Properties

Label 5625.2.a.bd.1.1
Level $5625$
Weight $2$
Character 5625.1
Self dual yes
Analytic conductor $44.916$
Analytic rank $1$
Dimension $8$
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] = [5625,2,Mod(1,5625)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5625, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("5625.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5625 = 3^{2} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5625.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: \(44.9158511370\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.5444000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 2x^{6} + 20x^{5} - 4x^{4} - 30x^{3} + 7x^{2} + 12x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 75)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.53767\) of defining polynomial
Character \(\chi\) \(=\) 5625.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.53767 q^{2} +0.364440 q^{4} -1.68601 q^{7} +2.51496 q^{8} +O(q^{10})\) \(q-1.53767 q^{2} +0.364440 q^{4} -1.68601 q^{7} +2.51496 q^{8} +2.97757 q^{11} -0.232892 q^{13} +2.59253 q^{14} -4.59606 q^{16} +7.45901 q^{17} +0.753527 q^{19} -4.57853 q^{22} +0.872721 q^{23} +0.358112 q^{26} -0.614448 q^{28} -6.87482 q^{29} -9.81929 q^{31} +2.03733 q^{32} -11.4695 q^{34} -10.1272 q^{37} -1.15868 q^{38} -3.79732 q^{41} +5.27322 q^{43} +1.08514 q^{44} -1.34196 q^{46} -8.56747 q^{47} -4.15738 q^{49} -0.0848751 q^{52} +5.97876 q^{53} -4.24024 q^{56} +10.5712 q^{58} +3.85114 q^{59} -4.39643 q^{61} +15.0989 q^{62} +6.05938 q^{64} -1.79282 q^{67} +2.71836 q^{68} +4.37450 q^{71} +15.0528 q^{73} +15.5723 q^{74} +0.274615 q^{76} -5.02021 q^{77} +7.37584 q^{79} +5.83904 q^{82} +4.34451 q^{83} -8.10849 q^{86} +7.48847 q^{88} -12.1032 q^{89} +0.392657 q^{91} +0.318054 q^{92} +13.1740 q^{94} -9.47426 q^{97} +6.39269 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} + 4 q^{4} - 8 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} + 4 q^{4} - 8 q^{7} + 12 q^{8} - 2 q^{11} - 16 q^{13} - 6 q^{14} + 16 q^{17} - 14 q^{19} - 12 q^{22} + 14 q^{23} - 6 q^{26} - 16 q^{28} - 2 q^{29} - 22 q^{31} - 2 q^{32} - 12 q^{34} - 28 q^{37} - 16 q^{38} - 8 q^{41} - 20 q^{43} - 22 q^{44} - 2 q^{46} + 10 q^{47} - 16 q^{52} + 44 q^{53} - 30 q^{56} - 8 q^{58} - 14 q^{59} - 20 q^{61} + 16 q^{62} + 6 q^{64} - 16 q^{67} - 2 q^{68} - 16 q^{71} - 24 q^{73} - 26 q^{74} - 16 q^{76} + 46 q^{77} - 30 q^{79} - 16 q^{82} + 12 q^{83} - 32 q^{86} - 32 q^{88} - 16 q^{89} - 12 q^{91} - 2 q^{92} + 14 q^{94} - 16 q^{97} + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.53767 −1.08730 −0.543650 0.839312i \(-0.682958\pi\)
−0.543650 + 0.839312i \(0.682958\pi\)
\(3\) 0 0
\(4\) 0.364440 0.182220
\(5\) 0 0
\(6\) 0 0
\(7\) −1.68601 −0.637251 −0.318625 0.947881i \(-0.603221\pi\)
−0.318625 + 0.947881i \(0.603221\pi\)
\(8\) 2.51496 0.889172
\(9\) 0 0
\(10\) 0 0
\(11\) 2.97757 0.897772 0.448886 0.893589i \(-0.351821\pi\)
0.448886 + 0.893589i \(0.351821\pi\)
\(12\) 0 0
\(13\) −0.232892 −0.0645926 −0.0322963 0.999478i \(-0.510282\pi\)
−0.0322963 + 0.999478i \(0.510282\pi\)
\(14\) 2.59253 0.692882
\(15\) 0 0
\(16\) −4.59606 −1.14902
\(17\) 7.45901 1.80908 0.904538 0.426392i \(-0.140216\pi\)
0.904538 + 0.426392i \(0.140216\pi\)
\(18\) 0 0
\(19\) 0.753527 0.172871 0.0864354 0.996257i \(-0.472452\pi\)
0.0864354 + 0.996257i \(0.472452\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −4.57853 −0.976146
\(23\) 0.872721 0.181975 0.0909874 0.995852i \(-0.470998\pi\)
0.0909874 + 0.995852i \(0.470998\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0.358112 0.0702315
\(27\) 0 0
\(28\) −0.614448 −0.116120
\(29\) −6.87482 −1.27662 −0.638311 0.769778i \(-0.720367\pi\)
−0.638311 + 0.769778i \(0.720367\pi\)
\(30\) 0 0
\(31\) −9.81929 −1.76360 −0.881798 0.471627i \(-0.843667\pi\)
−0.881798 + 0.471627i \(0.843667\pi\)
\(32\) 2.03733 0.360152
\(33\) 0 0
\(34\) −11.4695 −1.96701
\(35\) 0 0
\(36\) 0 0
\(37\) −10.1272 −1.66489 −0.832447 0.554105i \(-0.813061\pi\)
−0.832447 + 0.554105i \(0.813061\pi\)
\(38\) −1.15868 −0.187962
\(39\) 0 0
\(40\) 0 0
\(41\) −3.79732 −0.593042 −0.296521 0.955026i \(-0.595826\pi\)
−0.296521 + 0.955026i \(0.595826\pi\)
\(42\) 0 0
\(43\) 5.27322 0.804159 0.402079 0.915605i \(-0.368288\pi\)
0.402079 + 0.915605i \(0.368288\pi\)
\(44\) 1.08514 0.163592
\(45\) 0 0
\(46\) −1.34196 −0.197861
\(47\) −8.56747 −1.24969 −0.624847 0.780747i \(-0.714839\pi\)
−0.624847 + 0.780747i \(0.714839\pi\)
\(48\) 0 0
\(49\) −4.15738 −0.593911
\(50\) 0 0
\(51\) 0 0
\(52\) −0.0848751 −0.0117701
\(53\) 5.97876 0.821246 0.410623 0.911805i \(-0.365311\pi\)
0.410623 + 0.911805i \(0.365311\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −4.24024 −0.566625
\(57\) 0 0
\(58\) 10.5712 1.38807
\(59\) 3.85114 0.501376 0.250688 0.968068i \(-0.419343\pi\)
0.250688 + 0.968068i \(0.419343\pi\)
\(60\) 0 0
\(61\) −4.39643 −0.562905 −0.281452 0.959575i \(-0.590816\pi\)
−0.281452 + 0.959575i \(0.590816\pi\)
\(62\) 15.0989 1.91756
\(63\) 0 0
\(64\) 6.05938 0.757423
\(65\) 0 0
\(66\) 0 0
\(67\) −1.79282 −0.219028 −0.109514 0.993985i \(-0.534930\pi\)
−0.109514 + 0.993985i \(0.534930\pi\)
\(68\) 2.71836 0.329650
\(69\) 0 0
\(70\) 0 0
\(71\) 4.37450 0.519157 0.259579 0.965722i \(-0.416416\pi\)
0.259579 + 0.965722i \(0.416416\pi\)
\(72\) 0 0
\(73\) 15.0528 1.76180 0.880900 0.473303i \(-0.156938\pi\)
0.880900 + 0.473303i \(0.156938\pi\)
\(74\) 15.5723 1.81024
\(75\) 0 0
\(76\) 0.274615 0.0315005
\(77\) −5.02021 −0.572106
\(78\) 0 0
\(79\) 7.37584 0.829847 0.414924 0.909856i \(-0.363808\pi\)
0.414924 + 0.909856i \(0.363808\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 5.83904 0.644814
\(83\) 4.34451 0.476872 0.238436 0.971158i \(-0.423365\pi\)
0.238436 + 0.971158i \(0.423365\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −8.10849 −0.874361
\(87\) 0 0
\(88\) 7.48847 0.798273
\(89\) −12.1032 −1.28294 −0.641469 0.767149i \(-0.721675\pi\)
−0.641469 + 0.767149i \(0.721675\pi\)
\(90\) 0 0
\(91\) 0.392657 0.0411617
\(92\) 0.318054 0.0331594
\(93\) 0 0
\(94\) 13.1740 1.35879
\(95\) 0 0
\(96\) 0 0
\(97\) −9.47426 −0.961965 −0.480982 0.876730i \(-0.659720\pi\)
−0.480982 + 0.876730i \(0.659720\pi\)
\(98\) 6.39269 0.645760
\(99\) 0 0
\(100\) 0 0
\(101\) 6.54468 0.651220 0.325610 0.945504i \(-0.394430\pi\)
0.325610 + 0.945504i \(0.394430\pi\)
\(102\) 0 0
\(103\) 0.700804 0.0690522 0.0345261 0.999404i \(-0.489008\pi\)
0.0345261 + 0.999404i \(0.489008\pi\)
\(104\) −0.585713 −0.0574339
\(105\) 0 0
\(106\) −9.19338 −0.892940
\(107\) 12.5288 1.21120 0.605602 0.795768i \(-0.292933\pi\)
0.605602 + 0.795768i \(0.292933\pi\)
\(108\) 0 0
\(109\) 4.28348 0.410283 0.205141 0.978732i \(-0.434235\pi\)
0.205141 + 0.978732i \(0.434235\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 7.74899 0.732211
\(113\) 8.22761 0.773988 0.386994 0.922082i \(-0.373513\pi\)
0.386994 + 0.922082i \(0.373513\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −2.50546 −0.232626
\(117\) 0 0
\(118\) −5.92180 −0.545146
\(119\) −12.5760 −1.15284
\(120\) 0 0
\(121\) −2.13407 −0.194006
\(122\) 6.76027 0.612046
\(123\) 0 0
\(124\) −3.57854 −0.321362
\(125\) 0 0
\(126\) 0 0
\(127\) −11.9482 −1.06023 −0.530114 0.847926i \(-0.677851\pi\)
−0.530114 + 0.847926i \(0.677851\pi\)
\(128\) −13.3920 −1.18370
\(129\) 0 0
\(130\) 0 0
\(131\) −21.4573 −1.87474 −0.937369 0.348339i \(-0.886746\pi\)
−0.937369 + 0.348339i \(0.886746\pi\)
\(132\) 0 0
\(133\) −1.27045 −0.110162
\(134\) 2.75678 0.238149
\(135\) 0 0
\(136\) 18.7591 1.60858
\(137\) 10.0192 0.855998 0.427999 0.903779i \(-0.359219\pi\)
0.427999 + 0.903779i \(0.359219\pi\)
\(138\) 0 0
\(139\) −6.57467 −0.557656 −0.278828 0.960341i \(-0.589946\pi\)
−0.278828 + 0.960341i \(0.589946\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −6.72655 −0.564479
\(143\) −0.693452 −0.0579894
\(144\) 0 0
\(145\) 0 0
\(146\) −23.1463 −1.91560
\(147\) 0 0
\(148\) −3.69074 −0.303377
\(149\) −10.9143 −0.894132 −0.447066 0.894501i \(-0.647531\pi\)
−0.447066 + 0.894501i \(0.647531\pi\)
\(150\) 0 0
\(151\) 20.4128 1.66117 0.830584 0.556894i \(-0.188007\pi\)
0.830584 + 0.556894i \(0.188007\pi\)
\(152\) 1.89509 0.153712
\(153\) 0 0
\(154\) 7.71944 0.622050
\(155\) 0 0
\(156\) 0 0
\(157\) −3.49944 −0.279286 −0.139643 0.990202i \(-0.544595\pi\)
−0.139643 + 0.990202i \(0.544595\pi\)
\(158\) −11.3416 −0.902292
\(159\) 0 0
\(160\) 0 0
\(161\) −1.47141 −0.115964
\(162\) 0 0
\(163\) −5.97357 −0.467886 −0.233943 0.972250i \(-0.575163\pi\)
−0.233943 + 0.972250i \(0.575163\pi\)
\(164\) −1.38389 −0.108064
\(165\) 0 0
\(166\) −6.68044 −0.518503
\(167\) −2.33767 −0.180895 −0.0904473 0.995901i \(-0.528830\pi\)
−0.0904473 + 0.995901i \(0.528830\pi\)
\(168\) 0 0
\(169\) −12.9458 −0.995828
\(170\) 0 0
\(171\) 0 0
\(172\) 1.92177 0.146534
\(173\) 6.07099 0.461569 0.230784 0.973005i \(-0.425871\pi\)
0.230784 + 0.973005i \(0.425871\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −13.6851 −1.03155
\(177\) 0 0
\(178\) 18.6108 1.39494
\(179\) −11.4235 −0.853835 −0.426917 0.904291i \(-0.640401\pi\)
−0.426917 + 0.904291i \(0.640401\pi\)
\(180\) 0 0
\(181\) −7.85007 −0.583491 −0.291746 0.956496i \(-0.594236\pi\)
−0.291746 + 0.956496i \(0.594236\pi\)
\(182\) −0.603779 −0.0447551
\(183\) 0 0
\(184\) 2.19486 0.161807
\(185\) 0 0
\(186\) 0 0
\(187\) 22.2097 1.62414
\(188\) −3.12232 −0.227719
\(189\) 0 0
\(190\) 0 0
\(191\) −12.5641 −0.909110 −0.454555 0.890719i \(-0.650202\pi\)
−0.454555 + 0.890719i \(0.650202\pi\)
\(192\) 0 0
\(193\) 10.1437 0.730161 0.365081 0.930976i \(-0.381041\pi\)
0.365081 + 0.930976i \(0.381041\pi\)
\(194\) 14.5683 1.04594
\(195\) 0 0
\(196\) −1.51511 −0.108222
\(197\) −2.04703 −0.145845 −0.0729223 0.997338i \(-0.523233\pi\)
−0.0729223 + 0.997338i \(0.523233\pi\)
\(198\) 0 0
\(199\) 3.57125 0.253159 0.126580 0.991956i \(-0.459600\pi\)
0.126580 + 0.991956i \(0.459600\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −10.0636 −0.708071
\(203\) 11.5910 0.813529
\(204\) 0 0
\(205\) 0 0
\(206\) −1.07761 −0.0750805
\(207\) 0 0
\(208\) 1.07039 0.0742179
\(209\) 2.24368 0.155199
\(210\) 0 0
\(211\) 3.72643 0.256538 0.128269 0.991739i \(-0.459058\pi\)
0.128269 + 0.991739i \(0.459058\pi\)
\(212\) 2.17890 0.149647
\(213\) 0 0
\(214\) −19.2652 −1.31694
\(215\) 0 0
\(216\) 0 0
\(217\) 16.5554 1.12385
\(218\) −6.58659 −0.446100
\(219\) 0 0
\(220\) 0 0
\(221\) −1.73714 −0.116853
\(222\) 0 0
\(223\) −26.2488 −1.75775 −0.878875 0.477051i \(-0.841706\pi\)
−0.878875 + 0.477051i \(0.841706\pi\)
\(224\) −3.43495 −0.229507
\(225\) 0 0
\(226\) −12.6514 −0.841557
\(227\) −4.49111 −0.298085 −0.149043 0.988831i \(-0.547619\pi\)
−0.149043 + 0.988831i \(0.547619\pi\)
\(228\) 0 0
\(229\) −7.53935 −0.498214 −0.249107 0.968476i \(-0.580137\pi\)
−0.249107 + 0.968476i \(0.580137\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −17.2899 −1.13514
\(233\) 10.3640 0.678971 0.339485 0.940611i \(-0.389747\pi\)
0.339485 + 0.940611i \(0.389747\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 1.40351 0.0913607
\(237\) 0 0
\(238\) 19.3377 1.25348
\(239\) 18.5738 1.20144 0.600719 0.799460i \(-0.294881\pi\)
0.600719 + 0.799460i \(0.294881\pi\)
\(240\) 0 0
\(241\) −5.43543 −0.350127 −0.175063 0.984557i \(-0.556013\pi\)
−0.175063 + 0.984557i \(0.556013\pi\)
\(242\) 3.28150 0.210943
\(243\) 0 0
\(244\) −1.60223 −0.102572
\(245\) 0 0
\(246\) 0 0
\(247\) −0.175490 −0.0111662
\(248\) −24.6951 −1.56814
\(249\) 0 0
\(250\) 0 0
\(251\) −23.3577 −1.47432 −0.737162 0.675716i \(-0.763834\pi\)
−0.737162 + 0.675716i \(0.763834\pi\)
\(252\) 0 0
\(253\) 2.59859 0.163372
\(254\) 18.3724 1.15279
\(255\) 0 0
\(256\) 8.47377 0.529611
\(257\) 4.48380 0.279692 0.139846 0.990173i \(-0.455339\pi\)
0.139846 + 0.990173i \(0.455339\pi\)
\(258\) 0 0
\(259\) 17.0745 1.06095
\(260\) 0 0
\(261\) 0 0
\(262\) 32.9944 2.03840
\(263\) −7.72550 −0.476375 −0.238187 0.971219i \(-0.576553\pi\)
−0.238187 + 0.971219i \(0.576553\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 1.95354 0.119779
\(267\) 0 0
\(268\) −0.653376 −0.0399113
\(269\) −10.7394 −0.654791 −0.327396 0.944887i \(-0.606171\pi\)
−0.327396 + 0.944887i \(0.606171\pi\)
\(270\) 0 0
\(271\) −25.7126 −1.56193 −0.780964 0.624576i \(-0.785272\pi\)
−0.780964 + 0.624576i \(0.785272\pi\)
\(272\) −34.2821 −2.07866
\(273\) 0 0
\(274\) −15.4063 −0.930726
\(275\) 0 0
\(276\) 0 0
\(277\) −9.62446 −0.578278 −0.289139 0.957287i \(-0.593369\pi\)
−0.289139 + 0.957287i \(0.593369\pi\)
\(278\) 10.1097 0.606340
\(279\) 0 0
\(280\) 0 0
\(281\) 2.83275 0.168988 0.0844939 0.996424i \(-0.473073\pi\)
0.0844939 + 0.996424i \(0.473073\pi\)
\(282\) 0 0
\(283\) −23.6634 −1.40664 −0.703321 0.710873i \(-0.748300\pi\)
−0.703321 + 0.710873i \(0.748300\pi\)
\(284\) 1.59424 0.0946007
\(285\) 0 0
\(286\) 1.06630 0.0630518
\(287\) 6.40231 0.377916
\(288\) 0 0
\(289\) 38.6369 2.27276
\(290\) 0 0
\(291\) 0 0
\(292\) 5.48584 0.321035
\(293\) 23.2376 1.35755 0.678777 0.734345i \(-0.262510\pi\)
0.678777 + 0.734345i \(0.262510\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −25.4694 −1.48038
\(297\) 0 0
\(298\) 16.7826 0.972189
\(299\) −0.203250 −0.0117542
\(300\) 0 0
\(301\) −8.89069 −0.512451
\(302\) −31.3882 −1.80619
\(303\) 0 0
\(304\) −3.46326 −0.198631
\(305\) 0 0
\(306\) 0 0
\(307\) 21.8131 1.24494 0.622470 0.782643i \(-0.286129\pi\)
0.622470 + 0.782643i \(0.286129\pi\)
\(308\) −1.82956 −0.104249
\(309\) 0 0
\(310\) 0 0
\(311\) −10.0414 −0.569394 −0.284697 0.958618i \(-0.591893\pi\)
−0.284697 + 0.958618i \(0.591893\pi\)
\(312\) 0 0
\(313\) −6.05366 −0.342173 −0.171087 0.985256i \(-0.554728\pi\)
−0.171087 + 0.985256i \(0.554728\pi\)
\(314\) 5.38099 0.303667
\(315\) 0 0
\(316\) 2.68805 0.151215
\(317\) −8.49907 −0.477355 −0.238678 0.971099i \(-0.576714\pi\)
−0.238678 + 0.971099i \(0.576714\pi\)
\(318\) 0 0
\(319\) −20.4703 −1.14612
\(320\) 0 0
\(321\) 0 0
\(322\) 2.26255 0.126087
\(323\) 5.62057 0.312737
\(324\) 0 0
\(325\) 0 0
\(326\) 9.18540 0.508733
\(327\) 0 0
\(328\) −9.55010 −0.527316
\(329\) 14.4448 0.796368
\(330\) 0 0
\(331\) 16.2945 0.895624 0.447812 0.894128i \(-0.352203\pi\)
0.447812 + 0.894128i \(0.352203\pi\)
\(332\) 1.58331 0.0868955
\(333\) 0 0
\(334\) 3.59458 0.196687
\(335\) 0 0
\(336\) 0 0
\(337\) −24.6962 −1.34529 −0.672644 0.739966i \(-0.734841\pi\)
−0.672644 + 0.739966i \(0.734841\pi\)
\(338\) 19.9064 1.08276
\(339\) 0 0
\(340\) 0 0
\(341\) −29.2376 −1.58331
\(342\) 0 0
\(343\) 18.8114 1.01572
\(344\) 13.2619 0.715035
\(345\) 0 0
\(346\) −9.33520 −0.501864
\(347\) −13.9466 −0.748695 −0.374347 0.927289i \(-0.622133\pi\)
−0.374347 + 0.927289i \(0.622133\pi\)
\(348\) 0 0
\(349\) 18.4966 0.990099 0.495049 0.868865i \(-0.335150\pi\)
0.495049 + 0.868865i \(0.335150\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 6.06629 0.323334
\(353\) −0.441733 −0.0235111 −0.0117555 0.999931i \(-0.503742\pi\)
−0.0117555 + 0.999931i \(0.503742\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −4.41089 −0.233777
\(357\) 0 0
\(358\) 17.5657 0.928374
\(359\) −4.62127 −0.243901 −0.121951 0.992536i \(-0.538915\pi\)
−0.121951 + 0.992536i \(0.538915\pi\)
\(360\) 0 0
\(361\) −18.4322 −0.970116
\(362\) 12.0708 0.634429
\(363\) 0 0
\(364\) 0.143100 0.00750047
\(365\) 0 0
\(366\) 0 0
\(367\) −8.77696 −0.458153 −0.229077 0.973408i \(-0.573571\pi\)
−0.229077 + 0.973408i \(0.573571\pi\)
\(368\) −4.01108 −0.209092
\(369\) 0 0
\(370\) 0 0
\(371\) −10.0802 −0.523339
\(372\) 0 0
\(373\) 15.4955 0.802328 0.401164 0.916006i \(-0.368606\pi\)
0.401164 + 0.916006i \(0.368606\pi\)
\(374\) −34.1513 −1.76592
\(375\) 0 0
\(376\) −21.5468 −1.11119
\(377\) 1.60109 0.0824604
\(378\) 0 0
\(379\) −27.2931 −1.40195 −0.700977 0.713184i \(-0.747252\pi\)
−0.700977 + 0.713184i \(0.747252\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 19.3196 0.988474
\(383\) −13.1042 −0.669592 −0.334796 0.942291i \(-0.608667\pi\)
−0.334796 + 0.942291i \(0.608667\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −15.5977 −0.793904
\(387\) 0 0
\(388\) −3.45279 −0.175289
\(389\) −16.9056 −0.857147 −0.428574 0.903507i \(-0.640984\pi\)
−0.428574 + 0.903507i \(0.640984\pi\)
\(390\) 0 0
\(391\) 6.50964 0.329207
\(392\) −10.4556 −0.528089
\(393\) 0 0
\(394\) 3.14766 0.158577
\(395\) 0 0
\(396\) 0 0
\(397\) −10.4078 −0.522353 −0.261177 0.965291i \(-0.584110\pi\)
−0.261177 + 0.965291i \(0.584110\pi\)
\(398\) −5.49142 −0.275260
\(399\) 0 0
\(400\) 0 0
\(401\) 0.694800 0.0346967 0.0173483 0.999850i \(-0.494478\pi\)
0.0173483 + 0.999850i \(0.494478\pi\)
\(402\) 0 0
\(403\) 2.28683 0.113915
\(404\) 2.38514 0.118665
\(405\) 0 0
\(406\) −17.8232 −0.884549
\(407\) −30.1543 −1.49469
\(408\) 0 0
\(409\) −1.18910 −0.0587972 −0.0293986 0.999568i \(-0.509359\pi\)
−0.0293986 + 0.999568i \(0.509359\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0.255401 0.0125827
\(413\) −6.49306 −0.319502
\(414\) 0 0
\(415\) 0 0
\(416\) −0.474477 −0.0232632
\(417\) 0 0
\(418\) −3.45005 −0.168747
\(419\) −1.97841 −0.0966515 −0.0483258 0.998832i \(-0.515389\pi\)
−0.0483258 + 0.998832i \(0.515389\pi\)
\(420\) 0 0
\(421\) 11.0825 0.540126 0.270063 0.962843i \(-0.412955\pi\)
0.270063 + 0.962843i \(0.412955\pi\)
\(422\) −5.73003 −0.278933
\(423\) 0 0
\(424\) 15.0363 0.730229
\(425\) 0 0
\(426\) 0 0
\(427\) 7.41241 0.358712
\(428\) 4.56599 0.220705
\(429\) 0 0
\(430\) 0 0
\(431\) 10.4137 0.501608 0.250804 0.968038i \(-0.419305\pi\)
0.250804 + 0.968038i \(0.419305\pi\)
\(432\) 0 0
\(433\) −10.2643 −0.493272 −0.246636 0.969108i \(-0.579325\pi\)
−0.246636 + 0.969108i \(0.579325\pi\)
\(434\) −25.4568 −1.22196
\(435\) 0 0
\(436\) 1.56107 0.0747616
\(437\) 0.657618 0.0314582
\(438\) 0 0
\(439\) −24.7115 −1.17942 −0.589708 0.807616i \(-0.700757\pi\)
−0.589708 + 0.807616i \(0.700757\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 2.67116 0.127054
\(443\) −7.52935 −0.357730 −0.178865 0.983874i \(-0.557243\pi\)
−0.178865 + 0.983874i \(0.557243\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 40.3621 1.91120
\(447\) 0 0
\(448\) −10.2162 −0.482668
\(449\) 31.6627 1.49426 0.747128 0.664681i \(-0.231432\pi\)
0.747128 + 0.664681i \(0.231432\pi\)
\(450\) 0 0
\(451\) −11.3068 −0.532416
\(452\) 2.99847 0.141036
\(453\) 0 0
\(454\) 6.90585 0.324108
\(455\) 0 0
\(456\) 0 0
\(457\) 2.95742 0.138342 0.0691712 0.997605i \(-0.477965\pi\)
0.0691712 + 0.997605i \(0.477965\pi\)
\(458\) 11.5931 0.541708
\(459\) 0 0
\(460\) 0 0
\(461\) 17.6011 0.819765 0.409883 0.912138i \(-0.365570\pi\)
0.409883 + 0.912138i \(0.365570\pi\)
\(462\) 0 0
\(463\) −26.3421 −1.22422 −0.612110 0.790773i \(-0.709679\pi\)
−0.612110 + 0.790773i \(0.709679\pi\)
\(464\) 31.5971 1.46686
\(465\) 0 0
\(466\) −15.9365 −0.738244
\(467\) 7.35906 0.340537 0.170268 0.985398i \(-0.445537\pi\)
0.170268 + 0.985398i \(0.445537\pi\)
\(468\) 0 0
\(469\) 3.02271 0.139576
\(470\) 0 0
\(471\) 0 0
\(472\) 9.68547 0.445810
\(473\) 15.7014 0.721951
\(474\) 0 0
\(475\) 0 0
\(476\) −4.58317 −0.210069
\(477\) 0 0
\(478\) −28.5604 −1.30632
\(479\) 28.4670 1.30069 0.650346 0.759638i \(-0.274624\pi\)
0.650346 + 0.759638i \(0.274624\pi\)
\(480\) 0 0
\(481\) 2.35853 0.107540
\(482\) 8.35791 0.380692
\(483\) 0 0
\(484\) −0.777739 −0.0353518
\(485\) 0 0
\(486\) 0 0
\(487\) −2.38406 −0.108032 −0.0540161 0.998540i \(-0.517202\pi\)
−0.0540161 + 0.998540i \(0.517202\pi\)
\(488\) −11.0568 −0.500519
\(489\) 0 0
\(490\) 0 0
\(491\) 25.7231 1.16087 0.580434 0.814307i \(-0.302883\pi\)
0.580434 + 0.814307i \(0.302883\pi\)
\(492\) 0 0
\(493\) −51.2794 −2.30951
\(494\) 0.269847 0.0121410
\(495\) 0 0
\(496\) 45.1301 2.02640
\(497\) −7.37543 −0.330833
\(498\) 0 0
\(499\) 29.9989 1.34293 0.671467 0.741035i \(-0.265665\pi\)
0.671467 + 0.741035i \(0.265665\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 35.9165 1.60303
\(503\) −13.8250 −0.616426 −0.308213 0.951317i \(-0.599731\pi\)
−0.308213 + 0.951317i \(0.599731\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −3.99578 −0.177634
\(507\) 0 0
\(508\) −4.35439 −0.193195
\(509\) −31.9537 −1.41632 −0.708161 0.706051i \(-0.750475\pi\)
−0.708161 + 0.706051i \(0.750475\pi\)
\(510\) 0 0
\(511\) −25.3792 −1.12271
\(512\) 13.7541 0.607852
\(513\) 0 0
\(514\) −6.89462 −0.304109
\(515\) 0 0
\(516\) 0 0
\(517\) −25.5102 −1.12194
\(518\) −26.2549 −1.15358
\(519\) 0 0
\(520\) 0 0
\(521\) 38.7968 1.69972 0.849859 0.527010i \(-0.176687\pi\)
0.849859 + 0.527010i \(0.176687\pi\)
\(522\) 0 0
\(523\) −23.7143 −1.03695 −0.518477 0.855091i \(-0.673501\pi\)
−0.518477 + 0.855091i \(0.673501\pi\)
\(524\) −7.81991 −0.341614
\(525\) 0 0
\(526\) 11.8793 0.517962
\(527\) −73.2422 −3.19048
\(528\) 0 0
\(529\) −22.2384 −0.966885
\(530\) 0 0
\(531\) 0 0
\(532\) −0.463003 −0.0200737
\(533\) 0.884365 0.0383061
\(534\) 0 0
\(535\) 0 0
\(536\) −4.50888 −0.194754
\(537\) 0 0
\(538\) 16.5137 0.711954
\(539\) −12.3789 −0.533197
\(540\) 0 0
\(541\) 2.81765 0.121140 0.0605702 0.998164i \(-0.480708\pi\)
0.0605702 + 0.998164i \(0.480708\pi\)
\(542\) 39.5376 1.69828
\(543\) 0 0
\(544\) 15.1965 0.651543
\(545\) 0 0
\(546\) 0 0
\(547\) 4.42379 0.189148 0.0945739 0.995518i \(-0.469851\pi\)
0.0945739 + 0.995518i \(0.469851\pi\)
\(548\) 3.65139 0.155980
\(549\) 0 0
\(550\) 0 0
\(551\) −5.18036 −0.220691
\(552\) 0 0
\(553\) −12.4357 −0.528821
\(554\) 14.7993 0.628761
\(555\) 0 0
\(556\) −2.39607 −0.101616
\(557\) 7.20182 0.305151 0.152575 0.988292i \(-0.451243\pi\)
0.152575 + 0.988292i \(0.451243\pi\)
\(558\) 0 0
\(559\) −1.22809 −0.0519427
\(560\) 0 0
\(561\) 0 0
\(562\) −4.35585 −0.183740
\(563\) −23.2509 −0.979907 −0.489954 0.871749i \(-0.662986\pi\)
−0.489954 + 0.871749i \(0.662986\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 36.3865 1.52944
\(567\) 0 0
\(568\) 11.0017 0.461620
\(569\) 20.6237 0.864589 0.432295 0.901732i \(-0.357704\pi\)
0.432295 + 0.901732i \(0.357704\pi\)
\(570\) 0 0
\(571\) 1.80372 0.0754834 0.0377417 0.999288i \(-0.487984\pi\)
0.0377417 + 0.999288i \(0.487984\pi\)
\(572\) −0.252722 −0.0105668
\(573\) 0 0
\(574\) −9.84466 −0.410908
\(575\) 0 0
\(576\) 0 0
\(577\) 16.4863 0.686335 0.343168 0.939274i \(-0.388500\pi\)
0.343168 + 0.939274i \(0.388500\pi\)
\(578\) −59.4109 −2.47117
\(579\) 0 0
\(580\) 0 0
\(581\) −7.32488 −0.303887
\(582\) 0 0
\(583\) 17.8022 0.737291
\(584\) 37.8572 1.56654
\(585\) 0 0
\(586\) −35.7318 −1.47607
\(587\) −7.27161 −0.300131 −0.150066 0.988676i \(-0.547949\pi\)
−0.150066 + 0.988676i \(0.547949\pi\)
\(588\) 0 0
\(589\) −7.39909 −0.304874
\(590\) 0 0
\(591\) 0 0
\(592\) 46.5450 1.91299
\(593\) 2.47898 0.101800 0.0508998 0.998704i \(-0.483791\pi\)
0.0508998 + 0.998704i \(0.483791\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −3.97759 −0.162928
\(597\) 0 0
\(598\) 0.312532 0.0127804
\(599\) −30.2951 −1.23782 −0.618912 0.785460i \(-0.712426\pi\)
−0.618912 + 0.785460i \(0.712426\pi\)
\(600\) 0 0
\(601\) 4.46130 0.181980 0.0909900 0.995852i \(-0.470997\pi\)
0.0909900 + 0.995852i \(0.470997\pi\)
\(602\) 13.6710 0.557187
\(603\) 0 0
\(604\) 7.43922 0.302698
\(605\) 0 0
\(606\) 0 0
\(607\) −17.2931 −0.701906 −0.350953 0.936393i \(-0.614142\pi\)
−0.350953 + 0.936393i \(0.614142\pi\)
\(608\) 1.53518 0.0622598
\(609\) 0 0
\(610\) 0 0
\(611\) 1.99529 0.0807210
\(612\) 0 0
\(613\) −14.3129 −0.578094 −0.289047 0.957315i \(-0.593338\pi\)
−0.289047 + 0.957315i \(0.593338\pi\)
\(614\) −33.5415 −1.35362
\(615\) 0 0
\(616\) −12.6256 −0.508700
\(617\) −26.3569 −1.06109 −0.530544 0.847658i \(-0.678012\pi\)
−0.530544 + 0.847658i \(0.678012\pi\)
\(618\) 0 0
\(619\) −2.79825 −0.112471 −0.0562356 0.998418i \(-0.517910\pi\)
−0.0562356 + 0.998418i \(0.517910\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 15.4403 0.619101
\(623\) 20.4061 0.817553
\(624\) 0 0
\(625\) 0 0
\(626\) 9.30856 0.372045
\(627\) 0 0
\(628\) −1.27533 −0.0508914
\(629\) −75.5386 −3.01192
\(630\) 0 0
\(631\) 35.4035 1.40939 0.704696 0.709510i \(-0.251084\pi\)
0.704696 + 0.709510i \(0.251084\pi\)
\(632\) 18.5499 0.737877
\(633\) 0 0
\(634\) 13.0688 0.519028
\(635\) 0 0
\(636\) 0 0
\(637\) 0.968220 0.0383623
\(638\) 31.4766 1.24617
\(639\) 0 0
\(640\) 0 0
\(641\) −16.9334 −0.668829 −0.334415 0.942426i \(-0.608539\pi\)
−0.334415 + 0.942426i \(0.608539\pi\)
\(642\) 0 0
\(643\) −25.9118 −1.02186 −0.510931 0.859622i \(-0.670699\pi\)
−0.510931 + 0.859622i \(0.670699\pi\)
\(644\) −0.536241 −0.0211309
\(645\) 0 0
\(646\) −8.64260 −0.340038
\(647\) 10.9259 0.429542 0.214771 0.976664i \(-0.431100\pi\)
0.214771 + 0.976664i \(0.431100\pi\)
\(648\) 0 0
\(649\) 11.4671 0.450121
\(650\) 0 0
\(651\) 0 0
\(652\) −2.17701 −0.0852582
\(653\) 14.2580 0.557958 0.278979 0.960297i \(-0.410004\pi\)
0.278979 + 0.960297i \(0.410004\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 17.4527 0.681414
\(657\) 0 0
\(658\) −22.2114 −0.865890
\(659\) −22.8894 −0.891644 −0.445822 0.895122i \(-0.647088\pi\)
−0.445822 + 0.895122i \(0.647088\pi\)
\(660\) 0 0
\(661\) −39.9050 −1.55213 −0.776063 0.630656i \(-0.782786\pi\)
−0.776063 + 0.630656i \(0.782786\pi\)
\(662\) −25.0555 −0.973811
\(663\) 0 0
\(664\) 10.9263 0.424021
\(665\) 0 0
\(666\) 0 0
\(667\) −5.99980 −0.232313
\(668\) −0.851941 −0.0329626
\(669\) 0 0
\(670\) 0 0
\(671\) −13.0907 −0.505360
\(672\) 0 0
\(673\) 2.15847 0.0832029 0.0416015 0.999134i \(-0.486754\pi\)
0.0416015 + 0.999134i \(0.486754\pi\)
\(674\) 37.9747 1.46273
\(675\) 0 0
\(676\) −4.71795 −0.181460
\(677\) 41.9740 1.61319 0.806596 0.591103i \(-0.201307\pi\)
0.806596 + 0.591103i \(0.201307\pi\)
\(678\) 0 0
\(679\) 15.9737 0.613013
\(680\) 0 0
\(681\) 0 0
\(682\) 44.9579 1.72153
\(683\) −29.2062 −1.11754 −0.558772 0.829321i \(-0.688727\pi\)
−0.558772 + 0.829321i \(0.688727\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −28.9258 −1.10439
\(687\) 0 0
\(688\) −24.2361 −0.923991
\(689\) −1.39240 −0.0530464
\(690\) 0 0
\(691\) 44.5257 1.69384 0.846919 0.531721i \(-0.178455\pi\)
0.846919 + 0.531721i \(0.178455\pi\)
\(692\) 2.21251 0.0841070
\(693\) 0 0
\(694\) 21.4454 0.814055
\(695\) 0 0
\(696\) 0 0
\(697\) −28.3243 −1.07286
\(698\) −28.4417 −1.07653
\(699\) 0 0
\(700\) 0 0
\(701\) 4.50567 0.170177 0.0850884 0.996373i \(-0.472883\pi\)
0.0850884 + 0.996373i \(0.472883\pi\)
\(702\) 0 0
\(703\) −7.63108 −0.287812
\(704\) 18.0422 0.679992
\(705\) 0 0
\(706\) 0.679241 0.0255636
\(707\) −11.0344 −0.414990
\(708\) 0 0
\(709\) −51.2706 −1.92551 −0.962754 0.270380i \(-0.912851\pi\)
−0.962754 + 0.270380i \(0.912851\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −30.4390 −1.14075
\(713\) −8.56950 −0.320930
\(714\) 0 0
\(715\) 0 0
\(716\) −4.16319 −0.155586
\(717\) 0 0
\(718\) 7.10600 0.265194
\(719\) −35.2653 −1.31517 −0.657586 0.753379i \(-0.728422\pi\)
−0.657586 + 0.753379i \(0.728422\pi\)
\(720\) 0 0
\(721\) −1.18156 −0.0440036
\(722\) 28.3427 1.05481
\(723\) 0 0
\(724\) −2.86088 −0.106324
\(725\) 0 0
\(726\) 0 0
\(727\) −44.0038 −1.63201 −0.816005 0.578045i \(-0.803816\pi\)
−0.816005 + 0.578045i \(0.803816\pi\)
\(728\) 0.987517 0.0365998
\(729\) 0 0
\(730\) 0 0
\(731\) 39.3330 1.45478
\(732\) 0 0
\(733\) 27.8741 1.02955 0.514776 0.857325i \(-0.327875\pi\)
0.514776 + 0.857325i \(0.327875\pi\)
\(734\) 13.4961 0.498150
\(735\) 0 0
\(736\) 1.77802 0.0655386
\(737\) −5.33826 −0.196637
\(738\) 0 0
\(739\) −32.4413 −1.19337 −0.596687 0.802474i \(-0.703516\pi\)
−0.596687 + 0.802474i \(0.703516\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 15.5001 0.569027
\(743\) 9.09256 0.333574 0.166787 0.985993i \(-0.446661\pi\)
0.166787 + 0.985993i \(0.446661\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −23.8271 −0.872370
\(747\) 0 0
\(748\) 8.09411 0.295950
\(749\) −21.1236 −0.771840
\(750\) 0 0
\(751\) −49.8861 −1.82037 −0.910185 0.414202i \(-0.864061\pi\)
−0.910185 + 0.414202i \(0.864061\pi\)
\(752\) 39.3766 1.43592
\(753\) 0 0
\(754\) −2.46196 −0.0896591
\(755\) 0 0
\(756\) 0 0
\(757\) −31.1239 −1.13122 −0.565608 0.824674i \(-0.691358\pi\)
−0.565608 + 0.824674i \(0.691358\pi\)
\(758\) 41.9679 1.52434
\(759\) 0 0
\(760\) 0 0
\(761\) −39.1269 −1.41835 −0.709175 0.705032i \(-0.750933\pi\)
−0.709175 + 0.705032i \(0.750933\pi\)
\(762\) 0 0
\(763\) −7.22197 −0.261453
\(764\) −4.57887 −0.165658
\(765\) 0 0
\(766\) 20.1499 0.728047
\(767\) −0.896901 −0.0323852
\(768\) 0 0
\(769\) 14.3382 0.517050 0.258525 0.966005i \(-0.416764\pi\)
0.258525 + 0.966005i \(0.416764\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 3.69677 0.133050
\(773\) −4.24997 −0.152861 −0.0764304 0.997075i \(-0.524352\pi\)
−0.0764304 + 0.997075i \(0.524352\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −23.8274 −0.855352
\(777\) 0 0
\(778\) 25.9953 0.931975
\(779\) −2.86138 −0.102520
\(780\) 0 0
\(781\) 13.0254 0.466085
\(782\) −10.0097 −0.357946
\(783\) 0 0
\(784\) 19.1076 0.682414
\(785\) 0 0
\(786\) 0 0
\(787\) 27.1296 0.967067 0.483534 0.875326i \(-0.339353\pi\)
0.483534 + 0.875326i \(0.339353\pi\)
\(788\) −0.746017 −0.0265758
\(789\) 0 0
\(790\) 0 0
\(791\) −13.8718 −0.493225
\(792\) 0 0
\(793\) 1.02389 0.0363595
\(794\) 16.0038 0.567954
\(795\) 0 0
\(796\) 1.30151 0.0461306
\(797\) −12.6175 −0.446935 −0.223468 0.974711i \(-0.571738\pi\)
−0.223468 + 0.974711i \(0.571738\pi\)
\(798\) 0 0
\(799\) −63.9049 −2.26079
\(800\) 0 0
\(801\) 0 0
\(802\) −1.06838 −0.0377257
\(803\) 44.8209 1.58169
\(804\) 0 0
\(805\) 0 0
\(806\) −3.51640 −0.123860
\(807\) 0 0
\(808\) 16.4596 0.579047
\(809\) 37.4138 1.31540 0.657699 0.753280i \(-0.271530\pi\)
0.657699 + 0.753280i \(0.271530\pi\)
\(810\) 0 0
\(811\) 37.1992 1.30624 0.653120 0.757254i \(-0.273460\pi\)
0.653120 + 0.757254i \(0.273460\pi\)
\(812\) 4.22422 0.148241
\(813\) 0 0
\(814\) 46.3675 1.62518
\(815\) 0 0
\(816\) 0 0
\(817\) 3.97351 0.139016
\(818\) 1.82845 0.0639302
\(819\) 0 0
\(820\) 0 0
\(821\) −27.4740 −0.958850 −0.479425 0.877583i \(-0.659155\pi\)
−0.479425 + 0.877583i \(0.659155\pi\)
\(822\) 0 0
\(823\) 23.4267 0.816602 0.408301 0.912847i \(-0.366121\pi\)
0.408301 + 0.912847i \(0.366121\pi\)
\(824\) 1.76249 0.0613993
\(825\) 0 0
\(826\) 9.98420 0.347395
\(827\) 32.4570 1.12864 0.564320 0.825556i \(-0.309138\pi\)
0.564320 + 0.825556i \(0.309138\pi\)
\(828\) 0 0
\(829\) −0.154106 −0.00535231 −0.00267615 0.999996i \(-0.500852\pi\)
−0.00267615 + 0.999996i \(0.500852\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −1.41118 −0.0489239
\(833\) −31.0100 −1.07443
\(834\) 0 0
\(835\) 0 0
\(836\) 0.817686 0.0282802
\(837\) 0 0
\(838\) 3.04214 0.105089
\(839\) 35.4217 1.22289 0.611447 0.791285i \(-0.290588\pi\)
0.611447 + 0.791285i \(0.290588\pi\)
\(840\) 0 0
\(841\) 18.2632 0.629765
\(842\) −17.0412 −0.587279
\(843\) 0 0
\(844\) 1.35806 0.0467462
\(845\) 0 0
\(846\) 0 0
\(847\) 3.59806 0.123631
\(848\) −27.4788 −0.943624
\(849\) 0 0
\(850\) 0 0
\(851\) −8.83818 −0.302969
\(852\) 0 0
\(853\) −36.6066 −1.25339 −0.626694 0.779266i \(-0.715592\pi\)
−0.626694 + 0.779266i \(0.715592\pi\)
\(854\) −11.3979 −0.390027
\(855\) 0 0
\(856\) 31.5094 1.07697
\(857\) 2.04867 0.0699813 0.0349907 0.999388i \(-0.488860\pi\)
0.0349907 + 0.999388i \(0.488860\pi\)
\(858\) 0 0
\(859\) 14.1821 0.483888 0.241944 0.970290i \(-0.422215\pi\)
0.241944 + 0.970290i \(0.422215\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −16.0128 −0.545398
\(863\) −43.1358 −1.46836 −0.734180 0.678955i \(-0.762433\pi\)
−0.734180 + 0.678955i \(0.762433\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 15.7832 0.536334
\(867\) 0 0
\(868\) 6.03344 0.204788
\(869\) 21.9621 0.745013
\(870\) 0 0
\(871\) 0.417534 0.0141476
\(872\) 10.7728 0.364812
\(873\) 0 0
\(874\) −1.01120 −0.0342044
\(875\) 0 0
\(876\) 0 0
\(877\) −6.74680 −0.227823 −0.113912 0.993491i \(-0.536338\pi\)
−0.113912 + 0.993491i \(0.536338\pi\)
\(878\) 37.9983 1.28238
\(879\) 0 0
\(880\) 0 0
\(881\) −17.1783 −0.578752 −0.289376 0.957216i \(-0.593448\pi\)
−0.289376 + 0.957216i \(0.593448\pi\)
\(882\) 0 0
\(883\) −56.8617 −1.91355 −0.956774 0.290831i \(-0.906068\pi\)
−0.956774 + 0.290831i \(0.906068\pi\)
\(884\) −0.633084 −0.0212929
\(885\) 0 0
\(886\) 11.5777 0.388960
\(887\) −36.9297 −1.23998 −0.619989 0.784610i \(-0.712863\pi\)
−0.619989 + 0.784610i \(0.712863\pi\)
\(888\) 0 0
\(889\) 20.1447 0.675632
\(890\) 0 0
\(891\) 0 0
\(892\) −9.56611 −0.320297
\(893\) −6.45581 −0.216036
\(894\) 0 0
\(895\) 0 0
\(896\) 22.5790 0.754312
\(897\) 0 0
\(898\) −48.6869 −1.62470
\(899\) 67.5058 2.25145
\(900\) 0 0
\(901\) 44.5956 1.48570
\(902\) 17.3862 0.578895
\(903\) 0 0
\(904\) 20.6921 0.688209
\(905\) 0 0
\(906\) 0 0
\(907\) −25.7833 −0.856120 −0.428060 0.903750i \(-0.640803\pi\)
−0.428060 + 0.903750i \(0.640803\pi\)
\(908\) −1.63674 −0.0543170
\(909\) 0 0
\(910\) 0 0
\(911\) 27.0066 0.894768 0.447384 0.894342i \(-0.352356\pi\)
0.447384 + 0.894342i \(0.352356\pi\)
\(912\) 0 0
\(913\) 12.9361 0.428122
\(914\) −4.54755 −0.150420
\(915\) 0 0
\(916\) −2.74764 −0.0907845
\(917\) 36.1772 1.19468
\(918\) 0 0
\(919\) −25.9342 −0.855492 −0.427746 0.903899i \(-0.640692\pi\)
−0.427746 + 0.903899i \(0.640692\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −27.0647 −0.891330
\(923\) −1.01879 −0.0335337
\(924\) 0 0
\(925\) 0 0
\(926\) 40.5055 1.33109
\(927\) 0 0
\(928\) −14.0063 −0.459778
\(929\) −15.0866 −0.494974 −0.247487 0.968891i \(-0.579605\pi\)
−0.247487 + 0.968891i \(0.579605\pi\)
\(930\) 0 0
\(931\) −3.13270 −0.102670
\(932\) 3.77707 0.123722
\(933\) 0 0
\(934\) −11.3158 −0.370265
\(935\) 0 0
\(936\) 0 0
\(937\) 0.655563 0.0214163 0.0107082 0.999943i \(-0.496591\pi\)
0.0107082 + 0.999943i \(0.496591\pi\)
\(938\) −4.64795 −0.151761
\(939\) 0 0
\(940\) 0 0
\(941\) 36.4140 1.18706 0.593531 0.804811i \(-0.297733\pi\)
0.593531 + 0.804811i \(0.297733\pi\)
\(942\) 0 0
\(943\) −3.31400 −0.107919
\(944\) −17.7001 −0.576089
\(945\) 0 0
\(946\) −24.1436 −0.784976
\(947\) 45.4127 1.47571 0.737857 0.674957i \(-0.235838\pi\)
0.737857 + 0.674957i \(0.235838\pi\)
\(948\) 0 0
\(949\) −3.50568 −0.113799
\(950\) 0 0
\(951\) 0 0
\(952\) −31.6280 −1.02507
\(953\) −46.2536 −1.49830 −0.749151 0.662399i \(-0.769538\pi\)
−0.749151 + 0.662399i \(0.769538\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 6.76903 0.218926
\(957\) 0 0
\(958\) −43.7730 −1.41424
\(959\) −16.8924 −0.545485
\(960\) 0 0
\(961\) 65.4184 2.11027
\(962\) −3.62665 −0.116928
\(963\) 0 0
\(964\) −1.98088 −0.0638000
\(965\) 0 0
\(966\) 0 0
\(967\) 2.91740 0.0938172 0.0469086 0.998899i \(-0.485063\pi\)
0.0469086 + 0.998899i \(0.485063\pi\)
\(968\) −5.36709 −0.172505
\(969\) 0 0
\(970\) 0 0
\(971\) 1.21820 0.0390940 0.0195470 0.999809i \(-0.493778\pi\)
0.0195470 + 0.999809i \(0.493778\pi\)
\(972\) 0 0
\(973\) 11.0849 0.355367
\(974\) 3.66591 0.117463
\(975\) 0 0
\(976\) 20.2063 0.646787
\(977\) 44.6720 1.42918 0.714591 0.699542i \(-0.246613\pi\)
0.714591 + 0.699542i \(0.246613\pi\)
\(978\) 0 0
\(979\) −36.0382 −1.15178
\(980\) 0 0
\(981\) 0 0
\(982\) −39.5538 −1.26221
\(983\) −36.1310 −1.15240 −0.576201 0.817308i \(-0.695465\pi\)
−0.576201 + 0.817308i \(0.695465\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 78.8510 2.51113
\(987\) 0 0
\(988\) −0.0639556 −0.00203470
\(989\) 4.60205 0.146337
\(990\) 0 0
\(991\) −40.0195 −1.27126 −0.635631 0.771993i \(-0.719260\pi\)
−0.635631 + 0.771993i \(0.719260\pi\)
\(992\) −20.0051 −0.635163
\(993\) 0 0
\(994\) 11.3410 0.359715
\(995\) 0 0
\(996\) 0 0
\(997\) −3.24338 −0.102719 −0.0513595 0.998680i \(-0.516355\pi\)
−0.0513595 + 0.998680i \(0.516355\pi\)
\(998\) −46.1285 −1.46017
\(999\) 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 5625.2.a.bd.1.1 8
3.2 odd 2 1875.2.a.m.1.8 8
5.4 even 2 5625.2.a.t.1.8 8
15.2 even 4 1875.2.b.h.1249.14 16
15.8 even 4 1875.2.b.h.1249.3 16
15.14 odd 2 1875.2.a.p.1.1 8
25.3 odd 20 225.2.m.b.109.4 16
25.17 odd 20 225.2.m.b.64.4 16
75.8 even 20 375.2.i.c.199.4 16
75.17 even 20 75.2.i.a.64.1 yes 16
75.29 odd 10 375.2.g.d.76.4 16
75.44 odd 10 375.2.g.d.301.4 16
75.47 even 20 375.2.i.c.49.4 16
75.53 even 20 75.2.i.a.34.1 16
75.56 odd 10 375.2.g.e.301.1 16
75.71 odd 10 375.2.g.e.76.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.i.a.34.1 16 75.53 even 20
75.2.i.a.64.1 yes 16 75.17 even 20
225.2.m.b.64.4 16 25.17 odd 20
225.2.m.b.109.4 16 25.3 odd 20
375.2.g.d.76.4 16 75.29 odd 10
375.2.g.d.301.4 16 75.44 odd 10
375.2.g.e.76.1 16 75.71 odd 10
375.2.g.e.301.1 16 75.56 odd 10
375.2.i.c.49.4 16 75.47 even 20
375.2.i.c.199.4 16 75.8 even 20
1875.2.a.m.1.8 8 3.2 odd 2
1875.2.a.p.1.1 8 15.14 odd 2
1875.2.b.h.1249.3 16 15.8 even 4
1875.2.b.h.1249.14 16 15.2 even 4
5625.2.a.t.1.8 8 5.4 even 2
5625.2.a.bd.1.1 8 1.1 even 1 trivial