−0.380344 + 0.924845i \(0.624194\pi\)
\(908\) 10.5783 0.351053
\(909\) 0 0
\(910\) −63.3990 −2.10166
\(911\) 56.5383i 1.87320i −0.350403 0.936599i \(-0.613955\pi\)
0.350403 0.936599i \(-0.386045\pi\)
\(912\) 0 0
\(913\) 14.7123i 0.486905i
\(914\) 23.5282 0.778243
\(915\) 0 0
\(916\) −32.2073 −1.06416
\(917\) 64.5825 2.13270
\(918\) 0 0
\(919\) −5.96335 −0.196713 −0.0983564 0.995151i \(-0.531359\pi\)
−0.0983564 + 0.995151i \(0.531359\pi\)
\(920\) 25.3834i 0.836868i
\(921\) 0 0
\(922\) 0.147461 0.00485636
\(923\) 4.68362i 0.154163i
\(924\) 0 0
\(925\) 3.42534i 0.112625i
\(926\) 26.5936 0.873920
\(927\) 0 0
\(928\) −48.8053 −1.60211
\(929\) 21.0386i 0.690255i −0.938556 0.345127i \(-0.887836\pi\)
0.938556 0.345127i \(-0.112164\pi\)
\(930\) 0 0
\(931\) −14.6765 −0.481004
\(932\) 45.9425i 1.50490i
\(933\) 0 0
\(934\) −20.1734 −0.660094
\(935\) 8.69269i 0.284281i
\(936\) 0 0
\(937\) 38.7402i 1.26559i −0.774320 0.632794i \(-0.781908\pi\)
0.774320 0.632794i \(-0.218092\pi\)
\(938\) 34.7689 1.13524
\(939\) 0 0
\(940\) 12.4764i 0.406934i
\(941\) 40.2288 1.31142 0.655710 0.755013i \(-0.272369\pi\)
0.655710 + 0.755013i \(0.272369\pi\)
\(942\) 0 0
\(943\) 74.8466 2.43734
\(944\) −20.0402 −0.652254
\(945\) 0 0
\(946\) 37.3767i 1.21522i
\(947\) 2.31151i 0.0751139i −0.999294 0.0375570i \(-0.988042\pi\)
0.999294 0.0375570i \(-0.0119576\pi\)
\(948\) 0 0
\(949\) −28.6304 −0.929381
\(950\) 6.69258i 0.217136i
\(951\) 0 0
\(952\) 14.0658i 0.455875i
\(953\) −50.6620 −1.64110 −0.820552 0.571573i \(-0.806334\pi\)
−0.820552 + 0.571573i \(0.806334\pi\)
\(954\) 0 0
\(955\) 17.0777i 0.552621i
\(956\) 32.1347i 1.03931i
\(957\) 0 0
\(958\) 4.58883i 0.148258i
\(959\) 36.4330i 1.17648i
\(960\) 0 0
\(961\) −28.7367 −0.926991
\(962\) 21.1352i 0.681425i
\(963\) 0 0
\(964\) 17.5120i 0.564025i
\(965\) 49.0588i 1.57926i
\(966\) 0 0
\(967\) −21.8023 −0.701115 −0.350557 0.936541i \(-0.614008\pi\)
−0.350557 + 0.936541i \(0.614008\pi\)
\(968\) 11.4245i 0.367197i
\(969\) 0 0
\(970\) 52.9097i 1.69883i
\(971\) −6.51291 −0.209009 −0.104505 0.994524i \(-0.533326\pi\)
−0.104505 + 0.994524i \(0.533326\pi\)
\(972\) 0 0
\(973\) 31.7687i 1.01846i
\(974\) −14.4869 −0.464189
\(975\) 0 0
\(976\) −1.39923 −0.0447883
\(977\) 11.9767 0.383168 0.191584 0.981476i \(-0.438638\pi\)
0.191584 + 0.981476i \(0.438638\pi\)
\(978\) 0 0
\(979\) 28.4917 0.910600
\(980\) 35.2636 1.12645
\(981\) 0 0
\(982\) −71.9581 −2.29628
\(983\) 18.1753 0.579700 0.289850 0.957072i \(-0.406395\pi\)
0.289850 + 0.957072i \(0.406395\pi\)
\(984\) 0 0
\(985\) −30.4869 −0.971393
\(986\) −34.6153 −1.10238
\(987\) 0 0
\(988\) 23.7120i 0.754379i
\(989\) 83.0649i 2.64131i
\(990\) 0 0
\(991\) 26.7481i 0.849682i 0.905268 + 0.424841i \(0.139670\pi\)
−0.905268 + 0.424841i \(0.860330\pi\)
\(992\) 11.4573i 0.363769i
\(993\) 0 0
\(994\) 9.10536i 0.288805i
\(995\) 9.29543 0.294685
\(996\) 0 0
\(997\) 4.61861 0.146273 0.0731364 0.997322i \(-0.476699\pi\)
0.0731364 + 0.997322i \(0.476699\pi\)
\(998\) 22.2757 0.705126
\(999\) 0 0
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 1503.2.c.a.1502.16 yes 56
3.2 odd 2 inner 1503.2.c.a.1502.41 yes 56
167.166 odd 2 inner 1503.2.c.a.1502.42 yes 56
501.500 even 2 inner 1503.2.c.a.1502.15 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1503.2.c.a.1502.15 56 501.500 even 2 inner
1503.2.c.a.1502.16 yes 56 1.1 even 1 trivial
1503.2.c.a.1502.41 yes 56 3.2 odd 2 inner
1503.2.c.a.1502.42 yes 56 167.166 odd 2 inner