Properties

Label 1225.4.a.h.1.1
Level $1225$
Weight $4$
Character 1225.1
Self dual yes
Analytic conductor $72.277$
Analytic rank $0$
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] = [1225,4,Mod(1,1225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1225.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1225.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: \(72.2773397570\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 25)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +7.00000 q^{3} -7.00000 q^{4} -7.00000 q^{6} +15.0000 q^{8} +22.0000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +7.00000 q^{3} -7.00000 q^{4} -7.00000 q^{6} +15.0000 q^{8} +22.0000 q^{9} -43.0000 q^{11} -49.0000 q^{12} -28.0000 q^{13} +41.0000 q^{16} +91.0000 q^{17} -22.0000 q^{18} +35.0000 q^{19} +43.0000 q^{22} -162.000 q^{23} +105.000 q^{24} +28.0000 q^{26} -35.0000 q^{27} +160.000 q^{29} -42.0000 q^{31} -161.000 q^{32} -301.000 q^{33} -91.0000 q^{34} -154.000 q^{36} +314.000 q^{37} -35.0000 q^{38} -196.000 q^{39} +203.000 q^{41} -92.0000 q^{43} +301.000 q^{44} +162.000 q^{46} +196.000 q^{47} +287.000 q^{48} +637.000 q^{51} +196.000 q^{52} -82.0000 q^{53} +35.0000 q^{54} +245.000 q^{57} -160.000 q^{58} +280.000 q^{59} +518.000 q^{61} +42.0000 q^{62} -167.000 q^{64} +301.000 q^{66} -141.000 q^{67} -637.000 q^{68} -1134.00 q^{69} +412.000 q^{71} +330.000 q^{72} -763.000 q^{73} -314.000 q^{74} -245.000 q^{76} +196.000 q^{78} +510.000 q^{79} -839.000 q^{81} -203.000 q^{82} +777.000 q^{83} +92.0000 q^{86} +1120.00 q^{87} -645.000 q^{88} +945.000 q^{89} +1134.00 q^{92} -294.000 q^{93} -196.000 q^{94} -1127.00 q^{96} +1246.00 q^{97} -946.000 q^{99} +O(q^{100})\)

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 −0.353553 −0.176777 0.984251i \(-0.556567\pi\)
−0.176777 + 0.984251i \(0.556567\pi\)
\(3\) 7.00000 1.34715 0.673575 0.739119i \(-0.264758\pi\)
0.673575 + 0.739119i \(0.264758\pi\)
\(4\) −7.00000 −0.875000
\(5\) 0 0
\(6\) −7.00000 −0.476290
\(7\) 0 0
\(8\) 15.0000 0.662913
\(9\) 22.0000 0.814815
\(10\) 0 0
\(11\) −43.0000 −1.17864 −0.589318 0.807901i \(-0.700603\pi\)
−0.589318 + 0.807901i \(0.700603\pi\)
\(12\) −49.0000 −1.17876
\(13\) −28.0000 −0.597369 −0.298685 0.954352i \(-0.596548\pi\)
−0.298685 + 0.954352i \(0.596548\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 41.0000 0.640625
\(17\) 91.0000 1.29828 0.649139 0.760669i \(-0.275129\pi\)
0.649139 + 0.760669i \(0.275129\pi\)
\(18\) −22.0000 −0.288081
\(19\) 35.0000 0.422608 0.211304 0.977420i \(-0.432229\pi\)
0.211304 + 0.977420i \(0.432229\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 43.0000 0.416710
\(23\) −162.000 −1.46867 −0.734333 0.678789i \(-0.762505\pi\)
−0.734333 + 0.678789i \(0.762505\pi\)
\(24\) 105.000 0.893043
\(25\) 0 0
\(26\) 28.0000 0.211202
\(27\) −35.0000 −0.249472
\(28\) 0 0
\(29\) 160.000 1.02453 0.512263 0.858829i \(-0.328807\pi\)
0.512263 + 0.858829i \(0.328807\pi\)
\(30\) 0 0
\(31\) −42.0000 −0.243336 −0.121668 0.992571i \(-0.538824\pi\)
−0.121668 + 0.992571i \(0.538824\pi\)
\(32\) −161.000 −0.889408
\(33\) −301.000 −1.58780
\(34\) −91.0000 −0.459011
\(35\) 0 0
\(36\) −154.000 −0.712963
\(37\) 314.000 1.39517 0.697585 0.716502i \(-0.254258\pi\)
0.697585 + 0.716502i \(0.254258\pi\)
\(38\) −35.0000 −0.149414
\(39\) −196.000 −0.804747
\(40\) 0 0
\(41\) 203.000 0.773251 0.386625 0.922237i \(-0.373641\pi\)
0.386625 + 0.922237i \(0.373641\pi\)
\(42\) 0 0
\(43\) −92.0000 −0.326276 −0.163138 0.986603i \(-0.552162\pi\)
−0.163138 + 0.986603i \(0.552162\pi\)
\(44\) 301.000 1.03131
\(45\) 0 0
\(46\) 162.000 0.519252
\(47\) 196.000 0.608288 0.304144 0.952626i \(-0.401630\pi\)
0.304144 + 0.952626i \(0.401630\pi\)
\(48\) 287.000 0.863018
\(49\) 0 0
\(50\) 0 0
\(51\) 637.000 1.74898
\(52\) 196.000 0.522698
\(53\) −82.0000 −0.212520 −0.106260 0.994338i \(-0.533888\pi\)
−0.106260 + 0.994338i \(0.533888\pi\)
\(54\) 35.0000 0.0882018
\(55\) 0 0
\(56\) 0 0
\(57\) 245.000 0.569317
\(58\) −160.000 −0.362225
\(59\) 280.000 0.617846 0.308923 0.951087i \(-0.400032\pi\)
0.308923 + 0.951087i \(0.400032\pi\)
\(60\) 0 0
\(61\) 518.000 1.08726 0.543632 0.839324i \(-0.317049\pi\)
0.543632 + 0.839324i \(0.317049\pi\)
\(62\) 42.0000 0.0860323
\(63\) 0 0
\(64\) −167.000 −0.326172
\(65\) 0 0
\(66\) 301.000 0.561372
\(67\) −141.000 −0.257103 −0.128551 0.991703i \(-0.541033\pi\)
−0.128551 + 0.991703i \(0.541033\pi\)
\(68\) −637.000 −1.13599
\(69\) −1134.00 −1.97852
\(70\) 0 0
\(71\) 412.000 0.688668 0.344334 0.938847i \(-0.388105\pi\)
0.344334 + 0.938847i \(0.388105\pi\)
\(72\) 330.000 0.540151
\(73\) −763.000 −1.22332 −0.611660 0.791121i \(-0.709498\pi\)
−0.611660 + 0.791121i \(0.709498\pi\)
\(74\) −314.000 −0.493267
\(75\) 0 0
\(76\) −245.000 −0.369782
\(77\) 0 0
\(78\) 196.000 0.284521
\(79\) 510.000 0.726323 0.363161 0.931726i \(-0.381697\pi\)
0.363161 + 0.931726i \(0.381697\pi\)
\(80\) 0 0
\(81\) −839.000 −1.15089
\(82\) −203.000 −0.273385
\(83\) 777.000 1.02755 0.513776 0.857924i \(-0.328246\pi\)
0.513776 + 0.857924i \(0.328246\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 92.0000 0.115356
\(87\) 1120.00 1.38019
\(88\) −645.000 −0.781332
\(89\) 945.000 1.12550 0.562752 0.826626i \(-0.309743\pi\)
0.562752 + 0.826626i \(0.309743\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1134.00 1.28508
\(93\) −294.000 −0.327811
\(94\) −196.000 −0.215062
\(95\) 0 0
\(96\) −1127.00 −1.19817
\(97\) 1246.00 1.30425 0.652124 0.758112i \(-0.273878\pi\)
0.652124 + 0.758112i \(0.273878\pi\)
\(98\) 0 0
\(99\) −946.000 −0.960369
\(100\) 0 0
\(101\) −1302.00 −1.28271 −0.641356 0.767244i \(-0.721628\pi\)
−0.641356 + 0.767244i \(0.721628\pi\)
\(102\) −637.000 −0.618357
\(103\) 532.000 0.508927 0.254464 0.967082i \(-0.418101\pi\)
0.254464 + 0.967082i \(0.418101\pi\)
\(104\) −420.000 −0.396004
\(105\) 0 0
\(106\) 82.0000 0.0751372
\(107\) 1269.00 1.14653 0.573266 0.819370i \(-0.305676\pi\)
0.573266 + 0.819370i \(0.305676\pi\)
\(108\) 245.000 0.218288
\(109\) 1070.00 0.940251 0.470126 0.882599i \(-0.344209\pi\)
0.470126 + 0.882599i \(0.344209\pi\)
\(110\) 0 0
\(111\) 2198.00 1.87950
\(112\) 0 0
\(113\) 503.000 0.418746 0.209373 0.977836i \(-0.432858\pi\)
0.209373 + 0.977836i \(0.432858\pi\)
\(114\) −245.000 −0.201284
\(115\) 0 0
\(116\) −1120.00 −0.896460
\(117\) −616.000 −0.486745
\(118\) −280.000 −0.218441
\(119\) 0 0
\(120\) 0 0
\(121\) 518.000 0.389181
\(122\) −518.000 −0.384406
\(123\) 1421.00 1.04169
\(124\) 294.000 0.212919
\(125\) 0 0
\(126\) 0 0
\(127\) 874.000 0.610669 0.305334 0.952245i \(-0.401232\pi\)
0.305334 + 0.952245i \(0.401232\pi\)
\(128\) 1455.00 1.00473
\(129\) −644.000 −0.439543
\(130\) 0 0
\(131\) −1092.00 −0.728309 −0.364155 0.931339i \(-0.618642\pi\)
−0.364155 + 0.931339i \(0.618642\pi\)
\(132\) 2107.00 1.38932
\(133\) 0 0
\(134\) 141.000 0.0908996
\(135\) 0 0
\(136\) 1365.00 0.860645
\(137\) −411.000 −0.256307 −0.128154 0.991754i \(-0.540905\pi\)
−0.128154 + 0.991754i \(0.540905\pi\)
\(138\) 1134.00 0.699511
\(139\) 595.000 0.363074 0.181537 0.983384i \(-0.441893\pi\)
0.181537 + 0.983384i \(0.441893\pi\)
\(140\) 0 0
\(141\) 1372.00 0.819456
\(142\) −412.000 −0.243481
\(143\) 1204.00 0.704081
\(144\) 902.000 0.521991
\(145\) 0 0
\(146\) 763.000 0.432509
\(147\) 0 0
\(148\) −2198.00 −1.22077
\(149\) −3200.00 −1.75942 −0.879712 0.475507i \(-0.842265\pi\)
−0.879712 + 0.475507i \(0.842265\pi\)
\(150\) 0 0
\(151\) 202.000 0.108864 0.0544322 0.998517i \(-0.482665\pi\)
0.0544322 + 0.998517i \(0.482665\pi\)
\(152\) 525.000 0.280152
\(153\) 2002.00 1.05786
\(154\) 0 0
\(155\) 0 0
\(156\) 1372.00 0.704153
\(157\) 406.000 0.206384 0.103192 0.994661i \(-0.467094\pi\)
0.103192 + 0.994661i \(0.467094\pi\)
\(158\) −510.000 −0.256794
\(159\) −574.000 −0.286297
\(160\) 0 0
\(161\) 0 0
\(162\) 839.000 0.406902
\(163\) 3803.00 1.82745 0.913724 0.406336i \(-0.133194\pi\)
0.913724 + 0.406336i \(0.133194\pi\)
\(164\) −1421.00 −0.676594
\(165\) 0 0
\(166\) −777.000 −0.363295
\(167\) 4116.00 1.90722 0.953610 0.301046i \(-0.0973357\pi\)
0.953610 + 0.301046i \(0.0973357\pi\)
\(168\) 0 0
\(169\) −1413.00 −0.643150
\(170\) 0 0
\(171\) 770.000 0.344347
\(172\) 644.000 0.285492
\(173\) 1512.00 0.664481 0.332241 0.943195i \(-0.392195\pi\)
0.332241 + 0.943195i \(0.392195\pi\)
\(174\) −1120.00 −0.487971
\(175\) 0 0
\(176\) −1763.00 −0.755063
\(177\) 1960.00 0.832331
\(178\) −945.000 −0.397926
\(179\) 2585.00 1.07940 0.539698 0.841859i \(-0.318538\pi\)
0.539698 + 0.841859i \(0.318538\pi\)
\(180\) 0 0
\(181\) 2758.00 1.13260 0.566300 0.824199i \(-0.308374\pi\)
0.566300 + 0.824199i \(0.308374\pi\)
\(182\) 0 0
\(183\) 3626.00 1.46471
\(184\) −2430.00 −0.973598
\(185\) 0 0
\(186\) 294.000 0.115899
\(187\) −3913.00 −1.53020
\(188\) −1372.00 −0.532252
\(189\) 0 0
\(190\) 0 0
\(191\) −2378.00 −0.900869 −0.450435 0.892809i \(-0.648731\pi\)
−0.450435 + 0.892809i \(0.648731\pi\)
\(192\) −1169.00 −0.439403
\(193\) −3067.00 −1.14387 −0.571937 0.820298i \(-0.693808\pi\)
−0.571937 + 0.820298i \(0.693808\pi\)
\(194\) −1246.00 −0.461122
\(195\) 0 0
\(196\) 0 0
\(197\) −2346.00 −0.848455 −0.424227 0.905556i \(-0.639454\pi\)
−0.424227 + 0.905556i \(0.639454\pi\)
\(198\) 946.000 0.339542
\(199\) −4900.00 −1.74549 −0.872743 0.488180i \(-0.837661\pi\)
−0.872743 + 0.488180i \(0.837661\pi\)
\(200\) 0 0
\(201\) −987.000 −0.346356
\(202\) 1302.00 0.453507
\(203\) 0 0
\(204\) −4459.00 −1.53036
\(205\) 0 0
\(206\) −532.000 −0.179933
\(207\) −3564.00 −1.19669
\(208\) −1148.00 −0.382690
\(209\) −1505.00 −0.498101
\(210\) 0 0
\(211\) 4307.00 1.40524 0.702621 0.711564i \(-0.252013\pi\)
0.702621 + 0.711564i \(0.252013\pi\)
\(212\) 574.000 0.185955
\(213\) 2884.00 0.927739
\(214\) −1269.00 −0.405360
\(215\) 0 0
\(216\) −525.000 −0.165378
\(217\) 0 0
\(218\) −1070.00 −0.332429
\(219\) −5341.00 −1.64800
\(220\) 0 0
\(221\) −2548.00 −0.775552
\(222\) −2198.00 −0.664505
\(223\) 2212.00 0.664244 0.332122 0.943236i \(-0.392235\pi\)
0.332122 + 0.943236i \(0.392235\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −503.000 −0.148049
\(227\) 476.000 0.139177 0.0695886 0.997576i \(-0.477831\pi\)
0.0695886 + 0.997576i \(0.477831\pi\)
\(228\) −1715.00 −0.498152
\(229\) 2940.00 0.848387 0.424194 0.905572i \(-0.360558\pi\)
0.424194 + 0.905572i \(0.360558\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 2400.00 0.679171
\(233\) −1002.00 −0.281730 −0.140865 0.990029i \(-0.544988\pi\)
−0.140865 + 0.990029i \(0.544988\pi\)
\(234\) 616.000 0.172091
\(235\) 0 0
\(236\) −1960.00 −0.540615
\(237\) 3570.00 0.978466
\(238\) 0 0
\(239\) 2480.00 0.671204 0.335602 0.942004i \(-0.391060\pi\)
0.335602 + 0.942004i \(0.391060\pi\)
\(240\) 0 0
\(241\) −1897.00 −0.507039 −0.253520 0.967330i \(-0.581588\pi\)
−0.253520 + 0.967330i \(0.581588\pi\)
\(242\) −518.000 −0.137596
\(243\) −4928.00 −1.30095
\(244\) −3626.00 −0.951356
\(245\) 0 0
\(246\) −1421.00 −0.368291
\(247\) −980.000 −0.252453
\(248\) −630.000 −0.161311
\(249\) 5439.00 1.38427
\(250\) 0 0
\(251\) 2373.00 0.596743 0.298371 0.954450i \(-0.403557\pi\)
0.298371 + 0.954450i \(0.403557\pi\)
\(252\) 0 0
\(253\) 6966.00 1.73102
\(254\) −874.000 −0.215904
\(255\) 0 0
\(256\) −119.000 −0.0290527
\(257\) −4494.00 −1.09077 −0.545385 0.838185i \(-0.683617\pi\)
−0.545385 + 0.838185i \(0.683617\pi\)
\(258\) 644.000 0.155402
\(259\) 0 0
\(260\) 0 0
\(261\) 3520.00 0.834799
\(262\) 1092.00 0.257496
\(263\) −722.000 −0.169279 −0.0846396 0.996412i \(-0.526974\pi\)
−0.0846396 + 0.996412i \(0.526974\pi\)
\(264\) −4515.00 −1.05257
\(265\) 0 0
\(266\) 0 0
\(267\) 6615.00 1.51622
\(268\) 987.000 0.224965
\(269\) 6160.00 1.39621 0.698107 0.715993i \(-0.254026\pi\)
0.698107 + 0.715993i \(0.254026\pi\)
\(270\) 0 0
\(271\) 7238.00 1.62243 0.811213 0.584751i \(-0.198808\pi\)
0.811213 + 0.584751i \(0.198808\pi\)
\(272\) 3731.00 0.831710
\(273\) 0 0
\(274\) 411.000 0.0906183
\(275\) 0 0
\(276\) 7938.00 1.73120
\(277\) −1776.00 −0.385233 −0.192616 0.981274i \(-0.561697\pi\)
−0.192616 + 0.981274i \(0.561697\pi\)
\(278\) −595.000 −0.128366
\(279\) −924.000 −0.198274
\(280\) 0 0
\(281\) 4542.00 0.964246 0.482123 0.876104i \(-0.339866\pi\)
0.482123 + 0.876104i \(0.339866\pi\)
\(282\) −1372.00 −0.289721
\(283\) 7077.00 1.48652 0.743258 0.669005i \(-0.233280\pi\)
0.743258 + 0.669005i \(0.233280\pi\)
\(284\) −2884.00 −0.602584
\(285\) 0 0
\(286\) −1204.00 −0.248930
\(287\) 0 0
\(288\) −3542.00 −0.724703
\(289\) 3368.00 0.685528
\(290\) 0 0
\(291\) 8722.00 1.75702
\(292\) 5341.00 1.07041
\(293\) −4158.00 −0.829054 −0.414527 0.910037i \(-0.636053\pi\)
−0.414527 + 0.910037i \(0.636053\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 4710.00 0.924876
\(297\) 1505.00 0.294037
\(298\) 3200.00 0.622050
\(299\) 4536.00 0.877337
\(300\) 0 0
\(301\) 0 0
\(302\) −202.000 −0.0384894
\(303\) −9114.00 −1.72801
\(304\) 1435.00 0.270733
\(305\) 0 0
\(306\) −2002.00 −0.374009
\(307\) −2569.00 −0.477591 −0.238796 0.971070i \(-0.576753\pi\)
−0.238796 + 0.971070i \(0.576753\pi\)
\(308\) 0 0
\(309\) 3724.00 0.685602
\(310\) 0 0
\(311\) −2982.00 −0.543710 −0.271855 0.962338i \(-0.587637\pi\)
−0.271855 + 0.962338i \(0.587637\pi\)
\(312\) −2940.00 −0.533477
\(313\) 2422.00 0.437379 0.218689 0.975795i \(-0.429822\pi\)
0.218689 + 0.975795i \(0.429822\pi\)
\(314\) −406.000 −0.0729679
\(315\) 0 0
\(316\) −3570.00 −0.635532
\(317\) 9484.00 1.68036 0.840181 0.542307i \(-0.182449\pi\)
0.840181 + 0.542307i \(0.182449\pi\)
\(318\) 574.000 0.101221
\(319\) −6880.00 −1.20754
\(320\) 0 0
\(321\) 8883.00 1.54455
\(322\) 0 0
\(323\) 3185.00 0.548663
\(324\) 5873.00 1.00703
\(325\) 0 0
\(326\) −3803.00 −0.646100
\(327\) 7490.00 1.26666
\(328\) 3045.00 0.512598
\(329\) 0 0
\(330\) 0 0
\(331\) −183.000 −0.0303885 −0.0151942 0.999885i \(-0.504837\pi\)
−0.0151942 + 0.999885i \(0.504837\pi\)
\(332\) −5439.00 −0.899108
\(333\) 6908.00 1.13681
\(334\) −4116.00 −0.674304
\(335\) 0 0
\(336\) 0 0
\(337\) −2861.00 −0.462459 −0.231229 0.972899i \(-0.574275\pi\)
−0.231229 + 0.972899i \(0.574275\pi\)
\(338\) 1413.00 0.227388
\(339\) 3521.00 0.564113
\(340\) 0 0
\(341\) 1806.00 0.286805
\(342\) −770.000 −0.121745
\(343\) 0 0
\(344\) −1380.00 −0.216292
\(345\) 0 0
\(346\) −1512.00 −0.234930
\(347\) 629.000 0.0973098 0.0486549 0.998816i \(-0.484507\pi\)
0.0486549 + 0.998816i \(0.484507\pi\)
\(348\) −7840.00 −1.20767
\(349\) −5950.00 −0.912597 −0.456298 0.889827i \(-0.650825\pi\)
−0.456298 + 0.889827i \(0.650825\pi\)
\(350\) 0 0
\(351\) 980.000 0.149027
\(352\) 6923.00 1.04829
\(353\) −11718.0 −1.76682 −0.883408 0.468604i \(-0.844757\pi\)
−0.883408 + 0.468604i \(0.844757\pi\)
\(354\) −1960.00 −0.294274
\(355\) 0 0
\(356\) −6615.00 −0.984815
\(357\) 0 0
\(358\) −2585.00 −0.381624
\(359\) 8070.00 1.18640 0.593201 0.805054i \(-0.297864\pi\)
0.593201 + 0.805054i \(0.297864\pi\)
\(360\) 0 0
\(361\) −5634.00 −0.821403
\(362\) −2758.00 −0.400434
\(363\) 3626.00 0.524286
\(364\) 0 0
\(365\) 0 0
\(366\) −3626.00 −0.517853
\(367\) 8316.00 1.18281 0.591406 0.806374i \(-0.298573\pi\)
0.591406 + 0.806374i \(0.298573\pi\)
\(368\) −6642.00 −0.940865
\(369\) 4466.00 0.630056
\(370\) 0 0
\(371\) 0 0
\(372\) 2058.00 0.286834
\(373\) −12062.0 −1.67439 −0.837194 0.546906i \(-0.815805\pi\)
−0.837194 + 0.546906i \(0.815805\pi\)
\(374\) 3913.00 0.541006
\(375\) 0 0
\(376\) 2940.00 0.403242
\(377\) −4480.00 −0.612021
\(378\) 0 0
\(379\) 1735.00 0.235148 0.117574 0.993064i \(-0.462488\pi\)
0.117574 + 0.993064i \(0.462488\pi\)
\(380\) 0 0
\(381\) 6118.00 0.822663
\(382\) 2378.00 0.318505
\(383\) 7602.00 1.01421 0.507107 0.861883i \(-0.330715\pi\)
0.507107 + 0.861883i \(0.330715\pi\)
\(384\) 10185.0 1.35352
\(385\) 0 0
\(386\) 3067.00 0.404420
\(387\) −2024.00 −0.265855
\(388\) −8722.00 −1.14122
\(389\) 3030.00 0.394928 0.197464 0.980310i \(-0.436729\pi\)
0.197464 + 0.980310i \(0.436729\pi\)
\(390\) 0 0
\(391\) −14742.0 −1.90674
\(392\) 0 0
\(393\) −7644.00 −0.981142
\(394\) 2346.00 0.299974
\(395\) 0 0
\(396\) 6622.00 0.840323
\(397\) −1204.00 −0.152209 −0.0761046 0.997100i \(-0.524248\pi\)
−0.0761046 + 0.997100i \(0.524248\pi\)
\(398\) 4900.00 0.617123
\(399\) 0 0
\(400\) 0 0
\(401\) 1077.00 0.134122 0.0670609 0.997749i \(-0.478638\pi\)
0.0670609 + 0.997749i \(0.478638\pi\)
\(402\) 987.000 0.122455
\(403\) 1176.00 0.145362
\(404\) 9114.00 1.12237
\(405\) 0 0
\(406\) 0 0
\(407\) −13502.0 −1.64440
\(408\) 9555.00 1.15942
\(409\) 3955.00 0.478147 0.239074 0.971001i \(-0.423156\pi\)
0.239074 + 0.971001i \(0.423156\pi\)
\(410\) 0 0
\(411\) −2877.00 −0.345285
\(412\) −3724.00 −0.445311
\(413\) 0 0
\(414\) 3564.00 0.423094
\(415\) 0 0
\(416\) 4508.00 0.531305
\(417\) 4165.00 0.489115
\(418\) 1505.00 0.176105
\(419\) −6265.00 −0.730466 −0.365233 0.930916i \(-0.619011\pi\)
−0.365233 + 0.930916i \(0.619011\pi\)
\(420\) 0 0
\(421\) −3788.00 −0.438517 −0.219259 0.975667i \(-0.570364\pi\)
−0.219259 + 0.975667i \(0.570364\pi\)
\(422\) −4307.00 −0.496828
\(423\) 4312.00 0.495642
\(424\) −1230.00 −0.140882
\(425\) 0 0
\(426\) −2884.00 −0.328005
\(427\) 0 0
\(428\) −8883.00 −1.00321
\(429\) 8428.00 0.948503
\(430\) 0 0
\(431\) −15258.0 −1.70523 −0.852613 0.522544i \(-0.824983\pi\)
−0.852613 + 0.522544i \(0.824983\pi\)
\(432\) −1435.00 −0.159818
\(433\) −13573.0 −1.50641 −0.753206 0.657784i \(-0.771494\pi\)
−0.753206 + 0.657784i \(0.771494\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −7490.00 −0.822720
\(437\) −5670.00 −0.620670
\(438\) 5341.00 0.582655
\(439\) 8120.00 0.882794 0.441397 0.897312i \(-0.354483\pi\)
0.441397 + 0.897312i \(0.354483\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 2548.00 0.274199
\(443\) 6183.00 0.663122 0.331561 0.943434i \(-0.392425\pi\)
0.331561 + 0.943434i \(0.392425\pi\)
\(444\) −15386.0 −1.64457
\(445\) 0 0
\(446\) −2212.00 −0.234846
\(447\) −22400.0 −2.37021
\(448\) 0 0
\(449\) −1975.00 −0.207586 −0.103793 0.994599i \(-0.533098\pi\)
−0.103793 + 0.994599i \(0.533098\pi\)
\(450\) 0 0
\(451\) −8729.00 −0.911380
\(452\) −3521.00 −0.366402
\(453\) 1414.00 0.146657
\(454\) −476.000 −0.0492066
\(455\) 0 0
\(456\) 3675.00 0.377407
\(457\) −11831.0 −1.21101 −0.605504 0.795842i \(-0.707029\pi\)
−0.605504 + 0.795842i \(0.707029\pi\)
\(458\) −2940.00 −0.299950
\(459\) −3185.00 −0.323885
\(460\) 0 0
\(461\) −1932.00 −0.195189 −0.0975946 0.995226i \(-0.531115\pi\)
−0.0975946 + 0.995226i \(0.531115\pi\)
\(462\) 0 0
\(463\) 9228.00 0.926267 0.463133 0.886289i \(-0.346725\pi\)
0.463133 + 0.886289i \(0.346725\pi\)
\(464\) 6560.00 0.656337
\(465\) 0 0
\(466\) 1002.00 0.0996068
\(467\) 13916.0 1.37892 0.689460 0.724324i \(-0.257848\pi\)
0.689460 + 0.724324i \(0.257848\pi\)
\(468\) 4312.00 0.425902
\(469\) 0 0
\(470\) 0 0
\(471\) 2842.00 0.278031
\(472\) 4200.00 0.409578
\(473\) 3956.00 0.384560
\(474\) −3570.00 −0.345940
\(475\) 0 0
\(476\) 0 0
\(477\) −1804.00 −0.173165
\(478\) −2480.00 −0.237307
\(479\) −2310.00 −0.220348 −0.110174 0.993912i \(-0.535141\pi\)
−0.110174 + 0.993912i \(0.535141\pi\)
\(480\) 0 0
\(481\) −8792.00 −0.833432
\(482\) 1897.00 0.179266
\(483\) 0 0
\(484\) −3626.00 −0.340533
\(485\) 0 0
\(486\) 4928.00 0.459956
\(487\) 17114.0 1.59242 0.796211 0.605019i \(-0.206835\pi\)
0.796211 + 0.605019i \(0.206835\pi\)
\(488\) 7770.00 0.720761
\(489\) 26621.0 2.46185
\(490\) 0 0
\(491\) −17228.0 −1.58348 −0.791740 0.610858i \(-0.790825\pi\)
−0.791740 + 0.610858i \(0.790825\pi\)
\(492\) −9947.00 −0.911474
\(493\) 14560.0 1.33012
\(494\) 980.000 0.0892556
\(495\) 0 0
\(496\) −1722.00 −0.155887
\(497\) 0 0
\(498\) −5439.00 −0.489412
\(499\) −12500.0 −1.12140 −0.560698 0.828020i \(-0.689467\pi\)
−0.560698 + 0.828020i \(0.689467\pi\)
\(500\) 0 0
\(501\) 28812.0 2.56931
\(502\) −2373.00 −0.210980
\(503\) −868.000 −0.0769428 −0.0384714 0.999260i \(-0.512249\pi\)
−0.0384714 + 0.999260i \(0.512249\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −6966.00 −0.612009
\(507\) −9891.00 −0.866420
\(508\) −6118.00 −0.534335
\(509\) −13370.0 −1.16427 −0.582136 0.813091i \(-0.697783\pi\)
−0.582136 + 0.813091i \(0.697783\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −11521.0 −0.994455
\(513\) −1225.00 −0.105429
\(514\) 4494.00 0.385646
\(515\) 0 0
\(516\) 4508.00 0.384600
\(517\) −8428.00 −0.716950
\(518\) 0 0
\(519\) 10584.0 0.895156
\(520\) 0 0
\(521\) −21637.0 −1.81945 −0.909726 0.415210i \(-0.863708\pi\)
−0.909726 + 0.415210i \(0.863708\pi\)
\(522\) −3520.00 −0.295146
\(523\) 287.000 0.0239955 0.0119977 0.999928i \(-0.496181\pi\)
0.0119977 + 0.999928i \(0.496181\pi\)
\(524\) 7644.00 0.637270
\(525\) 0 0
\(526\) 722.000 0.0598492
\(527\) −3822.00 −0.315918
\(528\) −12341.0 −1.01718
\(529\) 14077.0 1.15698
\(530\) 0 0
\(531\) 6160.00 0.503430
\(532\) 0 0
\(533\) −5684.00 −0.461916
\(534\) −6615.00 −0.536066
\(535\) 0 0
\(536\) −2115.00 −0.170437
\(537\) 18095.0 1.45411
\(538\) −6160.00 −0.493637
\(539\) 0 0
\(540\) 0 0
\(541\) −5328.00 −0.423417 −0.211709 0.977333i \(-0.567903\pi\)
−0.211709 + 0.977333i \(0.567903\pi\)
\(542\) −7238.00 −0.573614
\(543\) 19306.0 1.52578
\(544\) −14651.0 −1.15470
\(545\) 0 0
\(546\) 0 0
\(547\) −71.0000 −0.00554980 −0.00277490 0.999996i \(-0.500883\pi\)
−0.00277490 + 0.999996i \(0.500883\pi\)
\(548\) 2877.00 0.224269
\(549\) 11396.0 0.885919
\(550\) 0 0
\(551\) 5600.00 0.432973
\(552\) −17010.0 −1.31158
\(553\) 0 0
\(554\) 1776.00 0.136200
\(555\) 0 0
\(556\) −4165.00 −0.317689
\(557\) 18444.0 1.40305 0.701524 0.712646i \(-0.252503\pi\)
0.701524 + 0.712646i \(0.252503\pi\)
\(558\) 924.000 0.0701004
\(559\) 2576.00 0.194907
\(560\) 0 0
\(561\) −27391.0 −2.06141
\(562\) −4542.00 −0.340912
\(563\) 672.000 0.0503045 0.0251522 0.999684i \(-0.491993\pi\)
0.0251522 + 0.999684i \(0.491993\pi\)
\(564\) −9604.00 −0.717024
\(565\) 0 0
\(566\) −7077.00 −0.525563
\(567\) 0 0
\(568\) 6180.00 0.456526
\(569\) −10935.0 −0.805657 −0.402829 0.915275i \(-0.631973\pi\)
−0.402829 + 0.915275i \(0.631973\pi\)
\(570\) 0 0
\(571\) −13588.0 −0.995867 −0.497934 0.867215i \(-0.665908\pi\)
−0.497934 + 0.867215i \(0.665908\pi\)
\(572\) −8428.00 −0.616071
\(573\) −16646.0 −1.21361
\(574\) 0 0
\(575\) 0 0
\(576\) −3674.00 −0.265770
\(577\) 8701.00 0.627777 0.313889 0.949460i \(-0.398368\pi\)
0.313889 + 0.949460i \(0.398368\pi\)
\(578\) −3368.00 −0.242371
\(579\) −21469.0 −1.54097
\(580\) 0 0
\(581\) 0 0
\(582\) −8722.00 −0.621200
\(583\) 3526.00 0.250484
\(584\) −11445.0 −0.810955
\(585\) 0 0
\(586\) 4158.00 0.293115
\(587\) 11361.0 0.798839 0.399420 0.916768i \(-0.369212\pi\)
0.399420 + 0.916768i \(0.369212\pi\)
\(588\) 0 0
\(589\) −1470.00 −0.102836
\(590\) 0 0
\(591\) −16422.0 −1.14300
\(592\) 12874.0 0.893781
\(593\) 11417.0 0.790624 0.395312 0.918547i \(-0.370636\pi\)
0.395312 + 0.918547i \(0.370636\pi\)
\(594\) −1505.00 −0.103958
\(595\) 0 0
\(596\) 22400.0 1.53950
\(597\) −34300.0 −2.35143
\(598\) −4536.00 −0.310185
\(599\) −21050.0 −1.43586 −0.717930 0.696116i \(-0.754910\pi\)
−0.717930 + 0.696116i \(0.754910\pi\)
\(600\) 0 0
\(601\) −7427.00 −0.504083 −0.252041 0.967716i \(-0.581102\pi\)
−0.252041 + 0.967716i \(0.581102\pi\)
\(602\) 0 0
\(603\) −3102.00 −0.209491
\(604\) −1414.00 −0.0952564
\(605\) 0 0
\(606\) 9114.00 0.610942
\(607\) −4144.00 −0.277100 −0.138550 0.990355i \(-0.544244\pi\)
−0.138550 + 0.990355i \(0.544244\pi\)
\(608\) −5635.00 −0.375871
\(609\) 0 0
\(610\) 0 0
\(611\) −5488.00 −0.363373
\(612\) −14014.0 −0.925625
\(613\) −30122.0 −1.98469 −0.992346 0.123489i \(-0.960592\pi\)
−0.992346 + 0.123489i \(0.960592\pi\)
\(614\) 2569.00 0.168854
\(615\) 0 0
\(616\) 0 0
\(617\) 11934.0 0.778679 0.389339 0.921094i \(-0.372703\pi\)
0.389339 + 0.921094i \(0.372703\pi\)
\(618\) −3724.00 −0.242397
\(619\) −8540.00 −0.554526 −0.277263 0.960794i \(-0.589427\pi\)
−0.277263 + 0.960794i \(0.589427\pi\)
\(620\) 0 0
\(621\) 5670.00 0.366392
\(622\) 2982.00 0.192230
\(623\) 0 0
\(624\) −8036.00 −0.515541
\(625\) 0 0
\(626\) −2422.00 −0.154637
\(627\) −10535.0 −0.671017
\(628\) −2842.00 −0.180586
\(629\) 28574.0 1.81132
\(630\) 0 0
\(631\) −3158.00 −0.199236 −0.0996181 0.995026i \(-0.531762\pi\)
−0.0996181 + 0.995026i \(0.531762\pi\)
\(632\) 7650.00 0.481488
\(633\) 30149.0 1.89307
\(634\) −9484.00 −0.594097
\(635\) 0 0
\(636\) 4018.00 0.250510
\(637\) 0 0
\(638\) 6880.00 0.426931
\(639\) 9064.00 0.561137
\(640\) 0 0
\(641\) −4278.00 −0.263605 −0.131803 0.991276i \(-0.542076\pi\)
−0.131803 + 0.991276i \(0.542076\pi\)
\(642\) −8883.00 −0.546081
\(643\) −11508.0 −0.705803 −0.352901 0.935661i \(-0.614805\pi\)
−0.352901 + 0.935661i \(0.614805\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −3185.00 −0.193982
\(647\) −8204.00 −0.498505 −0.249252 0.968439i \(-0.580185\pi\)
−0.249252 + 0.968439i \(0.580185\pi\)
\(648\) −12585.0 −0.762941
\(649\) −12040.0 −0.728215
\(650\) 0 0
\(651\) 0 0
\(652\) −26621.0 −1.59902
\(653\) 5518.00 0.330683 0.165342 0.986236i \(-0.447127\pi\)
0.165342 + 0.986236i \(0.447127\pi\)
\(654\) −7490.00 −0.447832
\(655\) 0 0
\(656\) 8323.00 0.495364
\(657\) −16786.0 −0.996780
\(658\) 0 0
\(659\) 13295.0 0.785887 0.392944 0.919563i \(-0.371457\pi\)
0.392944 + 0.919563i \(0.371457\pi\)
\(660\) 0 0
\(661\) 9968.00 0.586551 0.293276 0.956028i \(-0.405255\pi\)
0.293276 + 0.956028i \(0.405255\pi\)
\(662\) 183.000 0.0107440
\(663\) −17836.0 −1.04479
\(664\) 11655.0 0.681177
\(665\) 0 0
\(666\) −6908.00 −0.401921
\(667\) −25920.0 −1.50469
\(668\) −28812.0 −1.66882
\(669\) 15484.0 0.894837
\(670\) 0 0
\(671\) −22274.0 −1.28149
\(672\) 0 0
\(673\) 15738.0 0.901419 0.450710 0.892671i \(-0.351171\pi\)
0.450710 + 0.892671i \(0.351171\pi\)
\(674\) 2861.00 0.163504
\(675\) 0 0
\(676\) 9891.00 0.562756
\(677\) −19824.0 −1.12540 −0.562702 0.826660i \(-0.690238\pi\)
−0.562702 + 0.826660i \(0.690238\pi\)
\(678\) −3521.00 −0.199444
\(679\) 0 0
\(680\) 0 0
\(681\) 3332.00 0.187493
\(682\) −1806.00 −0.101401
\(683\) 11073.0 0.620346 0.310173 0.950680i \(-0.399613\pi\)
0.310173 + 0.950680i \(0.399613\pi\)
\(684\) −5390.00 −0.301304
\(685\) 0 0
\(686\) 0 0
\(687\) 20580.0 1.14291
\(688\) −3772.00 −0.209021
\(689\) 2296.00 0.126953
\(690\) 0 0
\(691\) 6503.00 0.358011 0.179006 0.983848i \(-0.442712\pi\)
0.179006 + 0.983848i \(0.442712\pi\)
\(692\) −10584.0 −0.581421
\(693\) 0 0
\(694\) −629.000 −0.0344042
\(695\) 0 0
\(696\) 16800.0 0.914946
\(697\) 18473.0 1.00389
\(698\) 5950.00 0.322652
\(699\) −7014.00 −0.379533
\(700\) 0 0
\(701\) −10148.0 −0.546768 −0.273384 0.961905i \(-0.588143\pi\)
−0.273384 + 0.961905i \(0.588143\pi\)
\(702\) −980.000 −0.0526891
\(703\) 10990.0 0.589610
\(704\) 7181.00 0.384438
\(705\) 0 0
\(706\) 11718.0 0.624664
\(707\) 0 0
\(708\) −13720.0 −0.728290
\(709\) −9980.00 −0.528641 −0.264321 0.964435i \(-0.585148\pi\)
−0.264321 + 0.964435i \(0.585148\pi\)
\(710\) 0 0
\(711\) 11220.0 0.591818
\(712\) 14175.0 0.746110
\(713\) 6804.00 0.357380
\(714\) 0 0
\(715\) 0 0
\(716\) −18095.0 −0.944472
\(717\) 17360.0 0.904214
\(718\) −8070.00 −0.419456
\(719\) 27510.0 1.42691 0.713456 0.700700i \(-0.247129\pi\)
0.713456 + 0.700700i \(0.247129\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 5634.00 0.290410
\(723\) −13279.0 −0.683059
\(724\) −19306.0 −0.991025
\(725\) 0 0
\(726\) −3626.00 −0.185363
\(727\) −17024.0 −0.868480 −0.434240 0.900797i \(-0.642983\pi\)
−0.434240 + 0.900797i \(0.642983\pi\)
\(728\) 0 0
\(729\) −11843.0 −0.601687
\(730\) 0 0
\(731\) −8372.00 −0.423597
\(732\) −25382.0 −1.28162
\(733\) −34748.0 −1.75095 −0.875475 0.483263i \(-0.839451\pi\)
−0.875475 + 0.483263i \(0.839451\pi\)
\(734\) −8316.00 −0.418187
\(735\) 0 0
\(736\) 26082.0 1.30624
\(737\) 6063.00 0.303030
\(738\) −4466.00 −0.222758
\(739\) −12020.0 −0.598326 −0.299163 0.954202i \(-0.596707\pi\)
−0.299163 + 0.954202i \(0.596707\pi\)
\(740\) 0 0
\(741\) −6860.00 −0.340092
\(742\) 0 0
\(743\) −28642.0 −1.41423 −0.707115 0.707098i \(-0.750004\pi\)
−0.707115 + 0.707098i \(0.750004\pi\)
\(744\) −4410.00 −0.217310
\(745\) 0 0
\(746\) 12062.0 0.591986
\(747\) 17094.0 0.837265
\(748\) 27391.0 1.33892
\(749\) 0 0
\(750\) 0 0
\(751\) 8752.00 0.425253 0.212627 0.977134i \(-0.431798\pi\)
0.212627 + 0.977134i \(0.431798\pi\)
\(752\) 8036.00 0.389685
\(753\) 16611.0 0.803902
\(754\) 4480.00 0.216382
\(755\) 0 0
\(756\) 0 0
\(757\) −10256.0 −0.492418 −0.246209 0.969217i \(-0.579185\pi\)
−0.246209 + 0.969217i \(0.579185\pi\)
\(758\) −1735.00 −0.0831373
\(759\) 48762.0 2.33195
\(760\) 0 0
\(761\) −33957.0 −1.61753 −0.808765 0.588132i \(-0.799864\pi\)
−0.808765 + 0.588132i \(0.799864\pi\)
\(762\) −6118.00 −0.290855
\(763\) 0 0
\(764\) 16646.0 0.788261
\(765\) 0 0
\(766\) −7602.00 −0.358579
\(767\) −7840.00 −0.369082
\(768\) −833.000 −0.0391384
\(769\) −27965.0 −1.31137 −0.655685 0.755034i \(-0.727620\pi\)
−0.655685 + 0.755034i \(0.727620\pi\)
\(770\) 0 0
\(771\) −31458.0 −1.46943
\(772\) 21469.0 1.00089
\(773\) 9912.00 0.461203 0.230601 0.973048i \(-0.425931\pi\)
0.230601 + 0.973048i \(0.425931\pi\)
\(774\) 2024.00 0.0939938
\(775\) 0 0
\(776\) 18690.0 0.864603
\(777\) 0 0
\(778\) −3030.00 −0.139628
\(779\) 7105.00 0.326782
\(780\) 0 0
\(781\) −17716.0 −0.811688
\(782\) 14742.0 0.674134
\(783\) −5600.00 −0.255591
\(784\) 0 0
\(785\) 0 0
\(786\) 7644.00 0.346886
\(787\) −25564.0 −1.15789 −0.578944 0.815367i \(-0.696535\pi\)
−0.578944 + 0.815367i \(0.696535\pi\)
\(788\) 16422.0 0.742398
\(789\) −5054.00 −0.228045
\(790\) 0 0
\(791\) 0 0
\(792\) −14190.0 −0.636641
\(793\) −14504.0 −0.649498
\(794\) 1204.00 0.0538141
\(795\) 0 0
\(796\) 34300.0 1.52730
\(797\) 12446.0 0.553149 0.276575 0.960992i \(-0.410801\pi\)
0.276575 + 0.960992i \(0.410801\pi\)
\(798\) 0 0
\(799\) 17836.0 0.789728
\(800\) 0 0
\(801\) 20790.0 0.917077
\(802\) −1077.00 −0.0474192
\(803\) 32809.0 1.44185
\(804\) 6909.00 0.303062
\(805\) 0 0
\(806\) −1176.00 −0.0513931
\(807\) 43120.0 1.88091
\(808\) −19530.0 −0.850325
\(809\) 33970.0 1.47629 0.738147 0.674640i \(-0.235701\pi\)
0.738147 + 0.674640i \(0.235701\pi\)
\(810\) 0 0
\(811\) −18732.0 −0.811060 −0.405530 0.914082i \(-0.632913\pi\)
−0.405530 + 0.914082i \(0.632913\pi\)
\(812\) 0 0
\(813\) 50666.0 2.18565
\(814\) 13502.0 0.581382
\(815\) 0 0
\(816\) 26117.0 1.12044
\(817\) −3220.00 −0.137887
\(818\) −3955.00 −0.169051
\(819\) 0 0
\(820\) 0 0
\(821\) 6162.00 0.261943 0.130972 0.991386i \(-0.458190\pi\)
0.130972 + 0.991386i \(0.458190\pi\)
\(822\) 2877.00 0.122077
\(823\) 25388.0 1.07530 0.537649 0.843169i \(-0.319313\pi\)
0.537649 + 0.843169i \(0.319313\pi\)
\(824\) 7980.00 0.337374
\(825\) 0 0
\(826\) 0 0
\(827\) −25201.0 −1.05964 −0.529821 0.848109i \(-0.677741\pi\)
−0.529821 + 0.848109i \(0.677741\pi\)
\(828\) 24948.0 1.04710
\(829\) 19740.0 0.827019 0.413509 0.910500i \(-0.364303\pi\)
0.413509 + 0.910500i \(0.364303\pi\)
\(830\) 0 0
\(831\) −12432.0 −0.518967
\(832\) 4676.00 0.194845
\(833\) 0 0
\(834\) −4165.00 −0.172928
\(835\) 0 0
\(836\) 10535.0 0.435838
\(837\) 1470.00 0.0607057
\(838\) 6265.00 0.258259
\(839\) −29680.0 −1.22130 −0.610648 0.791902i \(-0.709091\pi\)
−0.610648 + 0.791902i \(0.709091\pi\)
\(840\) 0 0
\(841\) 1211.00 0.0496535
\(842\) 3788.00 0.155039
\(843\) 31794.0 1.29898
\(844\) −30149.0 −1.22959
\(845\) 0 0
\(846\) −4312.00 −0.175236
\(847\) 0 0
\(848\) −3362.00 −0.136146
\(849\) 49539.0 2.00256
\(850\) 0 0
\(851\) −50868.0 −2.04904
\(852\) −20188.0 −0.811772
\(853\) −1218.00 −0.0488904 −0.0244452 0.999701i \(-0.507782\pi\)
−0.0244452 + 0.999701i \(0.507782\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 19035.0 0.760050
\(857\) 38731.0 1.54379 0.771894 0.635752i \(-0.219310\pi\)
0.771894 + 0.635752i \(0.219310\pi\)
\(858\) −8428.00 −0.335346
\(859\) 23555.0 0.935607 0.467803 0.883833i \(-0.345046\pi\)
0.467803 + 0.883833i \(0.345046\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 15258.0 0.602888
\(863\) −24872.0 −0.981058 −0.490529 0.871425i \(-0.663196\pi\)
−0.490529 + 0.871425i \(0.663196\pi\)
\(864\) 5635.00 0.221883
\(865\) 0 0
\(866\) 13573.0 0.532597
\(867\) 23576.0 0.923510
\(868\) 0 0
\(869\) −21930.0 −0.856069
\(870\) 0 0
\(871\) 3948.00 0.153585
\(872\) 16050.0 0.623305
\(873\) 27412.0 1.06272
\(874\) 5670.00 0.219440
\(875\) 0 0
\(876\) 37387.0 1.44200
\(877\) 17124.0 0.659335 0.329667 0.944097i \(-0.393063\pi\)
0.329667 + 0.944097i \(0.393063\pi\)
\(878\) −8120.00 −0.312115
\(879\) −29106.0 −1.11686
\(880\) 0 0
\(881\) 658.000 0.0251630 0.0125815 0.999921i \(-0.495995\pi\)
0.0125815 + 0.999921i \(0.495995\pi\)
\(882\) 0 0
\(883\) −33727.0 −1.28540 −0.642698 0.766120i \(-0.722185\pi\)
−0.642698 + 0.766120i \(0.722185\pi\)
\(884\) 17836.0 0.678608
\(885\) 0 0
\(886\) −6183.00 −0.234449
\(887\) 36036.0 1.36412 0.682058 0.731298i \(-0.261085\pi\)
0.682058 + 0.731298i \(0.261085\pi\)
\(888\) 32970.0 1.24595
\(889\) 0 0
\(890\) 0 0
\(891\) 36077.0 1.35648
\(892\) −15484.0 −0.581214
\(893\) 6860.00 0.257067
\(894\) 22400.0 0.837996
\(895\) 0 0
\(896\) 0 0
\(897\) 31752.0 1.18190
\(898\) 1975.00 0.0733927
\(899\) −6720.00 −0.249304
\(900\) 0 0
\(901\) −7462.00 −0.275910
\(902\) 8729.00 0.322222
\(903\) 0 0
\(904\) 7545.00 0.277592
\(905\) 0 0
\(906\) −1414.00 −0.0518510
\(907\) −39156.0 −1.43347 −0.716733 0.697348i \(-0.754363\pi\)
−0.716733 + 0.697348i \(0.754363\pi\)
\(908\) −3332.00 −0.121780
\(909\) −28644.0 −1.04517
\(910\) 0 0
\(911\) 43532.0 1.58318 0.791591 0.611051i \(-0.209253\pi\)
0.791591 + 0.611051i \(0.209253\pi\)
\(912\) 10045.0 0.364718
\(913\) −33411.0 −1.21111
\(914\) 11831.0 0.428156
\(915\) 0 0
\(916\) −20580.0 −0.742339
\(917\) 0 0
\(918\) 3185.00 0.114511
\(919\) −28610.0 −1.02694 −0.513469 0.858108i \(-0.671640\pi\)
−0.513469 + 0.858108i \(0.671640\pi\)
\(920\) 0 0
\(921\) −17983.0 −0.643388
\(922\) 1932.00 0.0690098
\(923\) −11536.0 −0.411389
\(924\) 0 0
\(925\) 0 0
\(926\) −9228.00 −0.327485
\(927\) 11704.0 0.414682
\(928\) −25760.0 −0.911221
\(929\) 24290.0 0.857835 0.428918 0.903344i \(-0.358895\pi\)
0.428918 + 0.903344i \(0.358895\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 7014.00 0.246514
\(933\) −20874.0 −0.732459
\(934\) −13916.0 −0.487522
\(935\) 0 0
\(936\) −9240.00 −0.322670
\(937\) 34461.0 1.20149 0.600743 0.799442i \(-0.294872\pi\)
0.600743 + 0.799442i \(0.294872\pi\)
\(938\) 0 0
\(939\) 16954.0 0.589215
\(940\) 0 0
\(941\) 40628.0 1.40748 0.703738 0.710460i \(-0.251513\pi\)
0.703738 + 0.710460i \(0.251513\pi\)
\(942\) −2842.00 −0.0982987
\(943\) −32886.0 −1.13565
\(944\) 11480.0 0.395807
\(945\) 0 0
\(946\) −3956.00 −0.135963
\(947\) 20904.0 0.717306 0.358653 0.933471i \(-0.383236\pi\)
0.358653 + 0.933471i \(0.383236\pi\)
\(948\) −24990.0 −0.856158
\(949\) 21364.0 0.730774
\(950\) 0 0
\(951\) 66388.0 2.26370
\(952\) 0 0
\(953\) −1807.00 −0.0614213 −0.0307106 0.999528i \(-0.509777\pi\)
−0.0307106 + 0.999528i \(0.509777\pi\)
\(954\) 1804.00 0.0612229
\(955\) 0 0
\(956\) −17360.0 −0.587304
\(957\) −48160.0 −1.62674
\(958\) 2310.00 0.0779047
\(959\) 0 0
\(960\) 0 0
\(961\) −28027.0 −0.940787
\(962\) 8792.00 0.294663
\(963\) 27918.0 0.934211
\(964\) 13279.0 0.443660
\(965\) 0 0
\(966\) 0 0
\(967\) 57584.0 1.91497 0.957485 0.288482i \(-0.0931505\pi\)
0.957485 + 0.288482i \(0.0931505\pi\)
\(968\) 7770.00 0.257993
\(969\) 22295.0 0.739132
\(970\) 0 0
\(971\) −27237.0 −0.900182 −0.450091 0.892983i \(-0.648608\pi\)
−0.450091 + 0.892983i \(0.648608\pi\)
\(972\) 34496.0 1.13833
\(973\) 0 0
\(974\) −17114.0 −0.563006
\(975\) 0 0
\(976\) 21238.0 0.696528
\(977\) 13649.0 0.446950 0.223475 0.974710i \(-0.428260\pi\)
0.223475 + 0.974710i \(0.428260\pi\)
\(978\) −26621.0 −0.870394
\(979\) −40635.0 −1.32656
\(980\) 0 0
\(981\) 23540.0 0.766131
\(982\) 17228.0 0.559845
\(983\) 16002.0 0.519211 0.259606 0.965715i \(-0.416407\pi\)
0.259606 + 0.965715i \(0.416407\pi\)
\(984\) 21315.0 0.690546
\(985\) 0 0
\(986\) −14560.0 −0.470269
\(987\) 0 0
\(988\) 6860.00 0.220896
\(989\) 14904.0 0.479191
\(990\) 0 0
\(991\) 37022.0 1.18672 0.593362 0.804936i \(-0.297800\pi\)
0.593362 + 0.804936i \(0.297800\pi\)
\(992\) 6762.00 0.216425
\(993\) −1281.00 −0.0409379
\(994\) 0 0
\(995\) 0 0
\(996\) −38073.0 −1.21123
\(997\) 18396.0 0.584360 0.292180 0.956363i \(-0.405619\pi\)
0.292180 + 0.956363i \(0.405619\pi\)
\(998\) 12500.0 0.396474
\(999\) −10990.0 −0.348056
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 1225.4.a.h.1.1 1
5.4 even 2 1225.4.a.i.1.1 1
7.6 odd 2 25.4.a.a.1.1 1
21.20 even 2 225.4.a.e.1.1 1
28.27 even 2 400.4.a.s.1.1 1
35.13 even 4 25.4.b.b.24.2 2
35.27 even 4 25.4.b.b.24.1 2
35.34 odd 2 25.4.a.b.1.1 yes 1
56.13 odd 2 1600.4.a.bt.1.1 1
56.27 even 2 1600.4.a.h.1.1 1
105.62 odd 4 225.4.b.f.199.2 2
105.83 odd 4 225.4.b.f.199.1 2
105.104 even 2 225.4.a.c.1.1 1
140.27 odd 4 400.4.c.e.49.1 2
140.83 odd 4 400.4.c.e.49.2 2
140.139 even 2 400.4.a.c.1.1 1
280.69 odd 2 1600.4.a.i.1.1 1
280.139 even 2 1600.4.a.bs.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.4.a.a.1.1 1 7.6 odd 2
25.4.a.b.1.1 yes 1 35.34 odd 2
25.4.b.b.24.1 2 35.27 even 4
25.4.b.b.24.2 2 35.13 even 4
225.4.a.c.1.1 1 105.104 even 2
225.4.a.e.1.1 1 21.20 even 2
225.4.b.f.199.1 2 105.83 odd 4
225.4.b.f.199.2 2 105.62 odd 4
400.4.a.c.1.1 1 140.139 even 2
400.4.a.s.1.1 1 28.27 even 2
400.4.c.e.49.1 2 140.27 odd 4
400.4.c.e.49.2 2 140.83 odd 4
1225.4.a.h.1.1 1 1.1 even 1 trivial
1225.4.a.i.1.1 1 5.4 even 2
1600.4.a.h.1.1 1 56.27 even 2
1600.4.a.i.1.1 1 280.69 odd 2
1600.4.a.bs.1.1 1 280.139 even 2
1600.4.a.bt.1.1 1 56.13 odd 2