Properties

Label 1425.2.a.b.1.1
Level $1425$
Weight $2$
Character 1425.1
Self dual yes
Analytic conductor $11.379$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1425,2,Mod(1,1425)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1425, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1425.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1425.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: \(11.3786822880\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1425.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} +5.00000 q^{11} +1.00000 q^{12} +4.00000 q^{13} -1.00000 q^{16} -4.00000 q^{17} -1.00000 q^{18} -1.00000 q^{19} -5.00000 q^{22} -9.00000 q^{23} -3.00000 q^{24} -4.00000 q^{26} -1.00000 q^{27} +7.00000 q^{29} +3.00000 q^{31} -5.00000 q^{32} -5.00000 q^{33} +4.00000 q^{34} -1.00000 q^{36} -10.0000 q^{37} +1.00000 q^{38} -4.00000 q^{39} -2.00000 q^{41} +4.00000 q^{43} -5.00000 q^{44} +9.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} -7.00000 q^{49} +4.00000 q^{51} -4.00000 q^{52} +11.0000 q^{53} +1.00000 q^{54} +1.00000 q^{57} -7.00000 q^{58} +8.00000 q^{59} +13.0000 q^{61} -3.00000 q^{62} +7.00000 q^{64} +5.00000 q^{66} +9.00000 q^{67} +4.00000 q^{68} +9.00000 q^{69} +10.0000 q^{71} +3.00000 q^{72} -5.00000 q^{73} +10.0000 q^{74} +1.00000 q^{76} +4.00000 q^{78} -15.0000 q^{79} +1.00000 q^{81} +2.00000 q^{82} +9.00000 q^{83} -4.00000 q^{86} -7.00000 q^{87} +15.0000 q^{88} +3.00000 q^{89} +9.00000 q^{92} -3.00000 q^{93} -8.00000 q^{94} +5.00000 q^{96} -10.0000 q^{97} +7.00000 q^{98} +5.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107 −0.353553 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 3.00000 1.06066
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 1.00000 0.288675
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) −5.00000 −1.06600
\(23\) −9.00000 −1.87663 −0.938315 0.345782i \(-0.887614\pi\)
−0.938315 + 0.345782i \(0.887614\pi\)
\(24\) −3.00000 −0.612372
\(25\) 0 0
\(26\) −4.00000 −0.784465
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 7.00000 1.29987 0.649934 0.759991i \(-0.274797\pi\)
0.649934 + 0.759991i \(0.274797\pi\)
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) −5.00000 −0.883883
\(33\) −5.00000 −0.870388
\(34\) 4.00000 0.685994
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 1.00000 0.162221
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −5.00000 −0.753778
\(45\) 0 0
\(46\) 9.00000 1.32698
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 1.00000 0.144338
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) −4.00000 −0.554700
\(53\) 11.0000 1.51097 0.755483 0.655168i \(-0.227402\pi\)
0.755483 + 0.655168i \(0.227402\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) −7.00000 −0.919145
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) 13.0000 1.66448 0.832240 0.554416i \(-0.187058\pi\)
0.832240 + 0.554416i \(0.187058\pi\)
\(62\) −3.00000 −0.381000
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) 5.00000 0.615457
\(67\) 9.00000 1.09952 0.549762 0.835321i \(-0.314718\pi\)
0.549762 + 0.835321i \(0.314718\pi\)
\(68\) 4.00000 0.485071
\(69\) 9.00000 1.08347
\(70\) 0 0
\(71\) 10.0000 1.18678 0.593391 0.804914i \(-0.297789\pi\)
0.593391 + 0.804914i \(0.297789\pi\)
\(72\) 3.00000 0.353553
\(73\) −5.00000 −0.585206 −0.292603 0.956234i \(-0.594521\pi\)
−0.292603 + 0.956234i \(0.594521\pi\)
\(74\) 10.0000 1.16248
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) 4.00000 0.452911
\(79\) −15.0000 −1.68763 −0.843816 0.536633i \(-0.819696\pi\)
−0.843816 + 0.536633i \(0.819696\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) 9.00000 0.987878 0.493939 0.869496i \(-0.335557\pi\)
0.493939 + 0.869496i \(0.335557\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) −7.00000 −0.750479
\(88\) 15.0000 1.59901
\(89\) 3.00000 0.317999 0.159000 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 9.00000 0.938315
\(93\) −3.00000 −0.311086
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) 5.00000 0.510310
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 7.00000 0.707107
\(99\) 5.00000 0.502519
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) −4.00000 −0.396059
\(103\) −13.0000 −1.28093 −0.640464 0.767988i \(-0.721258\pi\)
−0.640464 + 0.767988i \(0.721258\pi\)
\(104\) 12.0000 1.17670
\(105\) 0 0
\(106\) −11.0000 −1.06841
\(107\) 6.00000 0.580042 0.290021 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 1.00000 0.0962250
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 10.0000 0.949158
\(112\) 0 0
\(113\) −1.00000 −0.0940721 −0.0470360 0.998893i \(-0.514978\pi\)
−0.0470360 + 0.998893i \(0.514978\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 0 0
\(116\) −7.00000 −0.649934
\(117\) 4.00000 0.369800
\(118\) −8.00000 −0.736460
\(119\) 0 0
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) −13.0000 −1.17696
\(123\) 2.00000 0.180334
\(124\) −3.00000 −0.269408
\(125\) 0 0
\(126\) 0 0
\(127\) 13.0000 1.15356 0.576782 0.816898i \(-0.304308\pi\)
0.576782 + 0.816898i \(0.304308\pi\)
\(128\) 3.00000 0.265165
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 3.00000 0.262111 0.131056 0.991375i \(-0.458163\pi\)
0.131056 + 0.991375i \(0.458163\pi\)
\(132\) 5.00000 0.435194
\(133\) 0 0
\(134\) −9.00000 −0.777482
\(135\) 0 0
\(136\) −12.0000 −1.02899
\(137\) 12.0000 1.02523 0.512615 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) −9.00000 −0.766131
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) −10.0000 −0.839181
\(143\) 20.0000 1.67248
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 5.00000 0.413803
\(147\) 7.00000 0.577350
\(148\) 10.0000 0.821995
\(149\) 16.0000 1.31077 0.655386 0.755295i \(-0.272506\pi\)
0.655386 + 0.755295i \(0.272506\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) −3.00000 −0.243332
\(153\) −4.00000 −0.323381
\(154\) 0 0
\(155\) 0 0
\(156\) 4.00000 0.320256
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 15.0000 1.19334
\(159\) −11.0000 −0.872357
\(160\) 0 0
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 14.0000 1.09656 0.548282 0.836293i \(-0.315282\pi\)
0.548282 + 0.836293i \(0.315282\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) −9.00000 −0.698535
\(167\) 6.00000 0.464294 0.232147 0.972681i \(-0.425425\pi\)
0.232147 + 0.972681i \(0.425425\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) −4.00000 −0.304997
\(173\) −3.00000 −0.228086 −0.114043 0.993476i \(-0.536380\pi\)
−0.114043 + 0.993476i \(0.536380\pi\)
\(174\) 7.00000 0.530669
\(175\) 0 0
\(176\) −5.00000 −0.376889
\(177\) −8.00000 −0.601317
\(178\) −3.00000 −0.224860
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −16.0000 −1.18927 −0.594635 0.803996i \(-0.702704\pi\)
−0.594635 + 0.803996i \(0.702704\pi\)
\(182\) 0 0
\(183\) −13.0000 −0.960988
\(184\) −27.0000 −1.99047
\(185\) 0 0
\(186\) 3.00000 0.219971
\(187\) −20.0000 −1.46254
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) 0 0
\(191\) −17.0000 −1.23008 −0.615038 0.788497i \(-0.710860\pi\)
−0.615038 + 0.788497i \(0.710860\pi\)
\(192\) −7.00000 −0.505181
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) 7.00000 0.500000
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) −5.00000 −0.355335
\(199\) 22.0000 1.55954 0.779769 0.626067i \(-0.215336\pi\)
0.779769 + 0.626067i \(0.215336\pi\)
\(200\) 0 0
\(201\) −9.00000 −0.634811
\(202\) 0 0
\(203\) 0 0
\(204\) −4.00000 −0.280056
\(205\) 0 0
\(206\) 13.0000 0.905753
\(207\) −9.00000 −0.625543
\(208\) −4.00000 −0.277350
\(209\) −5.00000 −0.345857
\(210\) 0 0
\(211\) 11.0000 0.757271 0.378636 0.925546i \(-0.376393\pi\)
0.378636 + 0.925546i \(0.376393\pi\)
\(212\) −11.0000 −0.755483
\(213\) −10.0000 −0.685189
\(214\) −6.00000 −0.410152
\(215\) 0 0
\(216\) −3.00000 −0.204124
\(217\) 0 0
\(218\) 10.0000 0.677285
\(219\) 5.00000 0.337869
\(220\) 0 0
\(221\) −16.0000 −1.07628
\(222\) −10.0000 −0.671156
\(223\) −1.00000 −0.0669650 −0.0334825 0.999439i \(-0.510660\pi\)
−0.0334825 + 0.999439i \(0.510660\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 1.00000 0.0665190
\(227\) 22.0000 1.46019 0.730096 0.683345i \(-0.239475\pi\)
0.730096 + 0.683345i \(0.239475\pi\)
\(228\) −1.00000 −0.0662266
\(229\) 21.0000 1.38772 0.693860 0.720110i \(-0.255909\pi\)
0.693860 + 0.720110i \(0.255909\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 21.0000 1.37872
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) −4.00000 −0.261488
\(235\) 0 0
\(236\) −8.00000 −0.520756
\(237\) 15.0000 0.974355
\(238\) 0 0
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) 4.00000 0.257663 0.128831 0.991667i \(-0.458877\pi\)
0.128831 + 0.991667i \(0.458877\pi\)
\(242\) −14.0000 −0.899954
\(243\) −1.00000 −0.0641500
\(244\) −13.0000 −0.832240
\(245\) 0 0
\(246\) −2.00000 −0.127515
\(247\) −4.00000 −0.254514
\(248\) 9.00000 0.571501
\(249\) −9.00000 −0.570352
\(250\) 0 0
\(251\) −24.0000 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 0 0
\(253\) −45.0000 −2.82913
\(254\) −13.0000 −0.815693
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 21.0000 1.30994 0.654972 0.755653i \(-0.272680\pi\)
0.654972 + 0.755653i \(0.272680\pi\)
\(258\) 4.00000 0.249029
\(259\) 0 0
\(260\) 0 0
\(261\) 7.00000 0.433289
\(262\) −3.00000 −0.185341
\(263\) −17.0000 −1.04826 −0.524132 0.851637i \(-0.675610\pi\)
−0.524132 + 0.851637i \(0.675610\pi\)
\(264\) −15.0000 −0.923186
\(265\) 0 0
\(266\) 0 0
\(267\) −3.00000 −0.183597
\(268\) −9.00000 −0.549762
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 0 0
\(271\) 10.0000 0.607457 0.303728 0.952759i \(-0.401768\pi\)
0.303728 + 0.952759i \(0.401768\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) −12.0000 −0.724947
\(275\) 0 0
\(276\) −9.00000 −0.541736
\(277\) 33.0000 1.98278 0.991389 0.130950i \(-0.0418029\pi\)
0.991389 + 0.130950i \(0.0418029\pi\)
\(278\) −14.0000 −0.839664
\(279\) 3.00000 0.179605
\(280\) 0 0
\(281\) −15.0000 −0.894825 −0.447412 0.894328i \(-0.647654\pi\)
−0.447412 + 0.894328i \(0.647654\pi\)
\(282\) 8.00000 0.476393
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) −10.0000 −0.593391
\(285\) 0 0
\(286\) −20.0000 −1.18262
\(287\) 0 0
\(288\) −5.00000 −0.294628
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 10.0000 0.586210
\(292\) 5.00000 0.292603
\(293\) 9.00000 0.525786 0.262893 0.964825i \(-0.415323\pi\)
0.262893 + 0.964825i \(0.415323\pi\)
\(294\) −7.00000 −0.408248
\(295\) 0 0
\(296\) −30.0000 −1.74371
\(297\) −5.00000 −0.290129
\(298\) −16.0000 −0.926855
\(299\) −36.0000 −2.08193
\(300\) 0 0
\(301\) 0 0
\(302\) 8.00000 0.460348
\(303\) 0 0
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 4.00000 0.228665
\(307\) −23.0000 −1.31268 −0.656340 0.754466i \(-0.727896\pi\)
−0.656340 + 0.754466i \(0.727896\pi\)
\(308\) 0 0
\(309\) 13.0000 0.739544
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) −12.0000 −0.679366
\(313\) −13.0000 −0.734803 −0.367402 0.930062i \(-0.619753\pi\)
−0.367402 + 0.930062i \(0.619753\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) 15.0000 0.843816
\(317\) 1.00000 0.0561656 0.0280828 0.999606i \(-0.491060\pi\)
0.0280828 + 0.999606i \(0.491060\pi\)
\(318\) 11.0000 0.616849
\(319\) 35.0000 1.95962
\(320\) 0 0
\(321\) −6.00000 −0.334887
\(322\) 0 0
\(323\) 4.00000 0.222566
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −14.0000 −0.775388
\(327\) 10.0000 0.553001
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) −3.00000 −0.164895 −0.0824475 0.996595i \(-0.526274\pi\)
−0.0824475 + 0.996595i \(0.526274\pi\)
\(332\) −9.00000 −0.493939
\(333\) −10.0000 −0.547997
\(334\) −6.00000 −0.328305
\(335\) 0 0
\(336\) 0 0
\(337\) −8.00000 −0.435788 −0.217894 0.975972i \(-0.569919\pi\)
−0.217894 + 0.975972i \(0.569919\pi\)
\(338\) −3.00000 −0.163178
\(339\) 1.00000 0.0543125
\(340\) 0 0
\(341\) 15.0000 0.812296
\(342\) 1.00000 0.0540738
\(343\) 0 0
\(344\) 12.0000 0.646997
\(345\) 0 0
\(346\) 3.00000 0.161281
\(347\) 20.0000 1.07366 0.536828 0.843692i \(-0.319622\pi\)
0.536828 + 0.843692i \(0.319622\pi\)
\(348\) 7.00000 0.375239
\(349\) −17.0000 −0.909989 −0.454995 0.890494i \(-0.650359\pi\)
−0.454995 + 0.890494i \(0.650359\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) −25.0000 −1.33250
\(353\) 12.0000 0.638696 0.319348 0.947638i \(-0.396536\pi\)
0.319348 + 0.947638i \(0.396536\pi\)
\(354\) 8.00000 0.425195
\(355\) 0 0
\(356\) −3.00000 −0.159000
\(357\) 0 0
\(358\) 4.00000 0.211407
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 16.0000 0.840941
\(363\) −14.0000 −0.734809
\(364\) 0 0
\(365\) 0 0
\(366\) 13.0000 0.679521
\(367\) 18.0000 0.939592 0.469796 0.882775i \(-0.344327\pi\)
0.469796 + 0.882775i \(0.344327\pi\)
\(368\) 9.00000 0.469157
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) 0 0
\(372\) 3.00000 0.155543
\(373\) −20.0000 −1.03556 −0.517780 0.855514i \(-0.673242\pi\)
−0.517780 + 0.855514i \(0.673242\pi\)
\(374\) 20.0000 1.03418
\(375\) 0 0
\(376\) 24.0000 1.23771
\(377\) 28.0000 1.44207
\(378\) 0 0
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 0 0
\(381\) −13.0000 −0.666010
\(382\) 17.0000 0.869796
\(383\) −18.0000 −0.919757 −0.459879 0.887982i \(-0.652107\pi\)
−0.459879 + 0.887982i \(0.652107\pi\)
\(384\) −3.00000 −0.153093
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) 4.00000 0.203331
\(388\) 10.0000 0.507673
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) 36.0000 1.82060
\(392\) −21.0000 −1.06066
\(393\) −3.00000 −0.151330
\(394\) −12.0000 −0.604551
\(395\) 0 0
\(396\) −5.00000 −0.251259
\(397\) −29.0000 −1.45547 −0.727734 0.685859i \(-0.759427\pi\)
−0.727734 + 0.685859i \(0.759427\pi\)
\(398\) −22.0000 −1.10276
\(399\) 0 0
\(400\) 0 0
\(401\) 3.00000 0.149813 0.0749064 0.997191i \(-0.476134\pi\)
0.0749064 + 0.997191i \(0.476134\pi\)
\(402\) 9.00000 0.448879
\(403\) 12.0000 0.597763
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −50.0000 −2.47841
\(408\) 12.0000 0.594089
\(409\) −34.0000 −1.68119 −0.840596 0.541663i \(-0.817795\pi\)
−0.840596 + 0.541663i \(0.817795\pi\)
\(410\) 0 0
\(411\) −12.0000 −0.591916
\(412\) 13.0000 0.640464
\(413\) 0 0
\(414\) 9.00000 0.442326
\(415\) 0 0
\(416\) −20.0000 −0.980581
\(417\) −14.0000 −0.685583
\(418\) 5.00000 0.244558
\(419\) 28.0000 1.36789 0.683945 0.729534i \(-0.260263\pi\)
0.683945 + 0.729534i \(0.260263\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) −11.0000 −0.535472
\(423\) 8.00000 0.388973
\(424\) 33.0000 1.60262
\(425\) 0 0
\(426\) 10.0000 0.484502
\(427\) 0 0
\(428\) −6.00000 −0.290021
\(429\) −20.0000 −0.965609
\(430\) 0 0
\(431\) 6.00000 0.289010 0.144505 0.989504i \(-0.453841\pi\)
0.144505 + 0.989504i \(0.453841\pi\)
\(432\) 1.00000 0.0481125
\(433\) −26.0000 −1.24948 −0.624740 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 10.0000 0.478913
\(437\) 9.00000 0.430528
\(438\) −5.00000 −0.238909
\(439\) 37.0000 1.76591 0.882957 0.469454i \(-0.155549\pi\)
0.882957 + 0.469454i \(0.155549\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) 16.0000 0.761042
\(443\) −15.0000 −0.712672 −0.356336 0.934358i \(-0.615974\pi\)
−0.356336 + 0.934358i \(0.615974\pi\)
\(444\) −10.0000 −0.474579
\(445\) 0 0
\(446\) 1.00000 0.0473514
\(447\) −16.0000 −0.756774
\(448\) 0 0
\(449\) −7.00000 −0.330350 −0.165175 0.986264i \(-0.552819\pi\)
−0.165175 + 0.986264i \(0.552819\pi\)
\(450\) 0 0
\(451\) −10.0000 −0.470882
\(452\) 1.00000 0.0470360
\(453\) 8.00000 0.375873
\(454\) −22.0000 −1.03251
\(455\) 0 0
\(456\) 3.00000 0.140488
\(457\) −14.0000 −0.654892 −0.327446 0.944870i \(-0.606188\pi\)
−0.327446 + 0.944870i \(0.606188\pi\)
\(458\) −21.0000 −0.981266
\(459\) 4.00000 0.186704
\(460\) 0 0
\(461\) 12.0000 0.558896 0.279448 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) 0 0
\(463\) −12.0000 −0.557687 −0.278844 0.960337i \(-0.589951\pi\)
−0.278844 + 0.960337i \(0.589951\pi\)
\(464\) −7.00000 −0.324967
\(465\) 0 0
\(466\) 0 0
\(467\) 7.00000 0.323921 0.161961 0.986797i \(-0.448218\pi\)
0.161961 + 0.986797i \(0.448218\pi\)
\(468\) −4.00000 −0.184900
\(469\) 0 0
\(470\) 0 0
\(471\) −2.00000 −0.0921551
\(472\) 24.0000 1.10469
\(473\) 20.0000 0.919601
\(474\) −15.0000 −0.688973
\(475\) 0 0
\(476\) 0 0
\(477\) 11.0000 0.503655
\(478\) −8.00000 −0.365911
\(479\) 15.0000 0.685367 0.342684 0.939451i \(-0.388664\pi\)
0.342684 + 0.939451i \(0.388664\pi\)
\(480\) 0 0
\(481\) −40.0000 −1.82384
\(482\) −4.00000 −0.182195
\(483\) 0 0
\(484\) −14.0000 −0.636364
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 39.0000 1.76545
\(489\) −14.0000 −0.633102
\(490\) 0 0
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) −2.00000 −0.0901670
\(493\) −28.0000 −1.26106
\(494\) 4.00000 0.179969
\(495\) 0 0
\(496\) −3.00000 −0.134704
\(497\) 0 0
\(498\) 9.00000 0.403300
\(499\) −32.0000 −1.43252 −0.716258 0.697835i \(-0.754147\pi\)
−0.716258 + 0.697835i \(0.754147\pi\)
\(500\) 0 0
\(501\) −6.00000 −0.268060
\(502\) 24.0000 1.07117
\(503\) 28.0000 1.24846 0.624229 0.781241i \(-0.285413\pi\)
0.624229 + 0.781241i \(0.285413\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 45.0000 2.00049
\(507\) −3.00000 −0.133235
\(508\) −13.0000 −0.576782
\(509\) −7.00000 −0.310270 −0.155135 0.987893i \(-0.549581\pi\)
−0.155135 + 0.987893i \(0.549581\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 11.0000 0.486136
\(513\) 1.00000 0.0441511
\(514\) −21.0000 −0.926270
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 40.0000 1.75920
\(518\) 0 0
\(519\) 3.00000 0.131685
\(520\) 0 0
\(521\) −41.0000 −1.79624 −0.898121 0.439748i \(-0.855068\pi\)
−0.898121 + 0.439748i \(0.855068\pi\)
\(522\) −7.00000 −0.306382
\(523\) 28.0000 1.22435 0.612177 0.790721i \(-0.290294\pi\)
0.612177 + 0.790721i \(0.290294\pi\)
\(524\) −3.00000 −0.131056
\(525\) 0 0
\(526\) 17.0000 0.741235
\(527\) −12.0000 −0.522728
\(528\) 5.00000 0.217597
\(529\) 58.0000 2.52174
\(530\) 0 0
\(531\) 8.00000 0.347170
\(532\) 0 0
\(533\) −8.00000 −0.346518
\(534\) 3.00000 0.129823
\(535\) 0 0
\(536\) 27.0000 1.16622
\(537\) 4.00000 0.172613
\(538\) −14.0000 −0.603583
\(539\) −35.0000 −1.50756
\(540\) 0 0
\(541\) −21.0000 −0.902861 −0.451430 0.892306i \(-0.649086\pi\)
−0.451430 + 0.892306i \(0.649086\pi\)
\(542\) −10.0000 −0.429537
\(543\) 16.0000 0.686626
\(544\) 20.0000 0.857493
\(545\) 0 0
\(546\) 0 0
\(547\) 31.0000 1.32546 0.662732 0.748857i \(-0.269397\pi\)
0.662732 + 0.748857i \(0.269397\pi\)
\(548\) −12.0000 −0.512615
\(549\) 13.0000 0.554826
\(550\) 0 0
\(551\) −7.00000 −0.298210
\(552\) 27.0000 1.14920
\(553\) 0 0
\(554\) −33.0000 −1.40204
\(555\) 0 0
\(556\) −14.0000 −0.593732
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) −3.00000 −0.127000
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) 20.0000 0.844401
\(562\) 15.0000 0.632737
\(563\) 30.0000 1.26435 0.632175 0.774826i \(-0.282163\pi\)
0.632175 + 0.774826i \(0.282163\pi\)
\(564\) 8.00000 0.336861
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 30.0000 1.25877
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 14.0000 0.585882 0.292941 0.956131i \(-0.405366\pi\)
0.292941 + 0.956131i \(0.405366\pi\)
\(572\) −20.0000 −0.836242
\(573\) 17.0000 0.710185
\(574\) 0 0
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) −13.0000 −0.541197 −0.270599 0.962692i \(-0.587222\pi\)
−0.270599 + 0.962692i \(0.587222\pi\)
\(578\) 1.00000 0.0415945
\(579\) −14.0000 −0.581820
\(580\) 0 0
\(581\) 0 0
\(582\) −10.0000 −0.414513
\(583\) 55.0000 2.27787
\(584\) −15.0000 −0.620704
\(585\) 0 0
\(586\) −9.00000 −0.371787
\(587\) −3.00000 −0.123823 −0.0619116 0.998082i \(-0.519720\pi\)
−0.0619116 + 0.998082i \(0.519720\pi\)
\(588\) −7.00000 −0.288675
\(589\) −3.00000 −0.123613
\(590\) 0 0
\(591\) −12.0000 −0.493614
\(592\) 10.0000 0.410997
\(593\) −20.0000 −0.821302 −0.410651 0.911793i \(-0.634698\pi\)
−0.410651 + 0.911793i \(0.634698\pi\)
\(594\) 5.00000 0.205152
\(595\) 0 0
\(596\) −16.0000 −0.655386
\(597\) −22.0000 −0.900400
\(598\) 36.0000 1.47215
\(599\) −44.0000 −1.79779 −0.898896 0.438163i \(-0.855629\pi\)
−0.898896 + 0.438163i \(0.855629\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 0 0
\(603\) 9.00000 0.366508
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) 0 0
\(607\) 1.00000 0.0405887 0.0202944 0.999794i \(-0.493540\pi\)
0.0202944 + 0.999794i \(0.493540\pi\)
\(608\) 5.00000 0.202777
\(609\) 0 0
\(610\) 0 0
\(611\) 32.0000 1.29458
\(612\) 4.00000 0.161690
\(613\) −26.0000 −1.05013 −0.525065 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 23.0000 0.928204
\(615\) 0 0
\(616\) 0 0
\(617\) −42.0000 −1.69086 −0.845428 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) −13.0000 −0.522937
\(619\) 32.0000 1.28619 0.643094 0.765787i \(-0.277650\pi\)
0.643094 + 0.765787i \(0.277650\pi\)
\(620\) 0 0
\(621\) 9.00000 0.361158
\(622\) −24.0000 −0.962312
\(623\) 0 0
\(624\) 4.00000 0.160128
\(625\) 0 0
\(626\) 13.0000 0.519584
\(627\) 5.00000 0.199681
\(628\) −2.00000 −0.0798087
\(629\) 40.0000 1.59490
\(630\) 0 0
\(631\) 28.0000 1.11466 0.557331 0.830290i \(-0.311825\pi\)
0.557331 + 0.830290i \(0.311825\pi\)
\(632\) −45.0000 −1.79000
\(633\) −11.0000 −0.437211
\(634\) −1.00000 −0.0397151
\(635\) 0 0
\(636\) 11.0000 0.436178
\(637\) −28.0000 −1.10940
\(638\) −35.0000 −1.38566
\(639\) 10.0000 0.395594
\(640\) 0 0
\(641\) −42.0000 −1.65890 −0.829450 0.558581i \(-0.811346\pi\)
−0.829450 + 0.558581i \(0.811346\pi\)
\(642\) 6.00000 0.236801
\(643\) 30.0000 1.18308 0.591542 0.806274i \(-0.298519\pi\)
0.591542 + 0.806274i \(0.298519\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −4.00000 −0.157378
\(647\) −31.0000 −1.21874 −0.609368 0.792888i \(-0.708577\pi\)
−0.609368 + 0.792888i \(0.708577\pi\)
\(648\) 3.00000 0.117851
\(649\) 40.0000 1.57014
\(650\) 0 0
\(651\) 0 0
\(652\) −14.0000 −0.548282
\(653\) 16.0000 0.626128 0.313064 0.949732i \(-0.398644\pi\)
0.313064 + 0.949732i \(0.398644\pi\)
\(654\) −10.0000 −0.391031
\(655\) 0 0
\(656\) 2.00000 0.0780869
\(657\) −5.00000 −0.195069
\(658\) 0 0
\(659\) −4.00000 −0.155818 −0.0779089 0.996960i \(-0.524824\pi\)
−0.0779089 + 0.996960i \(0.524824\pi\)
\(660\) 0 0
\(661\) 16.0000 0.622328 0.311164 0.950356i \(-0.399281\pi\)
0.311164 + 0.950356i \(0.399281\pi\)
\(662\) 3.00000 0.116598
\(663\) 16.0000 0.621389
\(664\) 27.0000 1.04780
\(665\) 0 0
\(666\) 10.0000 0.387492
\(667\) −63.0000 −2.43937
\(668\) −6.00000 −0.232147
\(669\) 1.00000 0.0386622
\(670\) 0 0
\(671\) 65.0000 2.50930
\(672\) 0 0
\(673\) 36.0000 1.38770 0.693849 0.720121i \(-0.255914\pi\)
0.693849 + 0.720121i \(0.255914\pi\)
\(674\) 8.00000 0.308148
\(675\) 0 0
\(676\) −3.00000 −0.115385
\(677\) 27.0000 1.03769 0.518847 0.854867i \(-0.326361\pi\)
0.518847 + 0.854867i \(0.326361\pi\)
\(678\) −1.00000 −0.0384048
\(679\) 0 0
\(680\) 0 0
\(681\) −22.0000 −0.843042
\(682\) −15.0000 −0.574380
\(683\) −6.00000 −0.229584 −0.114792 0.993390i \(-0.536620\pi\)
−0.114792 + 0.993390i \(0.536620\pi\)
\(684\) 1.00000 0.0382360
\(685\) 0 0
\(686\) 0 0
\(687\) −21.0000 −0.801200
\(688\) −4.00000 −0.152499
\(689\) 44.0000 1.67627
\(690\) 0 0
\(691\) 10.0000 0.380418 0.190209 0.981744i \(-0.439083\pi\)
0.190209 + 0.981744i \(0.439083\pi\)
\(692\) 3.00000 0.114043
\(693\) 0 0
\(694\) −20.0000 −0.759190
\(695\) 0 0
\(696\) −21.0000 −0.796003
\(697\) 8.00000 0.303022
\(698\) 17.0000 0.643459
\(699\) 0 0
\(700\) 0 0
\(701\) −8.00000 −0.302156 −0.151078 0.988522i \(-0.548274\pi\)
−0.151078 + 0.988522i \(0.548274\pi\)
\(702\) 4.00000 0.150970
\(703\) 10.0000 0.377157
\(704\) 35.0000 1.31911
\(705\) 0 0
\(706\) −12.0000 −0.451626
\(707\) 0 0
\(708\) 8.00000 0.300658
\(709\) 13.0000 0.488225 0.244113 0.969747i \(-0.421503\pi\)
0.244113 + 0.969747i \(0.421503\pi\)
\(710\) 0 0
\(711\) −15.0000 −0.562544
\(712\) 9.00000 0.337289
\(713\) −27.0000 −1.01116
\(714\) 0 0
\(715\) 0 0
\(716\) 4.00000 0.149487
\(717\) −8.00000 −0.298765
\(718\) 0 0
\(719\) 15.0000 0.559406 0.279703 0.960087i \(-0.409764\pi\)
0.279703 + 0.960087i \(0.409764\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −1.00000 −0.0372161
\(723\) −4.00000 −0.148762
\(724\) 16.0000 0.594635
\(725\) 0 0
\(726\) 14.0000 0.519589
\(727\) 2.00000 0.0741759 0.0370879 0.999312i \(-0.488192\pi\)
0.0370879 + 0.999312i \(0.488192\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −16.0000 −0.591781
\(732\) 13.0000 0.480494
\(733\) −1.00000 −0.0369358 −0.0184679 0.999829i \(-0.505879\pi\)
−0.0184679 + 0.999829i \(0.505879\pi\)
\(734\) −18.0000 −0.664392
\(735\) 0 0
\(736\) 45.0000 1.65872
\(737\) 45.0000 1.65760
\(738\) 2.00000 0.0736210
\(739\) 4.00000 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 0 0
\(741\) 4.00000 0.146944
\(742\) 0 0
\(743\) −40.0000 −1.46746 −0.733729 0.679442i \(-0.762222\pi\)
−0.733729 + 0.679442i \(0.762222\pi\)
\(744\) −9.00000 −0.329956
\(745\) 0 0
\(746\) 20.0000 0.732252
\(747\) 9.00000 0.329293
\(748\) 20.0000 0.731272
\(749\) 0 0
\(750\) 0 0
\(751\) 16.0000 0.583848 0.291924 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) −8.00000 −0.291730
\(753\) 24.0000 0.874609
\(754\) −28.0000 −1.01970
\(755\) 0 0
\(756\) 0 0
\(757\) 7.00000 0.254419 0.127210 0.991876i \(-0.459398\pi\)
0.127210 + 0.991876i \(0.459398\pi\)
\(758\) 8.00000 0.290573
\(759\) 45.0000 1.63340
\(760\) 0 0
\(761\) −30.0000 −1.08750 −0.543750 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 13.0000 0.470940
\(763\) 0 0
\(764\) 17.0000 0.615038
\(765\) 0 0
\(766\) 18.0000 0.650366
\(767\) 32.0000 1.15545
\(768\) 17.0000 0.613435
\(769\) −37.0000 −1.33425 −0.667127 0.744944i \(-0.732476\pi\)
−0.667127 + 0.744944i \(0.732476\pi\)
\(770\) 0 0
\(771\) −21.0000 −0.756297
\(772\) −14.0000 −0.503871
\(773\) −18.0000 −0.647415 −0.323708 0.946157i \(-0.604929\pi\)
−0.323708 + 0.946157i \(0.604929\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) −30.0000 −1.07694
\(777\) 0 0
\(778\) 30.0000 1.07555
\(779\) 2.00000 0.0716574
\(780\) 0 0
\(781\) 50.0000 1.78914
\(782\) −36.0000 −1.28736
\(783\) −7.00000 −0.250160
\(784\) 7.00000 0.250000
\(785\) 0 0
\(786\) 3.00000 0.107006
\(787\) −29.0000 −1.03374 −0.516869 0.856064i \(-0.672903\pi\)
−0.516869 + 0.856064i \(0.672903\pi\)
\(788\) −12.0000 −0.427482
\(789\) 17.0000 0.605216
\(790\) 0 0
\(791\) 0 0
\(792\) 15.0000 0.533002
\(793\) 52.0000 1.84657
\(794\) 29.0000 1.02917
\(795\) 0 0
\(796\) −22.0000 −0.779769
\(797\) −14.0000 −0.495905 −0.247953 0.968772i \(-0.579758\pi\)
−0.247953 + 0.968772i \(0.579758\pi\)
\(798\) 0 0
\(799\) −32.0000 −1.13208
\(800\) 0 0
\(801\) 3.00000 0.106000
\(802\) −3.00000 −0.105934
\(803\) −25.0000 −0.882231
\(804\) 9.00000 0.317406
\(805\) 0 0
\(806\) −12.0000 −0.422682
\(807\) −14.0000 −0.492823
\(808\) 0 0
\(809\) −44.0000 −1.54696 −0.773479 0.633822i \(-0.781485\pi\)
−0.773479 + 0.633822i \(0.781485\pi\)
\(810\) 0 0
\(811\) −51.0000 −1.79085 −0.895426 0.445210i \(-0.853129\pi\)
−0.895426 + 0.445210i \(0.853129\pi\)
\(812\) 0 0
\(813\) −10.0000 −0.350715
\(814\) 50.0000 1.75250
\(815\) 0 0
\(816\) −4.00000 −0.140028
\(817\) −4.00000 −0.139942
\(818\) 34.0000 1.18878
\(819\) 0 0
\(820\) 0 0
\(821\) 18.0000 0.628204 0.314102 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 12.0000 0.418548
\(823\) 2.00000 0.0697156 0.0348578 0.999392i \(-0.488902\pi\)
0.0348578 + 0.999392i \(0.488902\pi\)
\(824\) −39.0000 −1.35863
\(825\) 0 0
\(826\) 0 0
\(827\) 8.00000 0.278187 0.139094 0.990279i \(-0.455581\pi\)
0.139094 + 0.990279i \(0.455581\pi\)
\(828\) 9.00000 0.312772
\(829\) 30.0000 1.04194 0.520972 0.853574i \(-0.325570\pi\)
0.520972 + 0.853574i \(0.325570\pi\)
\(830\) 0 0
\(831\) −33.0000 −1.14476
\(832\) 28.0000 0.970725
\(833\) 28.0000 0.970143
\(834\) 14.0000 0.484780
\(835\) 0 0
\(836\) 5.00000 0.172929
\(837\) −3.00000 −0.103695
\(838\) −28.0000 −0.967244
\(839\) −20.0000 −0.690477 −0.345238 0.938515i \(-0.612202\pi\)
−0.345238 + 0.938515i \(0.612202\pi\)
\(840\) 0 0
\(841\) 20.0000 0.689655
\(842\) 26.0000 0.896019
\(843\) 15.0000 0.516627
\(844\) −11.0000 −0.378636
\(845\) 0 0
\(846\) −8.00000 −0.275046
\(847\) 0 0
\(848\) −11.0000 −0.377742
\(849\) 0 0
\(850\) 0 0
\(851\) 90.0000 3.08516
\(852\) 10.0000 0.342594
\(853\) −42.0000 −1.43805 −0.719026 0.694983i \(-0.755412\pi\)
−0.719026 + 0.694983i \(0.755412\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 18.0000 0.615227
\(857\) −10.0000 −0.341593 −0.170797 0.985306i \(-0.554634\pi\)
−0.170797 + 0.985306i \(0.554634\pi\)
\(858\) 20.0000 0.682789
\(859\) 22.0000 0.750630 0.375315 0.926897i \(-0.377534\pi\)
0.375315 + 0.926897i \(0.377534\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −6.00000 −0.204361
\(863\) −20.0000 −0.680808 −0.340404 0.940279i \(-0.610564\pi\)
−0.340404 + 0.940279i \(0.610564\pi\)
\(864\) 5.00000 0.170103
\(865\) 0 0
\(866\) 26.0000 0.883516
\(867\) 1.00000 0.0339618
\(868\) 0 0
\(869\) −75.0000 −2.54420
\(870\) 0 0
\(871\) 36.0000 1.21981
\(872\) −30.0000 −1.01593
\(873\) −10.0000 −0.338449
\(874\) −9.00000 −0.304430
\(875\) 0 0
\(876\) −5.00000 −0.168934
\(877\) −6.00000 −0.202606 −0.101303 0.994856i \(-0.532301\pi\)
−0.101303 + 0.994856i \(0.532301\pi\)
\(878\) −37.0000 −1.24869
\(879\) −9.00000 −0.303562
\(880\) 0 0
\(881\) 32.0000 1.07811 0.539054 0.842271i \(-0.318782\pi\)
0.539054 + 0.842271i \(0.318782\pi\)
\(882\) 7.00000 0.235702
\(883\) −34.0000 −1.14419 −0.572096 0.820187i \(-0.693869\pi\)
−0.572096 + 0.820187i \(0.693869\pi\)
\(884\) 16.0000 0.538138
\(885\) 0 0
\(886\) 15.0000 0.503935
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 30.0000 1.00673
\(889\) 0 0
\(890\) 0 0
\(891\) 5.00000 0.167506
\(892\) 1.00000 0.0334825
\(893\) −8.00000 −0.267710
\(894\) 16.0000 0.535120
\(895\) 0 0
\(896\) 0 0
\(897\) 36.0000 1.20201
\(898\) 7.00000 0.233593
\(899\) 21.0000 0.700389
\(900\) 0 0
\(901\) −44.0000 −1.46585
\(902\) 10.0000 0.332964
\(903\) 0 0
\(904\) −3.00000 −0.0997785
\(905\) 0 0
\(906\) −8.00000 −0.265782
\(907\) −24.0000 −0.796907 −0.398453 0.917189i \(-0.630453\pi\)
−0.398453 + 0.917189i \(0.630453\pi\)
\(908\) −22.0000 −0.730096
\(909\) 0 0
\(910\) 0 0
\(911\) 14.0000 0.463841 0.231920 0.972735i \(-0.425499\pi\)
0.231920 + 0.972735i \(0.425499\pi\)
\(912\) −1.00000 −0.0331133
\(913\) 45.0000 1.48928
\(914\) 14.0000 0.463079
\(915\) 0 0
\(916\) −21.0000 −0.693860
\(917\) 0 0
\(918\) −4.00000 −0.132020
\(919\) −26.0000 −0.857661 −0.428830 0.903385i \(-0.641074\pi\)
−0.428830 + 0.903385i \(0.641074\pi\)
\(920\) 0 0
\(921\) 23.0000 0.757876
\(922\) −12.0000 −0.395199
\(923\) 40.0000 1.31662
\(924\) 0 0
\(925\) 0 0
\(926\) 12.0000 0.394344
\(927\) −13.0000 −0.426976
\(928\) −35.0000 −1.14893
\(929\) 4.00000 0.131236 0.0656179 0.997845i \(-0.479098\pi\)
0.0656179 + 0.997845i \(0.479098\pi\)
\(930\) 0 0
\(931\) 7.00000 0.229416
\(932\) 0 0
\(933\) −24.0000 −0.785725
\(934\) −7.00000 −0.229047
\(935\) 0 0
\(936\) 12.0000 0.392232
\(937\) 42.0000 1.37208 0.686040 0.727564i \(-0.259347\pi\)
0.686040 + 0.727564i \(0.259347\pi\)
\(938\) 0 0
\(939\) 13.0000 0.424239
\(940\) 0 0
\(941\) 13.0000 0.423788 0.211894 0.977293i \(-0.432037\pi\)
0.211894 + 0.977293i \(0.432037\pi\)
\(942\) 2.00000 0.0651635
\(943\) 18.0000 0.586161
\(944\) −8.00000 −0.260378
\(945\) 0 0
\(946\) −20.0000 −0.650256
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) −15.0000 −0.487177
\(949\) −20.0000 −0.649227
\(950\) 0 0
\(951\) −1.00000 −0.0324272
\(952\) 0 0
\(953\) 17.0000 0.550684 0.275342 0.961346i \(-0.411209\pi\)
0.275342 + 0.961346i \(0.411209\pi\)
\(954\) −11.0000 −0.356138
\(955\) 0 0
\(956\) −8.00000 −0.258738
\(957\) −35.0000 −1.13139
\(958\) −15.0000 −0.484628
\(959\) 0 0
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 40.0000 1.28965
\(963\) 6.00000 0.193347
\(964\) −4.00000 −0.128831
\(965\) 0 0
\(966\) 0 0
\(967\) 2.00000 0.0643157 0.0321578 0.999483i \(-0.489762\pi\)
0.0321578 + 0.999483i \(0.489762\pi\)
\(968\) 42.0000 1.34993
\(969\) −4.00000 −0.128499
\(970\) 0 0
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −8.00000 −0.256337
\(975\) 0 0
\(976\) −13.0000 −0.416120
\(977\) 2.00000 0.0639857 0.0319928 0.999488i \(-0.489815\pi\)
0.0319928 + 0.999488i \(0.489815\pi\)
\(978\) 14.0000 0.447671
\(979\) 15.0000 0.479402
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) −28.0000 −0.893516
\(983\) −42.0000 −1.33959 −0.669796 0.742545i \(-0.733618\pi\)
−0.669796 + 0.742545i \(0.733618\pi\)
\(984\) 6.00000 0.191273
\(985\) 0 0
\(986\) 28.0000 0.891702
\(987\) 0 0
\(988\) 4.00000 0.127257
\(989\) −36.0000 −1.14473
\(990\) 0 0
\(991\) 5.00000 0.158830 0.0794151 0.996842i \(-0.474695\pi\)
0.0794151 + 0.996842i \(0.474695\pi\)
\(992\) −15.0000 −0.476250
\(993\) 3.00000 0.0952021
\(994\) 0 0
\(995\) 0 0
\(996\) 9.00000 0.285176
\(997\) 3.00000 0.0950110 0.0475055 0.998871i \(-0.484873\pi\)
0.0475055 + 0.998871i \(0.484873\pi\)
\(998\) 32.0000 1.01294
\(999\) 10.0000 0.316386
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 1425.2.a.b.1.1 1
3.2 odd 2 4275.2.a.l.1.1 1
5.2 odd 4 1425.2.c.h.799.1 2
5.3 odd 4 1425.2.c.h.799.2 2
5.4 even 2 1425.2.a.h.1.1 yes 1
15.14 odd 2 4275.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1425.2.a.b.1.1 1 1.1 even 1 trivial
1425.2.a.h.1.1 yes 1 5.4 even 2
1425.2.c.h.799.1 2 5.2 odd 4
1425.2.c.h.799.2 2 5.3 odd 4
4275.2.a.f.1.1 1 15.14 odd 2
4275.2.a.l.1.1 1 3.2 odd 2