Properties

Label 5625.2.a.be.1.8
Level $5625$
Weight $2$
Character 5625.1
Self dual yes
Analytic conductor $44.916$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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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: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.6152203125.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} - 8x^{6} + 20x^{5} + 26x^{4} - 35x^{3} - 27x^{2} + 16x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 625)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-1.66501\) 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)
 
\(f(q)\) \(=\) \(q+2.66501 q^{2} +5.10229 q^{4} +2.04213 q^{7} +8.26764 q^{8} +O(q^{10})\) \(q+2.66501 q^{2} +5.10229 q^{4} +2.04213 q^{7} +8.26764 q^{8} -1.34867 q^{11} -1.31964 q^{13} +5.44230 q^{14} +11.8288 q^{16} +4.08717 q^{17} -4.88680 q^{19} -3.59423 q^{22} -2.73618 q^{23} -3.51685 q^{26} +10.4195 q^{28} +4.61914 q^{29} +7.15727 q^{31} +14.9886 q^{32} +10.8924 q^{34} +8.64640 q^{37} -13.0234 q^{38} +10.0996 q^{41} -2.43460 q^{43} -6.88131 q^{44} -7.29194 q^{46} +7.57192 q^{47} -2.82971 q^{49} -6.73317 q^{52} -0.621245 q^{53} +16.8836 q^{56} +12.3101 q^{58} +11.3412 q^{59} +0.647513 q^{61} +19.0742 q^{62} +16.2872 q^{64} -10.9389 q^{67} +20.8539 q^{68} +2.95230 q^{71} -13.5390 q^{73} +23.0428 q^{74} -24.9339 q^{76} -2.75416 q^{77} +1.88929 q^{79} +26.9155 q^{82} +2.37481 q^{83} -6.48823 q^{86} -11.1503 q^{88} -7.33882 q^{89} -2.69487 q^{91} -13.9608 q^{92} +20.1793 q^{94} -5.79915 q^{97} -7.54121 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 5 q^{2} + 11 q^{4} - 10 q^{7} + 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 5 q^{2} + 11 q^{4} - 10 q^{7} + 15 q^{8} - q^{11} - 10 q^{13} + 8 q^{14} + 13 q^{16} + 15 q^{17} - 10 q^{19} + 5 q^{22} + 30 q^{23} - 11 q^{26} + 5 q^{28} - 10 q^{29} - 9 q^{31} + 30 q^{32} + 7 q^{34} + 10 q^{37} + 20 q^{38} + 4 q^{41} + 18 q^{44} - 9 q^{46} + 30 q^{47} - 4 q^{49} - 5 q^{52} + 10 q^{53} + 30 q^{58} + 5 q^{59} + 6 q^{61} + 10 q^{62} - 9 q^{64} - 10 q^{67} + 40 q^{68} + 9 q^{71} + 18 q^{74} - 10 q^{76} + 5 q^{77} - 20 q^{79} + 45 q^{82} + 40 q^{83} + 24 q^{86} + 40 q^{88} + 5 q^{89} + 6 q^{91} + 15 q^{92} + 47 q^{94} - 30 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.66501 1.88445 0.942224 0.334983i \(-0.108731\pi\)
0.942224 + 0.334983i \(0.108731\pi\)
\(3\) 0 0
\(4\) 5.10229 2.55115
\(5\) 0 0
\(6\) 0 0
\(7\) 2.04213 0.771852 0.385926 0.922530i \(-0.373882\pi\)
0.385926 + 0.922530i \(0.373882\pi\)
\(8\) 8.26764 2.92305
\(9\) 0 0
\(10\) 0 0
\(11\) −1.34867 −0.406640 −0.203320 0.979112i \(-0.565173\pi\)
−0.203320 + 0.979112i \(0.565173\pi\)
\(12\) 0 0
\(13\) −1.31964 −0.366002 −0.183001 0.983113i \(-0.558581\pi\)
−0.183001 + 0.983113i \(0.558581\pi\)
\(14\) 5.44230 1.45452
\(15\) 0 0
\(16\) 11.8288 2.95720
\(17\) 4.08717 0.991285 0.495643 0.868527i \(-0.334933\pi\)
0.495643 + 0.868527i \(0.334933\pi\)
\(18\) 0 0
\(19\) −4.88680 −1.12111 −0.560554 0.828118i \(-0.689412\pi\)
−0.560554 + 0.828118i \(0.689412\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −3.59423 −0.766292
\(23\) −2.73618 −0.570532 −0.285266 0.958448i \(-0.592082\pi\)
−0.285266 + 0.958448i \(0.592082\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −3.51685 −0.689711
\(27\) 0 0
\(28\) 10.4195 1.96911
\(29\) 4.61914 0.857753 0.428876 0.903363i \(-0.358910\pi\)
0.428876 + 0.903363i \(0.358910\pi\)
\(30\) 0 0
\(31\) 7.15727 1.28548 0.642742 0.766083i \(-0.277797\pi\)
0.642742 + 0.766083i \(0.277797\pi\)
\(32\) 14.9886 2.64963
\(33\) 0 0
\(34\) 10.8924 1.86803
\(35\) 0 0
\(36\) 0 0
\(37\) 8.64640 1.42146 0.710730 0.703465i \(-0.248365\pi\)
0.710730 + 0.703465i \(0.248365\pi\)
\(38\) −13.0234 −2.11267
\(39\) 0 0
\(40\) 0 0
\(41\) 10.0996 1.57729 0.788645 0.614849i \(-0.210783\pi\)
0.788645 + 0.614849i \(0.210783\pi\)
\(42\) 0 0
\(43\) −2.43460 −0.371272 −0.185636 0.982619i \(-0.559435\pi\)
−0.185636 + 0.982619i \(0.559435\pi\)
\(44\) −6.88131 −1.03740
\(45\) 0 0
\(46\) −7.29194 −1.07514
\(47\) 7.57192 1.10448 0.552239 0.833686i \(-0.313774\pi\)
0.552239 + 0.833686i \(0.313774\pi\)
\(48\) 0 0
\(49\) −2.82971 −0.404244
\(50\) 0 0
\(51\) 0 0
\(52\) −6.73317 −0.933723
\(53\) −0.621245 −0.0853345 −0.0426673 0.999089i \(-0.513586\pi\)
−0.0426673 + 0.999089i \(0.513586\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 16.8836 2.25616
\(57\) 0 0
\(58\) 12.3101 1.61639
\(59\) 11.3412 1.47650 0.738248 0.674530i \(-0.235653\pi\)
0.738248 + 0.674530i \(0.235653\pi\)
\(60\) 0 0
\(61\) 0.647513 0.0829056 0.0414528 0.999140i \(-0.486801\pi\)
0.0414528 + 0.999140i \(0.486801\pi\)
\(62\) 19.0742 2.42243
\(63\) 0 0
\(64\) 16.2872 2.03590
\(65\) 0 0
\(66\) 0 0
\(67\) −10.9389 −1.33639 −0.668197 0.743984i \(-0.732934\pi\)
−0.668197 + 0.743984i \(0.732934\pi\)
\(68\) 20.8539 2.52891
\(69\) 0 0
\(70\) 0 0
\(71\) 2.95230 0.350374 0.175187 0.984535i \(-0.443947\pi\)
0.175187 + 0.984535i \(0.443947\pi\)
\(72\) 0 0
\(73\) −13.5390 −1.58462 −0.792309 0.610120i \(-0.791121\pi\)
−0.792309 + 0.610120i \(0.791121\pi\)
\(74\) 23.0428 2.67867
\(75\) 0 0
\(76\) −24.9339 −2.86011
\(77\) −2.75416 −0.313866
\(78\) 0 0
\(79\) 1.88929 0.212562 0.106281 0.994336i \(-0.466106\pi\)
0.106281 + 0.994336i \(0.466106\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 26.9155 2.97232
\(83\) 2.37481 0.260669 0.130335 0.991470i \(-0.458395\pi\)
0.130335 + 0.991470i \(0.458395\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −6.48823 −0.699644
\(87\) 0 0
\(88\) −11.1503 −1.18863
\(89\) −7.33882 −0.777913 −0.388957 0.921256i \(-0.627164\pi\)
−0.388957 + 0.921256i \(0.627164\pi\)
\(90\) 0 0
\(91\) −2.69487 −0.282499
\(92\) −13.9608 −1.45551
\(93\) 0 0
\(94\) 20.1793 2.08133
\(95\) 0 0
\(96\) 0 0
\(97\) −5.79915 −0.588814 −0.294407 0.955680i \(-0.595122\pi\)
−0.294407 + 0.955680i \(0.595122\pi\)
\(98\) −7.54121 −0.761777
\(99\) 0 0
\(100\) 0 0
\(101\) −11.5536 −1.14962 −0.574812 0.818285i \(-0.694925\pi\)
−0.574812 + 0.818285i \(0.694925\pi\)
\(102\) 0 0
\(103\) −11.6071 −1.14368 −0.571842 0.820364i \(-0.693771\pi\)
−0.571842 + 0.820364i \(0.693771\pi\)
\(104\) −10.9103 −1.06984
\(105\) 0 0
\(106\) −1.65562 −0.160808
\(107\) 10.1703 0.983204 0.491602 0.870820i \(-0.336412\pi\)
0.491602 + 0.870820i \(0.336412\pi\)
\(108\) 0 0
\(109\) 1.19475 0.114436 0.0572182 0.998362i \(-0.481777\pi\)
0.0572182 + 0.998362i \(0.481777\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 24.1559 2.28252
\(113\) 4.48257 0.421685 0.210842 0.977520i \(-0.432379\pi\)
0.210842 + 0.977520i \(0.432379\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 23.5682 2.18825
\(117\) 0 0
\(118\) 30.2244 2.78238
\(119\) 8.34653 0.765125
\(120\) 0 0
\(121\) −9.18108 −0.834644
\(122\) 1.72563 0.156231
\(123\) 0 0
\(124\) 36.5185 3.27946
\(125\) 0 0
\(126\) 0 0
\(127\) 4.17685 0.370635 0.185318 0.982679i \(-0.440669\pi\)
0.185318 + 0.982679i \(0.440669\pi\)
\(128\) 13.4284 1.18691
\(129\) 0 0
\(130\) 0 0
\(131\) −3.38914 −0.296111 −0.148055 0.988979i \(-0.547301\pi\)
−0.148055 + 0.988979i \(0.547301\pi\)
\(132\) 0 0
\(133\) −9.97947 −0.865330
\(134\) −29.1522 −2.51837
\(135\) 0 0
\(136\) 33.7913 2.89758
\(137\) −8.33597 −0.712190 −0.356095 0.934450i \(-0.615892\pi\)
−0.356095 + 0.934450i \(0.615892\pi\)
\(138\) 0 0
\(139\) −3.43355 −0.291230 −0.145615 0.989341i \(-0.546516\pi\)
−0.145615 + 0.989341i \(0.546516\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 7.86792 0.660261
\(143\) 1.77976 0.148831
\(144\) 0 0
\(145\) 0 0
\(146\) −36.0816 −2.98613
\(147\) 0 0
\(148\) 44.1164 3.62635
\(149\) −9.96023 −0.815974 −0.407987 0.912988i \(-0.633769\pi\)
−0.407987 + 0.912988i \(0.633769\pi\)
\(150\) 0 0
\(151\) −21.0404 −1.71225 −0.856123 0.516772i \(-0.827134\pi\)
−0.856123 + 0.516772i \(0.827134\pi\)
\(152\) −40.4023 −3.27706
\(153\) 0 0
\(154\) −7.33987 −0.591464
\(155\) 0 0
\(156\) 0 0
\(157\) −7.80843 −0.623181 −0.311590 0.950217i \(-0.600862\pi\)
−0.311590 + 0.950217i \(0.600862\pi\)
\(158\) 5.03499 0.400562
\(159\) 0 0
\(160\) 0 0
\(161\) −5.58762 −0.440366
\(162\) 0 0
\(163\) 11.6785 0.914734 0.457367 0.889278i \(-0.348793\pi\)
0.457367 + 0.889278i \(0.348793\pi\)
\(164\) 51.5310 4.02390
\(165\) 0 0
\(166\) 6.32890 0.491218
\(167\) 7.51198 0.581294 0.290647 0.956830i \(-0.406129\pi\)
0.290647 + 0.956830i \(0.406129\pi\)
\(168\) 0 0
\(169\) −11.2586 −0.866043
\(170\) 0 0
\(171\) 0 0
\(172\) −12.4220 −0.947170
\(173\) 5.31381 0.404002 0.202001 0.979385i \(-0.435256\pi\)
0.202001 + 0.979385i \(0.435256\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −15.9532 −1.20251
\(177\) 0 0
\(178\) −19.5580 −1.46594
\(179\) −1.30674 −0.0976705 −0.0488352 0.998807i \(-0.515551\pi\)
−0.0488352 + 0.998807i \(0.515551\pi\)
\(180\) 0 0
\(181\) −20.1627 −1.49868 −0.749341 0.662184i \(-0.769630\pi\)
−0.749341 + 0.662184i \(0.769630\pi\)
\(182\) −7.18186 −0.532355
\(183\) 0 0
\(184\) −22.6217 −1.66770
\(185\) 0 0
\(186\) 0 0
\(187\) −5.51225 −0.403096
\(188\) 38.6342 2.81769
\(189\) 0 0
\(190\) 0 0
\(191\) −21.2957 −1.54090 −0.770452 0.637498i \(-0.779970\pi\)
−0.770452 + 0.637498i \(0.779970\pi\)
\(192\) 0 0
\(193\) 7.99352 0.575386 0.287693 0.957723i \(-0.407112\pi\)
0.287693 + 0.957723i \(0.407112\pi\)
\(194\) −15.4548 −1.10959
\(195\) 0 0
\(196\) −14.4380 −1.03129
\(197\) −21.7014 −1.54616 −0.773081 0.634307i \(-0.781285\pi\)
−0.773081 + 0.634307i \(0.781285\pi\)
\(198\) 0 0
\(199\) 9.34240 0.662265 0.331133 0.943584i \(-0.392569\pi\)
0.331133 + 0.943584i \(0.392569\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −30.7904 −2.16641
\(203\) 9.43288 0.662058
\(204\) 0 0
\(205\) 0 0
\(206\) −30.9331 −2.15521
\(207\) 0 0
\(208\) −15.6097 −1.08234
\(209\) 6.59069 0.455887
\(210\) 0 0
\(211\) −6.88147 −0.473740 −0.236870 0.971541i \(-0.576122\pi\)
−0.236870 + 0.971541i \(0.576122\pi\)
\(212\) −3.16977 −0.217701
\(213\) 0 0
\(214\) 27.1041 1.85280
\(215\) 0 0
\(216\) 0 0
\(217\) 14.6161 0.992203
\(218\) 3.18402 0.215649
\(219\) 0 0
\(220\) 0 0
\(221\) −5.39359 −0.362812
\(222\) 0 0
\(223\) −8.51496 −0.570204 −0.285102 0.958497i \(-0.592028\pi\)
−0.285102 + 0.958497i \(0.592028\pi\)
\(224\) 30.6086 2.04512
\(225\) 0 0
\(226\) 11.9461 0.794643
\(227\) 20.9836 1.39273 0.696364 0.717689i \(-0.254800\pi\)
0.696364 + 0.717689i \(0.254800\pi\)
\(228\) 0 0
\(229\) 29.7629 1.96679 0.983394 0.181485i \(-0.0580904\pi\)
0.983394 + 0.181485i \(0.0580904\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 38.1894 2.50726
\(233\) −2.27154 −0.148813 −0.0744066 0.997228i \(-0.523706\pi\)
−0.0744066 + 0.997228i \(0.523706\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 57.8660 3.76676
\(237\) 0 0
\(238\) 22.2436 1.44184
\(239\) −15.4308 −0.998135 −0.499067 0.866563i \(-0.666324\pi\)
−0.499067 + 0.866563i \(0.666324\pi\)
\(240\) 0 0
\(241\) −4.84184 −0.311890 −0.155945 0.987766i \(-0.549842\pi\)
−0.155945 + 0.987766i \(0.549842\pi\)
\(242\) −24.4677 −1.57284
\(243\) 0 0
\(244\) 3.30380 0.211504
\(245\) 0 0
\(246\) 0 0
\(247\) 6.44880 0.410328
\(248\) 59.1737 3.75754
\(249\) 0 0
\(250\) 0 0
\(251\) 17.8293 1.12537 0.562687 0.826670i \(-0.309768\pi\)
0.562687 + 0.826670i \(0.309768\pi\)
\(252\) 0 0
\(253\) 3.69020 0.232001
\(254\) 11.1313 0.698443
\(255\) 0 0
\(256\) 3.21240 0.200775
\(257\) 5.88929 0.367364 0.183682 0.982986i \(-0.441198\pi\)
0.183682 + 0.982986i \(0.441198\pi\)
\(258\) 0 0
\(259\) 17.6571 1.09716
\(260\) 0 0
\(261\) 0 0
\(262\) −9.03210 −0.558005
\(263\) 23.6773 1.46000 0.730002 0.683445i \(-0.239519\pi\)
0.730002 + 0.683445i \(0.239519\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −26.5954 −1.63067
\(267\) 0 0
\(268\) −55.8132 −3.40934
\(269\) −13.7460 −0.838111 −0.419056 0.907961i \(-0.637639\pi\)
−0.419056 + 0.907961i \(0.637639\pi\)
\(270\) 0 0
\(271\) 7.49882 0.455521 0.227760 0.973717i \(-0.426860\pi\)
0.227760 + 0.973717i \(0.426860\pi\)
\(272\) 48.3463 2.93143
\(273\) 0 0
\(274\) −22.2155 −1.34209
\(275\) 0 0
\(276\) 0 0
\(277\) −2.83798 −0.170517 −0.0852587 0.996359i \(-0.527172\pi\)
−0.0852587 + 0.996359i \(0.527172\pi\)
\(278\) −9.15045 −0.548807
\(279\) 0 0
\(280\) 0 0
\(281\) 22.9609 1.36974 0.684868 0.728667i \(-0.259860\pi\)
0.684868 + 0.728667i \(0.259860\pi\)
\(282\) 0 0
\(283\) 25.1247 1.49351 0.746753 0.665101i \(-0.231612\pi\)
0.746753 + 0.665101i \(0.231612\pi\)
\(284\) 15.0635 0.893854
\(285\) 0 0
\(286\) 4.74308 0.280464
\(287\) 20.6246 1.21743
\(288\) 0 0
\(289\) −0.295018 −0.0173540
\(290\) 0 0
\(291\) 0 0
\(292\) −69.0798 −4.04259
\(293\) −28.8755 −1.68692 −0.843461 0.537190i \(-0.819486\pi\)
−0.843461 + 0.537190i \(0.819486\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 71.4853 4.15500
\(297\) 0 0
\(298\) −26.5441 −1.53766
\(299\) 3.61076 0.208816
\(300\) 0 0
\(301\) −4.97176 −0.286567
\(302\) −56.0730 −3.22664
\(303\) 0 0
\(304\) −57.8049 −3.31534
\(305\) 0 0
\(306\) 0 0
\(307\) −23.9526 −1.36704 −0.683522 0.729930i \(-0.739553\pi\)
−0.683522 + 0.729930i \(0.739553\pi\)
\(308\) −14.0525 −0.800717
\(309\) 0 0
\(310\) 0 0
\(311\) 9.88835 0.560717 0.280358 0.959895i \(-0.409547\pi\)
0.280358 + 0.959895i \(0.409547\pi\)
\(312\) 0 0
\(313\) 18.7786 1.06143 0.530713 0.847551i \(-0.321924\pi\)
0.530713 + 0.847551i \(0.321924\pi\)
\(314\) −20.8096 −1.17435
\(315\) 0 0
\(316\) 9.63973 0.542277
\(317\) −25.8818 −1.45367 −0.726834 0.686813i \(-0.759009\pi\)
−0.726834 + 0.686813i \(0.759009\pi\)
\(318\) 0 0
\(319\) −6.22970 −0.348796
\(320\) 0 0
\(321\) 0 0
\(322\) −14.8911 −0.829848
\(323\) −19.9732 −1.11134
\(324\) 0 0
\(325\) 0 0
\(326\) 31.1234 1.72377
\(327\) 0 0
\(328\) 83.4997 4.61050
\(329\) 15.4628 0.852494
\(330\) 0 0
\(331\) 1.29743 0.0713134 0.0356567 0.999364i \(-0.488648\pi\)
0.0356567 + 0.999364i \(0.488648\pi\)
\(332\) 12.1170 0.665006
\(333\) 0 0
\(334\) 20.0195 1.09542
\(335\) 0 0
\(336\) 0 0
\(337\) 11.2769 0.614293 0.307147 0.951662i \(-0.400626\pi\)
0.307147 + 0.951662i \(0.400626\pi\)
\(338\) −30.0042 −1.63201
\(339\) 0 0
\(340\) 0 0
\(341\) −9.65281 −0.522729
\(342\) 0 0
\(343\) −20.0735 −1.08387
\(344\) −20.1284 −1.08525
\(345\) 0 0
\(346\) 14.1614 0.761321
\(347\) 2.15501 0.115687 0.0578434 0.998326i \(-0.481578\pi\)
0.0578434 + 0.998326i \(0.481578\pi\)
\(348\) 0 0
\(349\) 5.60904 0.300245 0.150122 0.988667i \(-0.452033\pi\)
0.150122 + 0.988667i \(0.452033\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −20.2147 −1.07745
\(353\) −18.4969 −0.984492 −0.492246 0.870456i \(-0.663824\pi\)
−0.492246 + 0.870456i \(0.663824\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −37.4448 −1.98457
\(357\) 0 0
\(358\) −3.48248 −0.184055
\(359\) −11.2450 −0.593487 −0.296743 0.954957i \(-0.595901\pi\)
−0.296743 + 0.954957i \(0.595901\pi\)
\(360\) 0 0
\(361\) 4.88081 0.256885
\(362\) −53.7339 −2.82419
\(363\) 0 0
\(364\) −13.7500 −0.720696
\(365\) 0 0
\(366\) 0 0
\(367\) 28.5677 1.49122 0.745610 0.666383i \(-0.232158\pi\)
0.745610 + 0.666383i \(0.232158\pi\)
\(368\) −32.3656 −1.68718
\(369\) 0 0
\(370\) 0 0
\(371\) −1.26866 −0.0658656
\(372\) 0 0
\(373\) −14.0848 −0.729284 −0.364642 0.931148i \(-0.618809\pi\)
−0.364642 + 0.931148i \(0.618809\pi\)
\(374\) −14.6902 −0.759613
\(375\) 0 0
\(376\) 62.6020 3.22845
\(377\) −6.09559 −0.313939
\(378\) 0 0
\(379\) 21.9903 1.12957 0.564783 0.825239i \(-0.308960\pi\)
0.564783 + 0.825239i \(0.308960\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −56.7534 −2.90376
\(383\) −14.7436 −0.753363 −0.376682 0.926343i \(-0.622935\pi\)
−0.376682 + 0.926343i \(0.622935\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 21.3028 1.08429
\(387\) 0 0
\(388\) −29.5889 −1.50215
\(389\) 14.2568 0.722847 0.361423 0.932402i \(-0.382291\pi\)
0.361423 + 0.932402i \(0.382291\pi\)
\(390\) 0 0
\(391\) −11.1832 −0.565560
\(392\) −23.3950 −1.18163
\(393\) 0 0
\(394\) −57.8345 −2.91366
\(395\) 0 0
\(396\) 0 0
\(397\) 5.49065 0.275568 0.137784 0.990462i \(-0.456002\pi\)
0.137784 + 0.990462i \(0.456002\pi\)
\(398\) 24.8976 1.24800
\(399\) 0 0
\(400\) 0 0
\(401\) −26.7528 −1.33597 −0.667985 0.744175i \(-0.732843\pi\)
−0.667985 + 0.744175i \(0.732843\pi\)
\(402\) 0 0
\(403\) −9.44500 −0.470489
\(404\) −58.9497 −2.93286
\(405\) 0 0
\(406\) 25.1387 1.24761
\(407\) −11.6612 −0.578022
\(408\) 0 0
\(409\) 5.13389 0.253854 0.126927 0.991912i \(-0.459489\pi\)
0.126927 + 0.991912i \(0.459489\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −59.2230 −2.91771
\(413\) 23.1601 1.13964
\(414\) 0 0
\(415\) 0 0
\(416\) −19.7795 −0.969769
\(417\) 0 0
\(418\) 17.5643 0.859096
\(419\) −31.8516 −1.55605 −0.778026 0.628231i \(-0.783779\pi\)
−0.778026 + 0.628231i \(0.783779\pi\)
\(420\) 0 0
\(421\) 3.08044 0.150132 0.0750658 0.997179i \(-0.476083\pi\)
0.0750658 + 0.997179i \(0.476083\pi\)
\(422\) −18.3392 −0.892739
\(423\) 0 0
\(424\) −5.13623 −0.249437
\(425\) 0 0
\(426\) 0 0
\(427\) 1.32231 0.0639908
\(428\) 51.8920 2.50830
\(429\) 0 0
\(430\) 0 0
\(431\) −23.3471 −1.12459 −0.562295 0.826937i \(-0.690081\pi\)
−0.562295 + 0.826937i \(0.690081\pi\)
\(432\) 0 0
\(433\) −8.21415 −0.394747 −0.197374 0.980328i \(-0.563241\pi\)
−0.197374 + 0.980328i \(0.563241\pi\)
\(434\) 38.9520 1.86976
\(435\) 0 0
\(436\) 6.09596 0.291944
\(437\) 13.3711 0.639628
\(438\) 0 0
\(439\) −24.3117 −1.16034 −0.580168 0.814497i \(-0.697013\pi\)
−0.580168 + 0.814497i \(0.697013\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −14.3740 −0.683700
\(443\) 0.0631363 0.00299970 0.00149985 0.999999i \(-0.499523\pi\)
0.00149985 + 0.999999i \(0.499523\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −22.6925 −1.07452
\(447\) 0 0
\(448\) 33.2605 1.57141
\(449\) 11.5711 0.546074 0.273037 0.962004i \(-0.411972\pi\)
0.273037 + 0.962004i \(0.411972\pi\)
\(450\) 0 0
\(451\) −13.6210 −0.641389
\(452\) 22.8714 1.07578
\(453\) 0 0
\(454\) 55.9215 2.62452
\(455\) 0 0
\(456\) 0 0
\(457\) 41.3967 1.93646 0.968228 0.250071i \(-0.0804539\pi\)
0.968228 + 0.250071i \(0.0804539\pi\)
\(458\) 79.3185 3.70631
\(459\) 0 0
\(460\) 0 0
\(461\) −29.3707 −1.36793 −0.683965 0.729515i \(-0.739746\pi\)
−0.683965 + 0.729515i \(0.739746\pi\)
\(462\) 0 0
\(463\) 30.5501 1.41978 0.709891 0.704311i \(-0.248744\pi\)
0.709891 + 0.704311i \(0.248744\pi\)
\(464\) 54.6388 2.53654
\(465\) 0 0
\(466\) −6.05367 −0.280431
\(467\) 32.7149 1.51386 0.756932 0.653494i \(-0.226697\pi\)
0.756932 + 0.653494i \(0.226697\pi\)
\(468\) 0 0
\(469\) −22.3386 −1.03150
\(470\) 0 0
\(471\) 0 0
\(472\) 93.7648 4.31588
\(473\) 3.28347 0.150974
\(474\) 0 0
\(475\) 0 0
\(476\) 42.5864 1.95195
\(477\) 0 0
\(478\) −41.1233 −1.88093
\(479\) 12.3353 0.563613 0.281807 0.959471i \(-0.409066\pi\)
0.281807 + 0.959471i \(0.409066\pi\)
\(480\) 0 0
\(481\) −11.4101 −0.520256
\(482\) −12.9036 −0.587741
\(483\) 0 0
\(484\) −46.8446 −2.12930
\(485\) 0 0
\(486\) 0 0
\(487\) −35.2962 −1.59942 −0.799711 0.600385i \(-0.795014\pi\)
−0.799711 + 0.600385i \(0.795014\pi\)
\(488\) 5.35341 0.242337
\(489\) 0 0
\(490\) 0 0
\(491\) −15.3156 −0.691183 −0.345591 0.938385i \(-0.612322\pi\)
−0.345591 + 0.938385i \(0.612322\pi\)
\(492\) 0 0
\(493\) 18.8792 0.850278
\(494\) 17.1861 0.773241
\(495\) 0 0
\(496\) 84.6618 3.80143
\(497\) 6.02898 0.270437
\(498\) 0 0
\(499\) −35.1777 −1.57477 −0.787386 0.616461i \(-0.788566\pi\)
−0.787386 + 0.616461i \(0.788566\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 47.5152 2.12071
\(503\) 7.33494 0.327049 0.163524 0.986539i \(-0.447714\pi\)
0.163524 + 0.986539i \(0.447714\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 9.83443 0.437194
\(507\) 0 0
\(508\) 21.3115 0.945544
\(509\) −5.70697 −0.252957 −0.126479 0.991969i \(-0.540367\pi\)
−0.126479 + 0.991969i \(0.540367\pi\)
\(510\) 0 0
\(511\) −27.6484 −1.22309
\(512\) −18.2956 −0.808560
\(513\) 0 0
\(514\) 15.6950 0.692278
\(515\) 0 0
\(516\) 0 0
\(517\) −10.2120 −0.449125
\(518\) 47.0563 2.06753
\(519\) 0 0
\(520\) 0 0
\(521\) 33.0901 1.44970 0.724852 0.688905i \(-0.241908\pi\)
0.724852 + 0.688905i \(0.241908\pi\)
\(522\) 0 0
\(523\) 31.4719 1.37617 0.688086 0.725629i \(-0.258451\pi\)
0.688086 + 0.725629i \(0.258451\pi\)
\(524\) −17.2924 −0.755421
\(525\) 0 0
\(526\) 63.1002 2.75130
\(527\) 29.2530 1.27428
\(528\) 0 0
\(529\) −15.5133 −0.674493
\(530\) 0 0
\(531\) 0 0
\(532\) −50.9182 −2.20758
\(533\) −13.3278 −0.577291
\(534\) 0 0
\(535\) 0 0
\(536\) −90.4386 −3.90635
\(537\) 0 0
\(538\) −36.6334 −1.57938
\(539\) 3.81635 0.164382
\(540\) 0 0
\(541\) −22.8759 −0.983512 −0.491756 0.870733i \(-0.663645\pi\)
−0.491756 + 0.870733i \(0.663645\pi\)
\(542\) 19.9844 0.858405
\(543\) 0 0
\(544\) 61.2609 2.62654
\(545\) 0 0
\(546\) 0 0
\(547\) −17.7548 −0.759139 −0.379570 0.925163i \(-0.623928\pi\)
−0.379570 + 0.925163i \(0.623928\pi\)
\(548\) −42.5325 −1.81690
\(549\) 0 0
\(550\) 0 0
\(551\) −22.5728 −0.961634
\(552\) 0 0
\(553\) 3.85818 0.164067
\(554\) −7.56324 −0.321331
\(555\) 0 0
\(556\) −17.5190 −0.742969
\(557\) −2.27448 −0.0963727 −0.0481864 0.998838i \(-0.515344\pi\)
−0.0481864 + 0.998838i \(0.515344\pi\)
\(558\) 0 0
\(559\) 3.21278 0.135886
\(560\) 0 0
\(561\) 0 0
\(562\) 61.1912 2.58120
\(563\) −0.397699 −0.0167610 −0.00838050 0.999965i \(-0.502668\pi\)
−0.00838050 + 0.999965i \(0.502668\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 66.9576 2.81444
\(567\) 0 0
\(568\) 24.4086 1.02416
\(569\) 11.1876 0.469010 0.234505 0.972115i \(-0.424653\pi\)
0.234505 + 0.972115i \(0.424653\pi\)
\(570\) 0 0
\(571\) −18.9124 −0.791458 −0.395729 0.918367i \(-0.629508\pi\)
−0.395729 + 0.918367i \(0.629508\pi\)
\(572\) 9.08084 0.379689
\(573\) 0 0
\(574\) 54.9649 2.29419
\(575\) 0 0
\(576\) 0 0
\(577\) 6.01809 0.250537 0.125268 0.992123i \(-0.460021\pi\)
0.125268 + 0.992123i \(0.460021\pi\)
\(578\) −0.786228 −0.0327028
\(579\) 0 0
\(580\) 0 0
\(581\) 4.84967 0.201198
\(582\) 0 0
\(583\) 0.837855 0.0347004
\(584\) −111.935 −4.63192
\(585\) 0 0
\(586\) −76.9535 −3.17892
\(587\) −16.3326 −0.674118 −0.337059 0.941483i \(-0.609432\pi\)
−0.337059 + 0.941483i \(0.609432\pi\)
\(588\) 0 0
\(589\) −34.9761 −1.44117
\(590\) 0 0
\(591\) 0 0
\(592\) 102.276 4.20354
\(593\) 8.40604 0.345195 0.172597 0.984992i \(-0.444784\pi\)
0.172597 + 0.984992i \(0.444784\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −50.8200 −2.08167
\(597\) 0 0
\(598\) 9.62272 0.393502
\(599\) −13.1918 −0.539001 −0.269501 0.963000i \(-0.586859\pi\)
−0.269501 + 0.963000i \(0.586859\pi\)
\(600\) 0 0
\(601\) −6.06690 −0.247474 −0.123737 0.992315i \(-0.539488\pi\)
−0.123737 + 0.992315i \(0.539488\pi\)
\(602\) −13.2498 −0.540021
\(603\) 0 0
\(604\) −107.354 −4.36819
\(605\) 0 0
\(606\) 0 0
\(607\) 14.9141 0.605345 0.302672 0.953095i \(-0.402121\pi\)
0.302672 + 0.953095i \(0.402121\pi\)
\(608\) −73.2462 −2.97053
\(609\) 0 0
\(610\) 0 0
\(611\) −9.99219 −0.404241
\(612\) 0 0
\(613\) 36.4772 1.47330 0.736651 0.676273i \(-0.236406\pi\)
0.736651 + 0.676273i \(0.236406\pi\)
\(614\) −63.8338 −2.57612
\(615\) 0 0
\(616\) −22.7704 −0.917446
\(617\) 16.2341 0.653559 0.326780 0.945101i \(-0.394036\pi\)
0.326780 + 0.945101i \(0.394036\pi\)
\(618\) 0 0
\(619\) −10.0360 −0.403382 −0.201691 0.979449i \(-0.564644\pi\)
−0.201691 + 0.979449i \(0.564644\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 26.3526 1.05664
\(623\) −14.9868 −0.600434
\(624\) 0 0
\(625\) 0 0
\(626\) 50.0451 2.00020
\(627\) 0 0
\(628\) −39.8409 −1.58982
\(629\) 35.3393 1.40907
\(630\) 0 0
\(631\) 28.7580 1.14484 0.572420 0.819961i \(-0.306005\pi\)
0.572420 + 0.819961i \(0.306005\pi\)
\(632\) 15.6200 0.621330
\(633\) 0 0
\(634\) −68.9754 −2.73936
\(635\) 0 0
\(636\) 0 0
\(637\) 3.73419 0.147954
\(638\) −16.6022 −0.657289
\(639\) 0 0
\(640\) 0 0
\(641\) 18.2437 0.720582 0.360291 0.932840i \(-0.382677\pi\)
0.360291 + 0.932840i \(0.382677\pi\)
\(642\) 0 0
\(643\) 33.9734 1.33978 0.669890 0.742461i \(-0.266341\pi\)
0.669890 + 0.742461i \(0.266341\pi\)
\(644\) −28.5097 −1.12344
\(645\) 0 0
\(646\) −53.2288 −2.09426
\(647\) 47.3728 1.86242 0.931208 0.364488i \(-0.118756\pi\)
0.931208 + 0.364488i \(0.118756\pi\)
\(648\) 0 0
\(649\) −15.2955 −0.600402
\(650\) 0 0
\(651\) 0 0
\(652\) 59.5873 2.33362
\(653\) −37.3959 −1.46342 −0.731708 0.681618i \(-0.761277\pi\)
−0.731708 + 0.681618i \(0.761277\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 119.466 4.66436
\(657\) 0 0
\(658\) 41.2087 1.60648
\(659\) 9.48154 0.369348 0.184674 0.982800i \(-0.440877\pi\)
0.184674 + 0.982800i \(0.440877\pi\)
\(660\) 0 0
\(661\) −8.78089 −0.341537 −0.170768 0.985311i \(-0.554625\pi\)
−0.170768 + 0.985311i \(0.554625\pi\)
\(662\) 3.45768 0.134386
\(663\) 0 0
\(664\) 19.6341 0.761951
\(665\) 0 0
\(666\) 0 0
\(667\) −12.6388 −0.489375
\(668\) 38.3283 1.48297
\(669\) 0 0
\(670\) 0 0
\(671\) −0.873283 −0.0337127
\(672\) 0 0
\(673\) −48.0259 −1.85126 −0.925631 0.378427i \(-0.876465\pi\)
−0.925631 + 0.378427i \(0.876465\pi\)
\(674\) 30.0531 1.15760
\(675\) 0 0
\(676\) −57.4444 −2.20940
\(677\) −15.4151 −0.592451 −0.296226 0.955118i \(-0.595728\pi\)
−0.296226 + 0.955118i \(0.595728\pi\)
\(678\) 0 0
\(679\) −11.8426 −0.454478
\(680\) 0 0
\(681\) 0 0
\(682\) −25.7248 −0.985055
\(683\) −27.8221 −1.06458 −0.532292 0.846561i \(-0.678669\pi\)
−0.532292 + 0.846561i \(0.678669\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −53.4962 −2.04249
\(687\) 0 0
\(688\) −28.7983 −1.09793
\(689\) 0.819818 0.0312326
\(690\) 0 0
\(691\) −20.1261 −0.765634 −0.382817 0.923824i \(-0.625046\pi\)
−0.382817 + 0.923824i \(0.625046\pi\)
\(692\) 27.1126 1.03067
\(693\) 0 0
\(694\) 5.74312 0.218006
\(695\) 0 0
\(696\) 0 0
\(697\) 41.2787 1.56354
\(698\) 14.9481 0.565796
\(699\) 0 0
\(700\) 0 0
\(701\) −31.6216 −1.19433 −0.597166 0.802118i \(-0.703707\pi\)
−0.597166 + 0.802118i \(0.703707\pi\)
\(702\) 0 0
\(703\) −42.2532 −1.59361
\(704\) −21.9661 −0.827877
\(705\) 0 0
\(706\) −49.2945 −1.85522
\(707\) −23.5939 −0.887340
\(708\) 0 0
\(709\) −4.76008 −0.178768 −0.0893842 0.995997i \(-0.528490\pi\)
−0.0893842 + 0.995997i \(0.528490\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −60.6748 −2.27388
\(713\) −19.5835 −0.733409
\(714\) 0 0
\(715\) 0 0
\(716\) −6.66738 −0.249172
\(717\) 0 0
\(718\) −29.9680 −1.11840
\(719\) 23.0727 0.860467 0.430234 0.902718i \(-0.358431\pi\)
0.430234 + 0.902718i \(0.358431\pi\)
\(720\) 0 0
\(721\) −23.7033 −0.882755
\(722\) 13.0074 0.484086
\(723\) 0 0
\(724\) −102.876 −3.82336
\(725\) 0 0
\(726\) 0 0
\(727\) −15.5807 −0.577856 −0.288928 0.957351i \(-0.593299\pi\)
−0.288928 + 0.957351i \(0.593299\pi\)
\(728\) −22.2802 −0.825760
\(729\) 0 0
\(730\) 0 0
\(731\) −9.95061 −0.368037
\(732\) 0 0
\(733\) −17.2215 −0.636092 −0.318046 0.948075i \(-0.603027\pi\)
−0.318046 + 0.948075i \(0.603027\pi\)
\(734\) 76.1331 2.81013
\(735\) 0 0
\(736\) −41.0114 −1.51170
\(737\) 14.7529 0.543431
\(738\) 0 0
\(739\) −27.4878 −1.01116 −0.505578 0.862781i \(-0.668721\pi\)
−0.505578 + 0.862781i \(0.668721\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −3.38100 −0.124120
\(743\) 48.4801 1.77856 0.889280 0.457362i \(-0.151206\pi\)
0.889280 + 0.457362i \(0.151206\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −37.5362 −1.37430
\(747\) 0 0
\(748\) −28.1251 −1.02836
\(749\) 20.7691 0.758888
\(750\) 0 0
\(751\) 3.29720 0.120316 0.0601582 0.998189i \(-0.480839\pi\)
0.0601582 + 0.998189i \(0.480839\pi\)
\(752\) 89.5667 3.26616
\(753\) 0 0
\(754\) −16.2448 −0.591602
\(755\) 0 0
\(756\) 0 0
\(757\) 35.7934 1.30093 0.650466 0.759535i \(-0.274574\pi\)
0.650466 + 0.759535i \(0.274574\pi\)
\(758\) 58.6045 2.12861
\(759\) 0 0
\(760\) 0 0
\(761\) 0.664110 0.0240740 0.0120370 0.999928i \(-0.496168\pi\)
0.0120370 + 0.999928i \(0.496168\pi\)
\(762\) 0 0
\(763\) 2.43983 0.0883279
\(764\) −108.657 −3.93107
\(765\) 0 0
\(766\) −39.2919 −1.41967
\(767\) −14.9662 −0.540400
\(768\) 0 0
\(769\) 25.7090 0.927091 0.463545 0.886073i \(-0.346577\pi\)
0.463545 + 0.886073i \(0.346577\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 40.7853 1.46789
\(773\) −38.5458 −1.38640 −0.693199 0.720746i \(-0.743799\pi\)
−0.693199 + 0.720746i \(0.743799\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −47.9453 −1.72114
\(777\) 0 0
\(778\) 37.9944 1.36217
\(779\) −49.3546 −1.76831
\(780\) 0 0
\(781\) −3.98169 −0.142476
\(782\) −29.8034 −1.06577
\(783\) 0 0
\(784\) −33.4720 −1.19543
\(785\) 0 0
\(786\) 0 0
\(787\) 36.2312 1.29150 0.645751 0.763548i \(-0.276544\pi\)
0.645751 + 0.763548i \(0.276544\pi\)
\(788\) −110.727 −3.94448
\(789\) 0 0
\(790\) 0 0
\(791\) 9.15398 0.325478
\(792\) 0 0
\(793\) −0.854482 −0.0303436
\(794\) 14.6326 0.519293
\(795\) 0 0
\(796\) 47.6676 1.68953
\(797\) 33.8276 1.19823 0.599117 0.800662i \(-0.295518\pi\)
0.599117 + 0.800662i \(0.295518\pi\)
\(798\) 0 0
\(799\) 30.9478 1.09485
\(800\) 0 0
\(801\) 0 0
\(802\) −71.2965 −2.51757
\(803\) 18.2596 0.644369
\(804\) 0 0
\(805\) 0 0
\(806\) −25.1710 −0.886612
\(807\) 0 0
\(808\) −95.5209 −3.36041
\(809\) 1.46409 0.0514747 0.0257374 0.999669i \(-0.491807\pi\)
0.0257374 + 0.999669i \(0.491807\pi\)
\(810\) 0 0
\(811\) −40.6885 −1.42877 −0.714383 0.699755i \(-0.753293\pi\)
−0.714383 + 0.699755i \(0.753293\pi\)
\(812\) 48.1293 1.68901
\(813\) 0 0
\(814\) −31.0771 −1.08925
\(815\) 0 0
\(816\) 0 0
\(817\) 11.8974 0.416237
\(818\) 13.6819 0.478375
\(819\) 0 0
\(820\) 0 0
\(821\) 29.9533 1.04538 0.522689 0.852523i \(-0.324929\pi\)
0.522689 + 0.852523i \(0.324929\pi\)
\(822\) 0 0
\(823\) −18.9369 −0.660098 −0.330049 0.943964i \(-0.607065\pi\)
−0.330049 + 0.943964i \(0.607065\pi\)
\(824\) −95.9636 −3.34305
\(825\) 0 0
\(826\) 61.7221 2.14759
\(827\) 20.2639 0.704644 0.352322 0.935879i \(-0.385392\pi\)
0.352322 + 0.935879i \(0.385392\pi\)
\(828\) 0 0
\(829\) 16.6400 0.577932 0.288966 0.957339i \(-0.406689\pi\)
0.288966 + 0.957339i \(0.406689\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −21.4932 −0.745142
\(833\) −11.5655 −0.400721
\(834\) 0 0
\(835\) 0 0
\(836\) 33.6276 1.16304
\(837\) 0 0
\(838\) −84.8849 −2.93230
\(839\) 7.13666 0.246385 0.123192 0.992383i \(-0.460687\pi\)
0.123192 + 0.992383i \(0.460687\pi\)
\(840\) 0 0
\(841\) −7.66354 −0.264260
\(842\) 8.20942 0.282915
\(843\) 0 0
\(844\) −35.1113 −1.20858
\(845\) 0 0
\(846\) 0 0
\(847\) −18.7490 −0.644222
\(848\) −7.34857 −0.252351
\(849\) 0 0
\(850\) 0 0
\(851\) −23.6581 −0.810988
\(852\) 0 0
\(853\) −31.8252 −1.08968 −0.544838 0.838541i \(-0.683409\pi\)
−0.544838 + 0.838541i \(0.683409\pi\)
\(854\) 3.52396 0.120587
\(855\) 0 0
\(856\) 84.0847 2.87396
\(857\) 43.1535 1.47409 0.737047 0.675842i \(-0.236220\pi\)
0.737047 + 0.675842i \(0.236220\pi\)
\(858\) 0 0
\(859\) 30.8915 1.05400 0.527002 0.849864i \(-0.323316\pi\)
0.527002 + 0.849864i \(0.323316\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −62.2203 −2.11923
\(863\) 38.1910 1.30004 0.650019 0.759918i \(-0.274761\pi\)
0.650019 + 0.759918i \(0.274761\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −21.8908 −0.743880
\(867\) 0 0
\(868\) 74.5754 2.53125
\(869\) −2.54804 −0.0864362
\(870\) 0 0
\(871\) 14.4353 0.489123
\(872\) 9.87777 0.334504
\(873\) 0 0
\(874\) 35.6343 1.20535
\(875\) 0 0
\(876\) 0 0
\(877\) 31.7247 1.07127 0.535634 0.844450i \(-0.320073\pi\)
0.535634 + 0.844450i \(0.320073\pi\)
\(878\) −64.7911 −2.18659
\(879\) 0 0
\(880\) 0 0
\(881\) 38.4409 1.29511 0.647554 0.762020i \(-0.275792\pi\)
0.647554 + 0.762020i \(0.275792\pi\)
\(882\) 0 0
\(883\) 12.1497 0.408871 0.204435 0.978880i \(-0.434464\pi\)
0.204435 + 0.978880i \(0.434464\pi\)
\(884\) −27.5196 −0.925586
\(885\) 0 0
\(886\) 0.168259 0.00565278
\(887\) 24.2431 0.814005 0.407002 0.913427i \(-0.366574\pi\)
0.407002 + 0.913427i \(0.366574\pi\)
\(888\) 0 0
\(889\) 8.52966 0.286075
\(890\) 0 0
\(891\) 0 0
\(892\) −43.4458 −1.45467
\(893\) −37.0025 −1.23824
\(894\) 0 0
\(895\) 0 0
\(896\) 27.4225 0.916120
\(897\) 0 0
\(898\) 30.8371 1.02905
\(899\) 33.0604 1.10263
\(900\) 0 0
\(901\) −2.53913 −0.0845908
\(902\) −36.3002 −1.20866
\(903\) 0 0
\(904\) 37.0603 1.23261
\(905\) 0 0
\(906\) 0 0
\(907\) −32.5046 −1.07930 −0.539649 0.841890i \(-0.681443\pi\)
−0.539649 + 0.841890i \(0.681443\pi\)
\(908\) 107.064 3.55305
\(909\) 0 0
\(910\) 0 0
\(911\) −44.2258 −1.46527 −0.732633 0.680624i \(-0.761709\pi\)
−0.732633 + 0.680624i \(0.761709\pi\)
\(912\) 0 0
\(913\) −3.20284 −0.105999
\(914\) 110.323 3.64915
\(915\) 0 0
\(916\) 151.859 5.01756
\(917\) −6.92106 −0.228554
\(918\) 0 0
\(919\) −39.7936 −1.31267 −0.656335 0.754469i \(-0.727894\pi\)
−0.656335 + 0.754469i \(0.727894\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −78.2732 −2.57779
\(923\) −3.89597 −0.128237
\(924\) 0 0
\(925\) 0 0
\(926\) 81.4163 2.67551
\(927\) 0 0
\(928\) 69.2344 2.27273
\(929\) 6.64622 0.218055 0.109028 0.994039i \(-0.465226\pi\)
0.109028 + 0.994039i \(0.465226\pi\)
\(930\) 0 0
\(931\) 13.8282 0.453202
\(932\) −11.5900 −0.379644
\(933\) 0 0
\(934\) 87.1855 2.85280
\(935\) 0 0
\(936\) 0 0
\(937\) 0.229585 0.00750023 0.00375011 0.999993i \(-0.498806\pi\)
0.00375011 + 0.999993i \(0.498806\pi\)
\(938\) −59.5325 −1.94381
\(939\) 0 0
\(940\) 0 0
\(941\) −28.3476 −0.924106 −0.462053 0.886852i \(-0.652887\pi\)
−0.462053 + 0.886852i \(0.652887\pi\)
\(942\) 0 0
\(943\) −27.6342 −0.899894
\(944\) 134.152 4.36629
\(945\) 0 0
\(946\) 8.75049 0.284503
\(947\) −48.0036 −1.55991 −0.779954 0.625837i \(-0.784758\pi\)
−0.779954 + 0.625837i \(0.784758\pi\)
\(948\) 0 0
\(949\) 17.8666 0.579973
\(950\) 0 0
\(951\) 0 0
\(952\) 69.0062 2.23650
\(953\) 26.8885 0.871004 0.435502 0.900188i \(-0.356571\pi\)
0.435502 + 0.900188i \(0.356571\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −78.7324 −2.54639
\(957\) 0 0
\(958\) 32.8737 1.06210
\(959\) −17.0231 −0.549705
\(960\) 0 0
\(961\) 20.2265 0.652468
\(962\) −30.4081 −0.980396
\(963\) 0 0
\(964\) −24.7045 −0.795677
\(965\) 0 0
\(966\) 0 0
\(967\) −3.78016 −0.121562 −0.0607809 0.998151i \(-0.519359\pi\)
−0.0607809 + 0.998151i \(0.519359\pi\)
\(968\) −75.9059 −2.43971
\(969\) 0 0
\(970\) 0 0
\(971\) −11.2875 −0.362232 −0.181116 0.983462i \(-0.557971\pi\)
−0.181116 + 0.983462i \(0.557971\pi\)
\(972\) 0 0
\(973\) −7.01175 −0.224786
\(974\) −94.0648 −3.01403
\(975\) 0 0
\(976\) 7.65930 0.245168
\(977\) −20.2798 −0.648807 −0.324403 0.945919i \(-0.605164\pi\)
−0.324403 + 0.945919i \(0.605164\pi\)
\(978\) 0 0
\(979\) 9.89766 0.316331
\(980\) 0 0
\(981\) 0 0
\(982\) −40.8162 −1.30250
\(983\) 28.3127 0.903035 0.451517 0.892262i \(-0.350883\pi\)
0.451517 + 0.892262i \(0.350883\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 50.3134 1.60230
\(987\) 0 0
\(988\) 32.9037 1.04681
\(989\) 6.66148 0.211823
\(990\) 0 0
\(991\) 23.8610 0.757970 0.378985 0.925403i \(-0.376273\pi\)
0.378985 + 0.925403i \(0.376273\pi\)
\(992\) 107.277 3.40606
\(993\) 0 0
\(994\) 16.0673 0.509624
\(995\) 0 0
\(996\) 0 0
\(997\) 13.6720 0.432996 0.216498 0.976283i \(-0.430536\pi\)
0.216498 + 0.976283i \(0.430536\pi\)
\(998\) −93.7491 −2.96757
\(999\) 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.be.1.8 8
3.2 odd 2 625.2.a.e.1.1 8
5.4 even 2 5625.2.a.s.1.1 8
12.11 even 2 10000.2.a.bn.1.5 8
15.2 even 4 625.2.b.d.624.1 16
15.8 even 4 625.2.b.d.624.16 16
15.14 odd 2 625.2.a.g.1.8 yes 8
60.59 even 2 10000.2.a.be.1.4 8
75.2 even 20 625.2.e.j.124.1 32
75.8 even 20 625.2.e.k.374.1 32
75.11 odd 10 625.2.d.q.501.1 16
75.14 odd 10 625.2.d.m.501.4 16
75.17 even 20 625.2.e.k.374.8 32
75.23 even 20 625.2.e.j.124.8 32
75.29 odd 10 625.2.d.n.376.1 16
75.38 even 20 625.2.e.j.499.1 32
75.41 odd 10 625.2.d.q.126.1 16
75.44 odd 10 625.2.d.n.251.1 16
75.47 even 20 625.2.e.k.249.1 32
75.53 even 20 625.2.e.k.249.8 32
75.56 odd 10 625.2.d.p.251.4 16
75.59 odd 10 625.2.d.m.126.4 16
75.62 even 20 625.2.e.j.499.8 32
75.71 odd 10 625.2.d.p.376.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
625.2.a.e.1.1 8 3.2 odd 2
625.2.a.g.1.8 yes 8 15.14 odd 2
625.2.b.d.624.1 16 15.2 even 4
625.2.b.d.624.16 16 15.8 even 4
625.2.d.m.126.4 16 75.59 odd 10
625.2.d.m.501.4 16 75.14 odd 10
625.2.d.n.251.1 16 75.44 odd 10
625.2.d.n.376.1 16 75.29 odd 10
625.2.d.p.251.4 16 75.56 odd 10
625.2.d.p.376.4 16 75.71 odd 10
625.2.d.q.126.1 16 75.41 odd 10
625.2.d.q.501.1 16 75.11 odd 10
625.2.e.j.124.1 32 75.2 even 20
625.2.e.j.124.8 32 75.23 even 20
625.2.e.j.499.1 32 75.38 even 20
625.2.e.j.499.8 32 75.62 even 20
625.2.e.k.249.1 32 75.47 even 20
625.2.e.k.249.8 32 75.53 even 20
625.2.e.k.374.1 32 75.8 even 20
625.2.e.k.374.8 32 75.17 even 20
5625.2.a.s.1.1 8 5.4 even 2
5625.2.a.be.1.8 8 1.1 even 1 trivial
10000.2.a.be.1.4 8 60.59 even 2
10000.2.a.bn.1.5 8 12.11 even 2