Properties

Label 2475.2.a.be.1.2
Level $2475$
Weight $2$
Character 2475.1
Self dual yes
Analytic conductor $19.763$
Analytic rank $0$
Dimension $3$
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] = [2475,2,Mod(1,2475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2475, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2475.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.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: \(19.7629745003\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.17009\) of defining polynomial
Character \(\chi\) \(=\) 2475.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.53919 q^{2} +0.369102 q^{4} -0.290725 q^{7} -2.51026 q^{8} +O(q^{10})\) \(q+1.53919 q^{2} +0.369102 q^{4} -0.290725 q^{7} -2.51026 q^{8} +1.00000 q^{11} -6.97107 q^{13} -0.447480 q^{14} -4.60197 q^{16} +4.78765 q^{17} +7.75872 q^{19} +1.53919 q^{22} +4.00000 q^{23} -10.7298 q^{26} -0.107307 q^{28} +7.41855 q^{29} +6.34017 q^{31} -2.06278 q^{32} +7.36910 q^{34} +3.41855 q^{37} +11.9421 q^{38} +7.41855 q^{41} -0.290725 q^{43} +0.369102 q^{44} +6.15676 q^{46} +5.26180 q^{47} -6.91548 q^{49} -2.57304 q^{52} +5.75872 q^{53} +0.729794 q^{56} +11.4186 q^{58} -3.60197 q^{59} -6.68035 q^{61} +9.75872 q^{62} +6.02893 q^{64} -6.15676 q^{67} +1.76713 q^{68} +5.07838 q^{71} +1.12783 q^{73} +5.26180 q^{74} +2.86376 q^{76} -0.290725 q^{77} -0.921622 q^{79} +11.4186 q^{82} +1.70928 q^{83} -0.447480 q^{86} -2.51026 q^{88} -4.34017 q^{89} +2.02666 q^{91} +1.47641 q^{92} +8.09890 q^{94} -4.68035 q^{97} -10.6442 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 5 q^{4} - 8 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 5 q^{4} - 8 q^{7} + 9 q^{8} + 3 q^{11} - 6 q^{13} - 2 q^{14} + 5 q^{16} + 4 q^{17} - 2 q^{19} + 3 q^{22} + 12 q^{23} + 8 q^{26} - 12 q^{28} + 8 q^{29} + 8 q^{31} + 11 q^{32} + 26 q^{34} - 4 q^{37} + 6 q^{38} + 8 q^{41} - 8 q^{43} + 5 q^{44} + 12 q^{46} + 8 q^{47} + 11 q^{49} + 26 q^{52} - 8 q^{53} - 38 q^{56} + 20 q^{58} + 8 q^{59} + 2 q^{61} + 4 q^{62} + 33 q^{64} - 12 q^{67} + 28 q^{68} + 12 q^{71} - 18 q^{73} + 8 q^{74} - 18 q^{76} - 8 q^{77} - 6 q^{79} + 20 q^{82} - 2 q^{83} - 2 q^{86} + 9 q^{88} - 2 q^{89} + 8 q^{91} + 20 q^{92} - 12 q^{94} + 8 q^{97} - 29 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.53919 1.08837 0.544185 0.838965i \(-0.316839\pi\)
0.544185 + 0.838965i \(0.316839\pi\)
\(3\) 0 0
\(4\) 0.369102 0.184551
\(5\) 0 0
\(6\) 0 0
\(7\) −0.290725 −0.109884 −0.0549418 0.998490i \(-0.517497\pi\)
−0.0549418 + 0.998490i \(0.517497\pi\)
\(8\) −2.51026 −0.887511
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −6.97107 −1.93343 −0.966714 0.255861i \(-0.917641\pi\)
−0.966714 + 0.255861i \(0.917641\pi\)
\(14\) −0.447480 −0.119594
\(15\) 0 0
\(16\) −4.60197 −1.15049
\(17\) 4.78765 1.16118 0.580588 0.814197i \(-0.302823\pi\)
0.580588 + 0.814197i \(0.302823\pi\)
\(18\) 0 0
\(19\) 7.75872 1.77997 0.889987 0.455987i \(-0.150714\pi\)
0.889987 + 0.455987i \(0.150714\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.53919 0.328156
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −10.7298 −2.10429
\(27\) 0 0
\(28\) −0.107307 −0.0202791
\(29\) 7.41855 1.37759 0.688795 0.724956i \(-0.258140\pi\)
0.688795 + 0.724956i \(0.258140\pi\)
\(30\) 0 0
\(31\) 6.34017 1.13873 0.569364 0.822085i \(-0.307189\pi\)
0.569364 + 0.822085i \(0.307189\pi\)
\(32\) −2.06278 −0.364651
\(33\) 0 0
\(34\) 7.36910 1.26379
\(35\) 0 0
\(36\) 0 0
\(37\) 3.41855 0.562006 0.281003 0.959707i \(-0.409333\pi\)
0.281003 + 0.959707i \(0.409333\pi\)
\(38\) 11.9421 1.93727
\(39\) 0 0
\(40\) 0 0
\(41\) 7.41855 1.15858 0.579291 0.815120i \(-0.303329\pi\)
0.579291 + 0.815120i \(0.303329\pi\)
\(42\) 0 0
\(43\) −0.290725 −0.0443351 −0.0221675 0.999754i \(-0.507057\pi\)
−0.0221675 + 0.999754i \(0.507057\pi\)
\(44\) 0.369102 0.0556443
\(45\) 0 0
\(46\) 6.15676 0.907764
\(47\) 5.26180 0.767512 0.383756 0.923435i \(-0.374630\pi\)
0.383756 + 0.923435i \(0.374630\pi\)
\(48\) 0 0
\(49\) −6.91548 −0.987926
\(50\) 0 0
\(51\) 0 0
\(52\) −2.57304 −0.356816
\(53\) 5.75872 0.791022 0.395511 0.918461i \(-0.370568\pi\)
0.395511 + 0.918461i \(0.370568\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0.729794 0.0975229
\(57\) 0 0
\(58\) 11.4186 1.49933
\(59\) −3.60197 −0.468936 −0.234468 0.972124i \(-0.575335\pi\)
−0.234468 + 0.972124i \(0.575335\pi\)
\(60\) 0 0
\(61\) −6.68035 −0.855331 −0.427665 0.903937i \(-0.640664\pi\)
−0.427665 + 0.903937i \(0.640664\pi\)
\(62\) 9.75872 1.23936
\(63\) 0 0
\(64\) 6.02893 0.753616
\(65\) 0 0
\(66\) 0 0
\(67\) −6.15676 −0.752167 −0.376084 0.926586i \(-0.622729\pi\)
−0.376084 + 0.926586i \(0.622729\pi\)
\(68\) 1.76713 0.214296
\(69\) 0 0
\(70\) 0 0
\(71\) 5.07838 0.602693 0.301346 0.953515i \(-0.402564\pi\)
0.301346 + 0.953515i \(0.402564\pi\)
\(72\) 0 0
\(73\) 1.12783 0.132002 0.0660010 0.997820i \(-0.478976\pi\)
0.0660010 + 0.997820i \(0.478976\pi\)
\(74\) 5.26180 0.611671
\(75\) 0 0
\(76\) 2.86376 0.328496
\(77\) −0.290725 −0.0331311
\(78\) 0 0
\(79\) −0.921622 −0.103691 −0.0518453 0.998655i \(-0.516510\pi\)
−0.0518453 + 0.998655i \(0.516510\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 11.4186 1.26097
\(83\) 1.70928 0.187617 0.0938087 0.995590i \(-0.470096\pi\)
0.0938087 + 0.995590i \(0.470096\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −0.447480 −0.0482530
\(87\) 0 0
\(88\) −2.51026 −0.267595
\(89\) −4.34017 −0.460057 −0.230029 0.973184i \(-0.573882\pi\)
−0.230029 + 0.973184i \(0.573882\pi\)
\(90\) 0 0
\(91\) 2.02666 0.212452
\(92\) 1.47641 0.153926
\(93\) 0 0
\(94\) 8.09890 0.835337
\(95\) 0 0
\(96\) 0 0
\(97\) −4.68035 −0.475217 −0.237609 0.971361i \(-0.576364\pi\)
−0.237609 + 0.971361i \(0.576364\pi\)
\(98\) −10.6442 −1.07523
\(99\) 0 0
\(100\) 0 0
\(101\) 8.58145 0.853886 0.426943 0.904279i \(-0.359590\pi\)
0.426943 + 0.904279i \(0.359590\pi\)
\(102\) 0 0
\(103\) −6.73820 −0.663935 −0.331968 0.943291i \(-0.607712\pi\)
−0.331968 + 0.943291i \(0.607712\pi\)
\(104\) 17.4992 1.71594
\(105\) 0 0
\(106\) 8.86376 0.860925
\(107\) 12.2329 1.18260 0.591298 0.806453i \(-0.298616\pi\)
0.591298 + 0.806453i \(0.298616\pi\)
\(108\) 0 0
\(109\) −6.31351 −0.604725 −0.302362 0.953193i \(-0.597775\pi\)
−0.302362 + 0.953193i \(0.597775\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.33791 0.126420
\(113\) −16.4969 −1.55190 −0.775950 0.630794i \(-0.782729\pi\)
−0.775950 + 0.630794i \(0.782729\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 2.73820 0.254236
\(117\) 0 0
\(118\) −5.54411 −0.510377
\(119\) −1.39189 −0.127594
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −10.2823 −0.930917
\(123\) 0 0
\(124\) 2.34017 0.210154
\(125\) 0 0
\(126\) 0 0
\(127\) −4.87217 −0.432336 −0.216168 0.976356i \(-0.569356\pi\)
−0.216168 + 0.976356i \(0.569356\pi\)
\(128\) 13.4052 1.18487
\(129\) 0 0
\(130\) 0 0
\(131\) 8.68035 0.758405 0.379203 0.925314i \(-0.376198\pi\)
0.379203 + 0.925314i \(0.376198\pi\)
\(132\) 0 0
\(133\) −2.25565 −0.195590
\(134\) −9.47641 −0.818637
\(135\) 0 0
\(136\) −12.0183 −1.03056
\(137\) −12.5958 −1.07613 −0.538067 0.842902i \(-0.680845\pi\)
−0.538067 + 0.842902i \(0.680845\pi\)
\(138\) 0 0
\(139\) −9.91548 −0.841020 −0.420510 0.907288i \(-0.638149\pi\)
−0.420510 + 0.907288i \(0.638149\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 7.81658 0.655953
\(143\) −6.97107 −0.582950
\(144\) 0 0
\(145\) 0 0
\(146\) 1.73594 0.143667
\(147\) 0 0
\(148\) 1.26180 0.103719
\(149\) 1.26180 0.103370 0.0516851 0.998663i \(-0.483541\pi\)
0.0516851 + 0.998663i \(0.483541\pi\)
\(150\) 0 0
\(151\) 1.60197 0.130366 0.0651832 0.997873i \(-0.479237\pi\)
0.0651832 + 0.997873i \(0.479237\pi\)
\(152\) −19.4764 −1.57975
\(153\) 0 0
\(154\) −0.447480 −0.0360590
\(155\) 0 0
\(156\) 0 0
\(157\) 7.10504 0.567044 0.283522 0.958966i \(-0.408497\pi\)
0.283522 + 0.958966i \(0.408497\pi\)
\(158\) −1.41855 −0.112854
\(159\) 0 0
\(160\) 0 0
\(161\) −1.16290 −0.0916492
\(162\) 0 0
\(163\) 22.9360 1.79649 0.898243 0.439499i \(-0.144844\pi\)
0.898243 + 0.439499i \(0.144844\pi\)
\(164\) 2.73820 0.213818
\(165\) 0 0
\(166\) 2.63090 0.204197
\(167\) 4.81432 0.372543 0.186271 0.982498i \(-0.440360\pi\)
0.186271 + 0.982498i \(0.440360\pi\)
\(168\) 0 0
\(169\) 35.5958 2.73814
\(170\) 0 0
\(171\) 0 0
\(172\) −0.107307 −0.00818209
\(173\) −12.8865 −0.979746 −0.489873 0.871794i \(-0.662957\pi\)
−0.489873 + 0.871794i \(0.662957\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −4.60197 −0.346886
\(177\) 0 0
\(178\) −6.68035 −0.500713
\(179\) −1.84324 −0.137771 −0.0688853 0.997625i \(-0.521944\pi\)
−0.0688853 + 0.997625i \(0.521944\pi\)
\(180\) 0 0
\(181\) 10.2823 0.764278 0.382139 0.924105i \(-0.375187\pi\)
0.382139 + 0.924105i \(0.375187\pi\)
\(182\) 3.11942 0.231226
\(183\) 0 0
\(184\) −10.0410 −0.740235
\(185\) 0 0
\(186\) 0 0
\(187\) 4.78765 0.350108
\(188\) 1.94214 0.141645
\(189\) 0 0
\(190\) 0 0
\(191\) 11.5174 0.833373 0.416687 0.909050i \(-0.363191\pi\)
0.416687 + 0.909050i \(0.363191\pi\)
\(192\) 0 0
\(193\) −3.86603 −0.278283 −0.139141 0.990273i \(-0.544434\pi\)
−0.139141 + 0.990273i \(0.544434\pi\)
\(194\) −7.20394 −0.517212
\(195\) 0 0
\(196\) −2.55252 −0.182323
\(197\) −8.57304 −0.610804 −0.305402 0.952224i \(-0.598791\pi\)
−0.305402 + 0.952224i \(0.598791\pi\)
\(198\) 0 0
\(199\) 8.31351 0.589329 0.294665 0.955601i \(-0.404792\pi\)
0.294665 + 0.955601i \(0.404792\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 13.2085 0.929345
\(203\) −2.15676 −0.151375
\(204\) 0 0
\(205\) 0 0
\(206\) −10.3714 −0.722608
\(207\) 0 0
\(208\) 32.0806 2.22439
\(209\) 7.75872 0.536682
\(210\) 0 0
\(211\) 25.9155 1.78410 0.892048 0.451941i \(-0.149268\pi\)
0.892048 + 0.451941i \(0.149268\pi\)
\(212\) 2.12556 0.145984
\(213\) 0 0
\(214\) 18.8287 1.28710
\(215\) 0 0
\(216\) 0 0
\(217\) −1.84324 −0.125128
\(218\) −9.71769 −0.658165
\(219\) 0 0
\(220\) 0 0
\(221\) −33.3751 −2.24505
\(222\) 0 0
\(223\) 9.62863 0.644781 0.322390 0.946607i \(-0.395514\pi\)
0.322390 + 0.946607i \(0.395514\pi\)
\(224\) 0.599701 0.0400692
\(225\) 0 0
\(226\) −25.3919 −1.68904
\(227\) −18.3896 −1.22056 −0.610281 0.792185i \(-0.708943\pi\)
−0.610281 + 0.792185i \(0.708943\pi\)
\(228\) 0 0
\(229\) 17.1506 1.13334 0.566672 0.823943i \(-0.308231\pi\)
0.566672 + 0.823943i \(0.308231\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −18.6225 −1.22263
\(233\) −4.10731 −0.269079 −0.134539 0.990908i \(-0.542955\pi\)
−0.134539 + 0.990908i \(0.542955\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −1.32950 −0.0865428
\(237\) 0 0
\(238\) −2.14238 −0.138870
\(239\) 4.36683 0.282467 0.141234 0.989976i \(-0.454893\pi\)
0.141234 + 0.989976i \(0.454893\pi\)
\(240\) 0 0
\(241\) −5.20394 −0.335215 −0.167608 0.985854i \(-0.553604\pi\)
−0.167608 + 0.985854i \(0.553604\pi\)
\(242\) 1.53919 0.0989428
\(243\) 0 0
\(244\) −2.46573 −0.157852
\(245\) 0 0
\(246\) 0 0
\(247\) −54.0866 −3.44145
\(248\) −15.9155 −1.01063
\(249\) 0 0
\(250\) 0 0
\(251\) 8.28231 0.522775 0.261388 0.965234i \(-0.415820\pi\)
0.261388 + 0.965234i \(0.415820\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) −7.49920 −0.470541
\(255\) 0 0
\(256\) 8.57531 0.535957
\(257\) 11.0205 0.687441 0.343721 0.939072i \(-0.388313\pi\)
0.343721 + 0.939072i \(0.388313\pi\)
\(258\) 0 0
\(259\) −0.993857 −0.0617553
\(260\) 0 0
\(261\) 0 0
\(262\) 13.3607 0.825426
\(263\) −2.97107 −0.183204 −0.0916020 0.995796i \(-0.529199\pi\)
−0.0916020 + 0.995796i \(0.529199\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −3.47187 −0.212874
\(267\) 0 0
\(268\) −2.27247 −0.138813
\(269\) 15.8432 0.965980 0.482990 0.875626i \(-0.339551\pi\)
0.482990 + 0.875626i \(0.339551\pi\)
\(270\) 0 0
\(271\) −6.28231 −0.381623 −0.190812 0.981627i \(-0.561112\pi\)
−0.190812 + 0.981627i \(0.561112\pi\)
\(272\) −22.0326 −1.33592
\(273\) 0 0
\(274\) −19.3874 −1.17123
\(275\) 0 0
\(276\) 0 0
\(277\) −5.12783 −0.308101 −0.154051 0.988063i \(-0.549232\pi\)
−0.154051 + 0.988063i \(0.549232\pi\)
\(278\) −15.2618 −0.915342
\(279\) 0 0
\(280\) 0 0
\(281\) −21.8888 −1.30578 −0.652889 0.757454i \(-0.726443\pi\)
−0.652889 + 0.757454i \(0.726443\pi\)
\(282\) 0 0
\(283\) −25.9649 −1.54345 −0.771727 0.635954i \(-0.780607\pi\)
−0.771727 + 0.635954i \(0.780607\pi\)
\(284\) 1.87444 0.111228
\(285\) 0 0
\(286\) −10.7298 −0.634466
\(287\) −2.15676 −0.127309
\(288\) 0 0
\(289\) 5.92162 0.348331
\(290\) 0 0
\(291\) 0 0
\(292\) 0.416283 0.0243611
\(293\) 10.4163 0.608526 0.304263 0.952588i \(-0.401590\pi\)
0.304263 + 0.952588i \(0.401590\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −8.58145 −0.498787
\(297\) 0 0
\(298\) 1.94214 0.112505
\(299\) −27.8843 −1.61259
\(300\) 0 0
\(301\) 0.0845208 0.00487170
\(302\) 2.46573 0.141887
\(303\) 0 0
\(304\) −35.7054 −2.04785
\(305\) 0 0
\(306\) 0 0
\(307\) 29.0700 1.65911 0.829555 0.558425i \(-0.188594\pi\)
0.829555 + 0.558425i \(0.188594\pi\)
\(308\) −0.107307 −0.00611439
\(309\) 0 0
\(310\) 0 0
\(311\) −5.44521 −0.308770 −0.154385 0.988011i \(-0.549340\pi\)
−0.154385 + 0.988011i \(0.549340\pi\)
\(312\) 0 0
\(313\) −25.0928 −1.41833 −0.709163 0.705044i \(-0.750927\pi\)
−0.709163 + 0.705044i \(0.750927\pi\)
\(314\) 10.9360 0.617154
\(315\) 0 0
\(316\) −0.340173 −0.0191362
\(317\) 21.1773 1.18943 0.594717 0.803935i \(-0.297264\pi\)
0.594717 + 0.803935i \(0.297264\pi\)
\(318\) 0 0
\(319\) 7.41855 0.415359
\(320\) 0 0
\(321\) 0 0
\(322\) −1.78992 −0.0997484
\(323\) 37.1461 2.06686
\(324\) 0 0
\(325\) 0 0
\(326\) 35.3028 1.95524
\(327\) 0 0
\(328\) −18.6225 −1.02825
\(329\) −1.52973 −0.0843369
\(330\) 0 0
\(331\) −6.70701 −0.368650 −0.184325 0.982865i \(-0.559010\pi\)
−0.184325 + 0.982865i \(0.559010\pi\)
\(332\) 0.630898 0.0346250
\(333\) 0 0
\(334\) 7.41014 0.405465
\(335\) 0 0
\(336\) 0 0
\(337\) −23.7503 −1.29376 −0.646881 0.762591i \(-0.723927\pi\)
−0.646881 + 0.762591i \(0.723927\pi\)
\(338\) 54.7887 2.98011
\(339\) 0 0
\(340\) 0 0
\(341\) 6.34017 0.343340
\(342\) 0 0
\(343\) 4.04557 0.218440
\(344\) 0.729794 0.0393479
\(345\) 0 0
\(346\) −19.8348 −1.06633
\(347\) −31.1689 −1.67323 −0.836616 0.547790i \(-0.815469\pi\)
−0.836616 + 0.547790i \(0.815469\pi\)
\(348\) 0 0
\(349\) 11.0472 0.591342 0.295671 0.955290i \(-0.404457\pi\)
0.295671 + 0.955290i \(0.404457\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −2.06278 −0.109947
\(353\) −27.4329 −1.46011 −0.730054 0.683390i \(-0.760505\pi\)
−0.730054 + 0.683390i \(0.760505\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −1.60197 −0.0849041
\(357\) 0 0
\(358\) −2.83710 −0.149945
\(359\) 19.1506 1.01073 0.505365 0.862905i \(-0.331358\pi\)
0.505365 + 0.862905i \(0.331358\pi\)
\(360\) 0 0
\(361\) 41.1978 2.16830
\(362\) 15.8264 0.831818
\(363\) 0 0
\(364\) 0.748046 0.0392083
\(365\) 0 0
\(366\) 0 0
\(367\) 14.5692 0.760504 0.380252 0.924883i \(-0.375837\pi\)
0.380252 + 0.924883i \(0.375837\pi\)
\(368\) −18.4079 −0.959577
\(369\) 0 0
\(370\) 0 0
\(371\) −1.67420 −0.0869203
\(372\) 0 0
\(373\) 8.81432 0.456388 0.228194 0.973616i \(-0.426718\pi\)
0.228194 + 0.973616i \(0.426718\pi\)
\(374\) 7.36910 0.381047
\(375\) 0 0
\(376\) −13.2085 −0.681175
\(377\) −51.7152 −2.66347
\(378\) 0 0
\(379\) −23.5174 −1.20801 −0.604005 0.796980i \(-0.706429\pi\)
−0.604005 + 0.796980i \(0.706429\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 17.7275 0.907019
\(383\) −19.3028 −0.986329 −0.493164 0.869936i \(-0.664160\pi\)
−0.493164 + 0.869936i \(0.664160\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −5.95055 −0.302875
\(387\) 0 0
\(388\) −1.72753 −0.0877019
\(389\) 20.0410 1.01612 0.508060 0.861321i \(-0.330363\pi\)
0.508060 + 0.861321i \(0.330363\pi\)
\(390\) 0 0
\(391\) 19.1506 0.968488
\(392\) 17.3596 0.876795
\(393\) 0 0
\(394\) −13.1955 −0.664781
\(395\) 0 0
\(396\) 0 0
\(397\) 33.3607 1.67433 0.837163 0.546954i \(-0.184213\pi\)
0.837163 + 0.546954i \(0.184213\pi\)
\(398\) 12.7961 0.641409
\(399\) 0 0
\(400\) 0 0
\(401\) 37.6475 1.88003 0.940014 0.341135i \(-0.110811\pi\)
0.940014 + 0.341135i \(0.110811\pi\)
\(402\) 0 0
\(403\) −44.1978 −2.20165
\(404\) 3.16743 0.157586
\(405\) 0 0
\(406\) −3.31965 −0.164752
\(407\) 3.41855 0.169451
\(408\) 0 0
\(409\) −25.2039 −1.24625 −0.623127 0.782120i \(-0.714138\pi\)
−0.623127 + 0.782120i \(0.714138\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −2.48709 −0.122530
\(413\) 1.04718 0.0515284
\(414\) 0 0
\(415\) 0 0
\(416\) 14.3798 0.705027
\(417\) 0 0
\(418\) 11.9421 0.584109
\(419\) −17.8432 −0.871700 −0.435850 0.900019i \(-0.643552\pi\)
−0.435850 + 0.900019i \(0.643552\pi\)
\(420\) 0 0
\(421\) −11.8120 −0.575684 −0.287842 0.957678i \(-0.592938\pi\)
−0.287842 + 0.957678i \(0.592938\pi\)
\(422\) 39.8888 1.94176
\(423\) 0 0
\(424\) −14.4559 −0.702040
\(425\) 0 0
\(426\) 0 0
\(427\) 1.94214 0.0939868
\(428\) 4.51518 0.218249
\(429\) 0 0
\(430\) 0 0
\(431\) −16.6803 −0.803464 −0.401732 0.915757i \(-0.631592\pi\)
−0.401732 + 0.915757i \(0.631592\pi\)
\(432\) 0 0
\(433\) 28.3135 1.36066 0.680330 0.732906i \(-0.261836\pi\)
0.680330 + 0.732906i \(0.261836\pi\)
\(434\) −2.83710 −0.136185
\(435\) 0 0
\(436\) −2.33033 −0.111603
\(437\) 31.0349 1.48460
\(438\) 0 0
\(439\) −27.4452 −1.30989 −0.654944 0.755677i \(-0.727308\pi\)
−0.654944 + 0.755677i \(0.727308\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −51.3705 −2.44345
\(443\) 0.412408 0.0195941 0.00979704 0.999952i \(-0.496881\pi\)
0.00979704 + 0.999952i \(0.496881\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 14.8203 0.701761
\(447\) 0 0
\(448\) −1.75276 −0.0828100
\(449\) −1.33403 −0.0629568 −0.0314784 0.999504i \(-0.510022\pi\)
−0.0314784 + 0.999504i \(0.510022\pi\)
\(450\) 0 0
\(451\) 7.41855 0.349326
\(452\) −6.08906 −0.286405
\(453\) 0 0
\(454\) −28.3051 −1.32842
\(455\) 0 0
\(456\) 0 0
\(457\) −11.6514 −0.545030 −0.272515 0.962151i \(-0.587855\pi\)
−0.272515 + 0.962151i \(0.587855\pi\)
\(458\) 26.3980 1.23350
\(459\) 0 0
\(460\) 0 0
\(461\) 2.05786 0.0958440 0.0479220 0.998851i \(-0.484740\pi\)
0.0479220 + 0.998851i \(0.484740\pi\)
\(462\) 0 0
\(463\) −28.7792 −1.33748 −0.668742 0.743494i \(-0.733167\pi\)
−0.668742 + 0.743494i \(0.733167\pi\)
\(464\) −34.1399 −1.58491
\(465\) 0 0
\(466\) −6.32192 −0.292857
\(467\) 1.84324 0.0852952 0.0426476 0.999090i \(-0.486421\pi\)
0.0426476 + 0.999090i \(0.486421\pi\)
\(468\) 0 0
\(469\) 1.78992 0.0826509
\(470\) 0 0
\(471\) 0 0
\(472\) 9.04187 0.416186
\(473\) −0.290725 −0.0133675
\(474\) 0 0
\(475\) 0 0
\(476\) −0.513749 −0.0235477
\(477\) 0 0
\(478\) 6.72138 0.307429
\(479\) −26.8371 −1.22622 −0.613109 0.789998i \(-0.710081\pi\)
−0.613109 + 0.789998i \(0.710081\pi\)
\(480\) 0 0
\(481\) −23.8310 −1.08660
\(482\) −8.00984 −0.364838
\(483\) 0 0
\(484\) 0.369102 0.0167774
\(485\) 0 0
\(486\) 0 0
\(487\) −28.5646 −1.29439 −0.647193 0.762326i \(-0.724057\pi\)
−0.647193 + 0.762326i \(0.724057\pi\)
\(488\) 16.7694 0.759115
\(489\) 0 0
\(490\) 0 0
\(491\) −25.9877 −1.17281 −0.586405 0.810018i \(-0.699457\pi\)
−0.586405 + 0.810018i \(0.699457\pi\)
\(492\) 0 0
\(493\) 35.5174 1.59963
\(494\) −83.2495 −3.74557
\(495\) 0 0
\(496\) −29.1773 −1.31010
\(497\) −1.47641 −0.0662260
\(498\) 0 0
\(499\) −27.5174 −1.23185 −0.615925 0.787805i \(-0.711218\pi\)
−0.615925 + 0.787805i \(0.711218\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 12.7480 0.568973
\(503\) 22.6576 1.01025 0.505125 0.863046i \(-0.331446\pi\)
0.505125 + 0.863046i \(0.331446\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 6.15676 0.273701
\(507\) 0 0
\(508\) −1.79833 −0.0797880
\(509\) 27.8432 1.23413 0.617065 0.786912i \(-0.288322\pi\)
0.617065 + 0.786912i \(0.288322\pi\)
\(510\) 0 0
\(511\) −0.327887 −0.0145049
\(512\) −13.6114 −0.601546
\(513\) 0 0
\(514\) 16.9627 0.748191
\(515\) 0 0
\(516\) 0 0
\(517\) 5.26180 0.231413
\(518\) −1.52973 −0.0672126
\(519\) 0 0
\(520\) 0 0
\(521\) −29.7009 −1.30122 −0.650609 0.759413i \(-0.725486\pi\)
−0.650609 + 0.759413i \(0.725486\pi\)
\(522\) 0 0
\(523\) −24.7565 −1.08252 −0.541262 0.840854i \(-0.682053\pi\)
−0.541262 + 0.840854i \(0.682053\pi\)
\(524\) 3.20394 0.139965
\(525\) 0 0
\(526\) −4.57304 −0.199394
\(527\) 30.3545 1.32226
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) −0.832567 −0.0360963
\(533\) −51.7152 −2.24004
\(534\) 0 0
\(535\) 0 0
\(536\) 15.4551 0.667557
\(537\) 0 0
\(538\) 24.3857 1.05134
\(539\) −6.91548 −0.297871
\(540\) 0 0
\(541\) −12.5236 −0.538431 −0.269216 0.963080i \(-0.586764\pi\)
−0.269216 + 0.963080i \(0.586764\pi\)
\(542\) −9.66967 −0.415348
\(543\) 0 0
\(544\) −9.87587 −0.423425
\(545\) 0 0
\(546\) 0 0
\(547\) 14.2784 0.610502 0.305251 0.952272i \(-0.401260\pi\)
0.305251 + 0.952272i \(0.401260\pi\)
\(548\) −4.64915 −0.198602
\(549\) 0 0
\(550\) 0 0
\(551\) 57.5585 2.45207
\(552\) 0 0
\(553\) 0.267938 0.0113939
\(554\) −7.89269 −0.335328
\(555\) 0 0
\(556\) −3.65983 −0.155211
\(557\) −3.57918 −0.151655 −0.0758274 0.997121i \(-0.524160\pi\)
−0.0758274 + 0.997121i \(0.524160\pi\)
\(558\) 0 0
\(559\) 2.02666 0.0857187
\(560\) 0 0
\(561\) 0 0
\(562\) −33.6910 −1.42117
\(563\) 28.9588 1.22047 0.610234 0.792222i \(-0.291076\pi\)
0.610234 + 0.792222i \(0.291076\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −39.9649 −1.67985
\(567\) 0 0
\(568\) −12.7480 −0.534896
\(569\) 25.8264 1.08270 0.541350 0.840797i \(-0.317913\pi\)
0.541350 + 0.840797i \(0.317913\pi\)
\(570\) 0 0
\(571\) 17.1194 0.716425 0.358213 0.933640i \(-0.383386\pi\)
0.358213 + 0.933640i \(0.383386\pi\)
\(572\) −2.57304 −0.107584
\(573\) 0 0
\(574\) −3.31965 −0.138560
\(575\) 0 0
\(576\) 0 0
\(577\) −22.5692 −0.939567 −0.469783 0.882782i \(-0.655668\pi\)
−0.469783 + 0.882782i \(0.655668\pi\)
\(578\) 9.11450 0.379113
\(579\) 0 0
\(580\) 0 0
\(581\) −0.496928 −0.0206161
\(582\) 0 0
\(583\) 5.75872 0.238502
\(584\) −2.83114 −0.117153
\(585\) 0 0
\(586\) 16.0326 0.662302
\(587\) 3.63317 0.149957 0.0749784 0.997185i \(-0.476111\pi\)
0.0749784 + 0.997185i \(0.476111\pi\)
\(588\) 0 0
\(589\) 49.1917 2.02691
\(590\) 0 0
\(591\) 0 0
\(592\) −15.7321 −0.646584
\(593\) 12.3051 0.505310 0.252655 0.967556i \(-0.418696\pi\)
0.252655 + 0.967556i \(0.418696\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0.465732 0.0190771
\(597\) 0 0
\(598\) −42.9192 −1.75510
\(599\) −19.6865 −0.804368 −0.402184 0.915559i \(-0.631749\pi\)
−0.402184 + 0.915559i \(0.631749\pi\)
\(600\) 0 0
\(601\) 25.8843 1.05584 0.527921 0.849294i \(-0.322972\pi\)
0.527921 + 0.849294i \(0.322972\pi\)
\(602\) 0.130094 0.00530222
\(603\) 0 0
\(604\) 0.591290 0.0240593
\(605\) 0 0
\(606\) 0 0
\(607\) 41.5357 1.68588 0.842941 0.538006i \(-0.180822\pi\)
0.842941 + 0.538006i \(0.180822\pi\)
\(608\) −16.0045 −0.649070
\(609\) 0 0
\(610\) 0 0
\(611\) −36.6803 −1.48393
\(612\) 0 0
\(613\) 20.6453 0.833855 0.416927 0.908940i \(-0.363107\pi\)
0.416927 + 0.908940i \(0.363107\pi\)
\(614\) 44.7442 1.80573
\(615\) 0 0
\(616\) 0.729794 0.0294042
\(617\) 8.69472 0.350036 0.175018 0.984565i \(-0.444002\pi\)
0.175018 + 0.984565i \(0.444002\pi\)
\(618\) 0 0
\(619\) 2.65368 0.106661 0.0533303 0.998577i \(-0.483016\pi\)
0.0533303 + 0.998577i \(0.483016\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −8.38121 −0.336056
\(623\) 1.26180 0.0505528
\(624\) 0 0
\(625\) 0 0
\(626\) −38.6225 −1.54367
\(627\) 0 0
\(628\) 2.62249 0.104649
\(629\) 16.3668 0.652588
\(630\) 0 0
\(631\) −10.6393 −0.423544 −0.211772 0.977319i \(-0.567923\pi\)
−0.211772 + 0.977319i \(0.567923\pi\)
\(632\) 2.31351 0.0920265
\(633\) 0 0
\(634\) 32.5958 1.29455
\(635\) 0 0
\(636\) 0 0
\(637\) 48.2083 1.91008
\(638\) 11.4186 0.452065
\(639\) 0 0
\(640\) 0 0
\(641\) 35.8576 1.41629 0.708145 0.706067i \(-0.249532\pi\)
0.708145 + 0.706067i \(0.249532\pi\)
\(642\) 0 0
\(643\) −12.2146 −0.481697 −0.240849 0.970563i \(-0.577426\pi\)
−0.240849 + 0.970563i \(0.577426\pi\)
\(644\) −0.429229 −0.0169140
\(645\) 0 0
\(646\) 57.1748 2.24951
\(647\) 25.9421 1.01989 0.509945 0.860207i \(-0.329666\pi\)
0.509945 + 0.860207i \(0.329666\pi\)
\(648\) 0 0
\(649\) −3.60197 −0.141390
\(650\) 0 0
\(651\) 0 0
\(652\) 8.46573 0.331544
\(653\) −14.3402 −0.561174 −0.280587 0.959829i \(-0.590529\pi\)
−0.280587 + 0.959829i \(0.590529\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −34.1399 −1.33294
\(657\) 0 0
\(658\) −2.35455 −0.0917899
\(659\) −6.52359 −0.254123 −0.127062 0.991895i \(-0.540555\pi\)
−0.127062 + 0.991895i \(0.540555\pi\)
\(660\) 0 0
\(661\) 3.16290 0.123022 0.0615112 0.998106i \(-0.480408\pi\)
0.0615112 + 0.998106i \(0.480408\pi\)
\(662\) −10.3234 −0.401228
\(663\) 0 0
\(664\) −4.29072 −0.166512
\(665\) 0 0
\(666\) 0 0
\(667\) 29.6742 1.14899
\(668\) 1.77698 0.0687532
\(669\) 0 0
\(670\) 0 0
\(671\) −6.68035 −0.257892
\(672\) 0 0
\(673\) 18.4885 0.712680 0.356340 0.934356i \(-0.384024\pi\)
0.356340 + 0.934356i \(0.384024\pi\)
\(674\) −36.5562 −1.40809
\(675\) 0 0
\(676\) 13.1385 0.505327
\(677\) 15.0966 0.580211 0.290105 0.956995i \(-0.406310\pi\)
0.290105 + 0.956995i \(0.406310\pi\)
\(678\) 0 0
\(679\) 1.36069 0.0522186
\(680\) 0 0
\(681\) 0 0
\(682\) 9.75872 0.373681
\(683\) −12.4657 −0.476988 −0.238494 0.971144i \(-0.576654\pi\)
−0.238494 + 0.971144i \(0.576654\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 6.22690 0.237744
\(687\) 0 0
\(688\) 1.33791 0.0510072
\(689\) −40.1445 −1.52938
\(690\) 0 0
\(691\) −30.7214 −1.16870 −0.584348 0.811503i \(-0.698650\pi\)
−0.584348 + 0.811503i \(0.698650\pi\)
\(692\) −4.75646 −0.180813
\(693\) 0 0
\(694\) −47.9748 −1.82110
\(695\) 0 0
\(696\) 0 0
\(697\) 35.5174 1.34532
\(698\) 17.0037 0.643599
\(699\) 0 0
\(700\) 0 0
\(701\) −17.9955 −0.679679 −0.339840 0.940483i \(-0.610373\pi\)
−0.339840 + 0.940483i \(0.610373\pi\)
\(702\) 0 0
\(703\) 26.5236 1.00036
\(704\) 6.02893 0.227224
\(705\) 0 0
\(706\) −42.2245 −1.58914
\(707\) −2.49484 −0.0938281
\(708\) 0 0
\(709\) 23.5897 0.885929 0.442965 0.896539i \(-0.353927\pi\)
0.442965 + 0.896539i \(0.353927\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 10.8950 0.408306
\(713\) 25.3607 0.949765
\(714\) 0 0
\(715\) 0 0
\(716\) −0.680346 −0.0254257
\(717\) 0 0
\(718\) 29.4764 1.10005
\(719\) −24.5646 −0.916106 −0.458053 0.888925i \(-0.651453\pi\)
−0.458053 + 0.888925i \(0.651453\pi\)
\(720\) 0 0
\(721\) 1.95896 0.0729556
\(722\) 63.4112 2.35992
\(723\) 0 0
\(724\) 3.79523 0.141048
\(725\) 0 0
\(726\) 0 0
\(727\) −8.51130 −0.315667 −0.157833 0.987466i \(-0.550451\pi\)
−0.157833 + 0.987466i \(0.550451\pi\)
\(728\) −5.08745 −0.188553
\(729\) 0 0
\(730\) 0 0
\(731\) −1.39189 −0.0514809
\(732\) 0 0
\(733\) 29.8615 1.10296 0.551480 0.834188i \(-0.314063\pi\)
0.551480 + 0.834188i \(0.314063\pi\)
\(734\) 22.4247 0.827711
\(735\) 0 0
\(736\) −8.25112 −0.304140
\(737\) −6.15676 −0.226787
\(738\) 0 0
\(739\) −36.7526 −1.35197 −0.675983 0.736917i \(-0.736281\pi\)
−0.675983 + 0.736917i \(0.736281\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −2.57691 −0.0946015
\(743\) −2.17501 −0.0797933 −0.0398966 0.999204i \(-0.512703\pi\)
−0.0398966 + 0.999204i \(0.512703\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 13.5669 0.496719
\(747\) 0 0
\(748\) 1.76713 0.0646128
\(749\) −3.55640 −0.129948
\(750\) 0 0
\(751\) −44.4580 −1.62229 −0.811147 0.584842i \(-0.801157\pi\)
−0.811147 + 0.584842i \(0.801157\pi\)
\(752\) −24.2146 −0.883016
\(753\) 0 0
\(754\) −79.5995 −2.89884
\(755\) 0 0
\(756\) 0 0
\(757\) 26.4247 0.960422 0.480211 0.877153i \(-0.340560\pi\)
0.480211 + 0.877153i \(0.340560\pi\)
\(758\) −36.1978 −1.31476
\(759\) 0 0
\(760\) 0 0
\(761\) 27.6163 1.00109 0.500546 0.865710i \(-0.333133\pi\)
0.500546 + 0.865710i \(0.333133\pi\)
\(762\) 0 0
\(763\) 1.83549 0.0664493
\(764\) 4.25112 0.153800
\(765\) 0 0
\(766\) −29.7107 −1.07349
\(767\) 25.1096 0.906654
\(768\) 0 0
\(769\) 27.8432 1.00405 0.502027 0.864852i \(-0.332588\pi\)
0.502027 + 0.864852i \(0.332588\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.42696 −0.0513575
\(773\) −18.0267 −0.648374 −0.324187 0.945993i \(-0.605091\pi\)
−0.324187 + 0.945993i \(0.605091\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 11.7489 0.421760
\(777\) 0 0
\(778\) 30.8469 1.10592
\(779\) 57.5585 2.06225
\(780\) 0 0
\(781\) 5.07838 0.181719
\(782\) 29.4764 1.05407
\(783\) 0 0
\(784\) 31.8248 1.13660
\(785\) 0 0
\(786\) 0 0
\(787\) −35.6391 −1.27040 −0.635199 0.772349i \(-0.719082\pi\)
−0.635199 + 0.772349i \(0.719082\pi\)
\(788\) −3.16433 −0.112725
\(789\) 0 0
\(790\) 0 0
\(791\) 4.79606 0.170528
\(792\) 0 0
\(793\) 46.5692 1.65372
\(794\) 51.3484 1.82229
\(795\) 0 0
\(796\) 3.06854 0.108761
\(797\) −4.97948 −0.176382 −0.0881911 0.996104i \(-0.528109\pi\)
−0.0881911 + 0.996104i \(0.528109\pi\)
\(798\) 0 0
\(799\) 25.1917 0.891217
\(800\) 0 0
\(801\) 0 0
\(802\) 57.9467 2.04617
\(803\) 1.12783 0.0398001
\(804\) 0 0
\(805\) 0 0
\(806\) −68.0288 −2.39621
\(807\) 0 0
\(808\) −21.5417 −0.757833
\(809\) −31.6697 −1.11345 −0.556723 0.830698i \(-0.687942\pi\)
−0.556723 + 0.830698i \(0.687942\pi\)
\(810\) 0 0
\(811\) −21.7998 −0.765493 −0.382747 0.923853i \(-0.625022\pi\)
−0.382747 + 0.923853i \(0.625022\pi\)
\(812\) −0.796064 −0.0279364
\(813\) 0 0
\(814\) 5.26180 0.184426
\(815\) 0 0
\(816\) 0 0
\(817\) −2.25565 −0.0789153
\(818\) −38.7936 −1.35639
\(819\) 0 0
\(820\) 0 0
\(821\) 10.9360 0.381669 0.190834 0.981622i \(-0.438881\pi\)
0.190834 + 0.981622i \(0.438881\pi\)
\(822\) 0 0
\(823\) −45.9709 −1.60244 −0.801222 0.598367i \(-0.795817\pi\)
−0.801222 + 0.598367i \(0.795817\pi\)
\(824\) 16.9146 0.589249
\(825\) 0 0
\(826\) 1.61181 0.0560820
\(827\) −22.9711 −0.798782 −0.399391 0.916781i \(-0.630778\pi\)
−0.399391 + 0.916781i \(0.630778\pi\)
\(828\) 0 0
\(829\) −48.1543 −1.67247 −0.836234 0.548373i \(-0.815248\pi\)
−0.836234 + 0.548373i \(0.815248\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −42.0281 −1.45706
\(833\) −33.1089 −1.14716
\(834\) 0 0
\(835\) 0 0
\(836\) 2.86376 0.0990453
\(837\) 0 0
\(838\) −27.4641 −0.948732
\(839\) 17.9278 0.618935 0.309468 0.950910i \(-0.399849\pi\)
0.309468 + 0.950910i \(0.399849\pi\)
\(840\) 0 0
\(841\) 26.0349 0.897755
\(842\) −18.1810 −0.626558
\(843\) 0 0
\(844\) 9.56547 0.329257
\(845\) 0 0
\(846\) 0 0
\(847\) −0.290725 −0.00998942
\(848\) −26.5015 −0.910064
\(849\) 0 0
\(850\) 0 0
\(851\) 13.6742 0.468746
\(852\) 0 0
\(853\) 9.54799 0.326917 0.163458 0.986550i \(-0.447735\pi\)
0.163458 + 0.986550i \(0.447735\pi\)
\(854\) 2.98932 0.102292
\(855\) 0 0
\(856\) −30.7077 −1.04957
\(857\) 10.5776 0.361323 0.180662 0.983545i \(-0.442176\pi\)
0.180662 + 0.983545i \(0.442176\pi\)
\(858\) 0 0
\(859\) −31.3340 −1.06910 −0.534552 0.845136i \(-0.679520\pi\)
−0.534552 + 0.845136i \(0.679520\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −25.6742 −0.874467
\(863\) 56.7792 1.93279 0.966394 0.257066i \(-0.0827557\pi\)
0.966394 + 0.257066i \(0.0827557\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 43.5798 1.48090
\(867\) 0 0
\(868\) −0.680346 −0.0230924
\(869\) −0.921622 −0.0312639
\(870\) 0 0
\(871\) 42.9192 1.45426
\(872\) 15.8486 0.536700
\(873\) 0 0
\(874\) 47.7686 1.61580
\(875\) 0 0
\(876\) 0 0
\(877\) −5.87832 −0.198497 −0.0992483 0.995063i \(-0.531644\pi\)
−0.0992483 + 0.995063i \(0.531644\pi\)
\(878\) −42.2434 −1.42564
\(879\) 0 0
\(880\) 0 0
\(881\) −8.21008 −0.276605 −0.138302 0.990390i \(-0.544165\pi\)
−0.138302 + 0.990390i \(0.544165\pi\)
\(882\) 0 0
\(883\) 27.0349 0.909797 0.454898 0.890543i \(-0.349676\pi\)
0.454898 + 0.890543i \(0.349676\pi\)
\(884\) −12.3188 −0.414327
\(885\) 0 0
\(886\) 0.634773 0.0213256
\(887\) −48.7442 −1.63667 −0.818335 0.574742i \(-0.805102\pi\)
−0.818335 + 0.574742i \(0.805102\pi\)
\(888\) 0 0
\(889\) 1.41646 0.0475066
\(890\) 0 0
\(891\) 0 0
\(892\) 3.55395 0.118995
\(893\) 40.8248 1.36615
\(894\) 0 0
\(895\) 0 0
\(896\) −3.89723 −0.130197
\(897\) 0 0
\(898\) −2.05332 −0.0685203
\(899\) 47.0349 1.56870
\(900\) 0 0
\(901\) 27.5708 0.918516
\(902\) 11.4186 0.380196
\(903\) 0 0
\(904\) 41.4116 1.37733
\(905\) 0 0
\(906\) 0 0
\(907\) 32.4657 1.07801 0.539003 0.842304i \(-0.318801\pi\)
0.539003 + 0.842304i \(0.318801\pi\)
\(908\) −6.78765 −0.225256
\(909\) 0 0
\(910\) 0 0
\(911\) 4.84939 0.160667 0.0803337 0.996768i \(-0.474401\pi\)
0.0803337 + 0.996768i \(0.474401\pi\)
\(912\) 0 0
\(913\) 1.70928 0.0565688
\(914\) −17.9337 −0.593195
\(915\) 0 0
\(916\) 6.33033 0.209160
\(917\) −2.52359 −0.0833363
\(918\) 0 0
\(919\) −14.3980 −0.474947 −0.237474 0.971394i \(-0.576319\pi\)
−0.237474 + 0.971394i \(0.576319\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 3.16743 0.104314
\(923\) −35.4017 −1.16526
\(924\) 0 0
\(925\) 0 0
\(926\) −44.2967 −1.45568
\(927\) 0 0
\(928\) −15.3028 −0.502340
\(929\) 11.1629 0.366243 0.183121 0.983090i \(-0.441380\pi\)
0.183121 + 0.983090i \(0.441380\pi\)
\(930\) 0 0
\(931\) −53.6553 −1.75848
\(932\) −1.51602 −0.0496588
\(933\) 0 0
\(934\) 2.83710 0.0928328
\(935\) 0 0
\(936\) 0 0
\(937\) −48.6453 −1.58917 −0.794586 0.607152i \(-0.792312\pi\)
−0.794586 + 0.607152i \(0.792312\pi\)
\(938\) 2.75503 0.0899548
\(939\) 0 0
\(940\) 0 0
\(941\) −16.4124 −0.535029 −0.267515 0.963554i \(-0.586202\pi\)
−0.267515 + 0.963554i \(0.586202\pi\)
\(942\) 0 0
\(943\) 29.6742 0.966325
\(944\) 16.5761 0.539508
\(945\) 0 0
\(946\) −0.447480 −0.0145488
\(947\) −45.7198 −1.48569 −0.742847 0.669462i \(-0.766525\pi\)
−0.742847 + 0.669462i \(0.766525\pi\)
\(948\) 0 0
\(949\) −7.86216 −0.255216
\(950\) 0 0
\(951\) 0 0
\(952\) 3.49400 0.113241
\(953\) −26.4619 −0.857184 −0.428592 0.903498i \(-0.640990\pi\)
−0.428592 + 0.903498i \(0.640990\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 1.61181 0.0521296
\(957\) 0 0
\(958\) −41.3074 −1.33458
\(959\) 3.66192 0.118249
\(960\) 0 0
\(961\) 9.19779 0.296703
\(962\) −36.6803 −1.18262
\(963\) 0 0
\(964\) −1.92079 −0.0618643
\(965\) 0 0
\(966\) 0 0
\(967\) −50.5464 −1.62546 −0.812731 0.582639i \(-0.802020\pi\)
−0.812731 + 0.582639i \(0.802020\pi\)
\(968\) −2.51026 −0.0806828
\(969\) 0 0
\(970\) 0 0
\(971\) 57.3074 1.83908 0.919540 0.392995i \(-0.128561\pi\)
0.919540 + 0.392995i \(0.128561\pi\)
\(972\) 0 0
\(973\) 2.88267 0.0924143
\(974\) −43.9664 −1.40877
\(975\) 0 0
\(976\) 30.7427 0.984051
\(977\) −45.7875 −1.46487 −0.732436 0.680836i \(-0.761616\pi\)
−0.732436 + 0.680836i \(0.761616\pi\)
\(978\) 0 0
\(979\) −4.34017 −0.138713
\(980\) 0 0
\(981\) 0 0
\(982\) −40.0000 −1.27645
\(983\) 55.0805 1.75679 0.878397 0.477932i \(-0.158613\pi\)
0.878397 + 0.477932i \(0.158613\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 54.6681 1.74099
\(987\) 0 0
\(988\) −19.9635 −0.635123
\(989\) −1.16290 −0.0369780
\(990\) 0 0
\(991\) 40.6947 1.29271 0.646355 0.763037i \(-0.276292\pi\)
0.646355 + 0.763037i \(0.276292\pi\)
\(992\) −13.0784 −0.415239
\(993\) 0 0
\(994\) −2.27247 −0.0720785
\(995\) 0 0
\(996\) 0 0
\(997\) 32.7610 1.03755 0.518775 0.854911i \(-0.326388\pi\)
0.518775 + 0.854911i \(0.326388\pi\)
\(998\) −42.3545 −1.34071
\(999\) 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 2475.2.a.be.1.2 3
3.2 odd 2 825.2.a.h.1.2 3
5.2 odd 4 495.2.c.d.199.5 6
5.3 odd 4 495.2.c.d.199.2 6
5.4 even 2 2475.2.a.y.1.2 3
15.2 even 4 165.2.c.a.34.2 6
15.8 even 4 165.2.c.a.34.5 yes 6
15.14 odd 2 825.2.a.n.1.2 3
33.32 even 2 9075.2.a.ck.1.2 3
60.23 odd 4 2640.2.d.i.529.3 6
60.47 odd 4 2640.2.d.i.529.6 6
165.32 odd 4 1815.2.c.d.364.5 6
165.98 odd 4 1815.2.c.d.364.2 6
165.164 even 2 9075.2.a.cc.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.c.a.34.2 6 15.2 even 4
165.2.c.a.34.5 yes 6 15.8 even 4
495.2.c.d.199.2 6 5.3 odd 4
495.2.c.d.199.5 6 5.2 odd 4
825.2.a.h.1.2 3 3.2 odd 2
825.2.a.n.1.2 3 15.14 odd 2
1815.2.c.d.364.2 6 165.98 odd 4
1815.2.c.d.364.5 6 165.32 odd 4
2475.2.a.y.1.2 3 5.4 even 2
2475.2.a.be.1.2 3 1.1 even 1 trivial
2640.2.d.i.529.3 6 60.23 odd 4
2640.2.d.i.529.6 6 60.47 odd 4
9075.2.a.cc.1.2 3 165.164 even 2
9075.2.a.ck.1.2 3 33.32 even 2