Properties

Label 775.2.a.e.1.3
Level $775$
Weight $2$
Character 775.1
Self dual yes
Analytic conductor $6.188$
Analytic rank $1$
Dimension $4$
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] = [775,2,Mod(1,775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(775, 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("775.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.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: \(6.18840615665\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.8468.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 3x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 155)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.89122\) of defining polynomial
Character \(\chi\) \(=\) 775.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.576713 q^{2} +1.89122 q^{3} -1.66740 q^{4} +1.09069 q^{6} -4.24412 q^{7} -2.11504 q^{8} +0.576713 q^{9} +O(q^{10})\) \(q+0.576713 q^{2} +1.89122 q^{3} -1.66740 q^{4} +1.09069 q^{6} -4.24412 q^{7} -2.11504 q^{8} +0.576713 q^{9} +1.09069 q^{11} -3.15343 q^{12} -4.69175 q^{13} -2.44763 q^{14} +2.11504 q^{16} -2.80053 q^{17} +0.332597 q^{18} -4.20573 q^{19} -8.02655 q^{21} +0.629015 q^{22} +4.24412 q^{23} -4.00000 q^{24} -2.70579 q^{26} -4.58297 q^{27} +7.07665 q^{28} +6.02655 q^{29} -1.00000 q^{31} +5.44984 q^{32} +2.06273 q^{33} -1.61510 q^{34} -0.961612 q^{36} -6.53428 q^{37} -2.42550 q^{38} -8.87313 q^{39} -7.42189 q^{41} -4.62901 q^{42} -8.58297 q^{43} -1.81862 q^{44} +2.44763 q^{46} +10.0893 q^{47} +4.00000 q^{48} +11.0125 q^{49} -5.29642 q^{51} +7.82304 q^{52} +7.45610 q^{53} -2.64306 q^{54} +8.97646 q^{56} -7.95395 q^{57} +3.47559 q^{58} +0.716096 q^{59} +9.78244 q^{61} -0.576713 q^{62} -2.44763 q^{63} -1.08708 q^{64} +1.18961 q^{66} -3.27207 q^{67} +4.66961 q^{68} +8.02655 q^{69} -4.42329 q^{71} -1.21977 q^{72} -2.50772 q^{73} -3.76840 q^{74} +7.01264 q^{76} -4.62901 q^{77} -5.11725 q^{78} -17.1925 q^{79} -10.3975 q^{81} -4.28030 q^{82} +5.55501 q^{83} +13.3835 q^{84} -4.94991 q^{86} +11.3975 q^{87} -2.30685 q^{88} +4.55097 q^{89} +19.9123 q^{91} -7.07665 q^{92} -1.89122 q^{93} +5.81862 q^{94} +10.3069 q^{96} -3.42690 q^{97} +6.35105 q^{98} +0.629015 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} - q^{3} + 5 q^{4} - 4 q^{6} - 2 q^{7} - 3 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} - q^{3} + 5 q^{4} - 4 q^{6} - 2 q^{7} - 3 q^{8} - q^{9} - 4 q^{11} - 6 q^{12} - 10 q^{13} - 16 q^{14} + 3 q^{16} - 11 q^{17} + 13 q^{18} - 3 q^{19} - 8 q^{22} + 2 q^{23} - 16 q^{24} + 2 q^{26} - q^{27} + 24 q^{28} - 8 q^{29} - 4 q^{31} - 7 q^{32} + 10 q^{33} - 2 q^{34} - 5 q^{36} - 3 q^{37} + 22 q^{38} - 10 q^{39} - 11 q^{41} - 8 q^{42} - 17 q^{43} - 24 q^{44} + 16 q^{46} + 10 q^{47} + 16 q^{48} + 16 q^{49} + q^{51} - 22 q^{52} - 13 q^{53} + 4 q^{54} - 24 q^{56} - 25 q^{57} + 10 q^{58} - 3 q^{59} + 22 q^{61} + q^{62} - 16 q^{63} - 9 q^{64} + 32 q^{66} + 12 q^{67} - 28 q^{68} - 21 q^{71} + 13 q^{72} - 19 q^{73} - 2 q^{74} + 2 q^{76} - 8 q^{77} + 20 q^{78} - 2 q^{79} - 20 q^{81} - 36 q^{82} + 15 q^{83} + 36 q^{84} + 8 q^{86} + 24 q^{87} + 4 q^{88} - 10 q^{89} - 4 q^{91} - 24 q^{92} + q^{93} + 40 q^{94} + 28 q^{96} - 4 q^{97} - 37 q^{98} - 8 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.576713 0.407797 0.203899 0.978992i \(-0.434639\pi\)
0.203899 + 0.978992i \(0.434639\pi\)
\(3\) 1.89122 1.09190 0.545948 0.837819i \(-0.316170\pi\)
0.545948 + 0.837819i \(0.316170\pi\)
\(4\) −1.66740 −0.833701
\(5\) 0 0
\(6\) 1.09069 0.445272
\(7\) −4.24412 −1.60412 −0.802062 0.597240i \(-0.796264\pi\)
−0.802062 + 0.597240i \(0.796264\pi\)
\(8\) −2.11504 −0.747779
\(9\) 0.576713 0.192238
\(10\) 0 0
\(11\) 1.09069 0.328855 0.164428 0.986389i \(-0.447422\pi\)
0.164428 + 0.986389i \(0.447422\pi\)
\(12\) −3.15343 −0.910315
\(13\) −4.69175 −1.30126 −0.650629 0.759396i \(-0.725495\pi\)
−0.650629 + 0.759396i \(0.725495\pi\)
\(14\) −2.44763 −0.654158
\(15\) 0 0
\(16\) 2.11504 0.528759
\(17\) −2.80053 −0.679228 −0.339614 0.940565i \(-0.610296\pi\)
−0.339614 + 0.940565i \(0.610296\pi\)
\(18\) 0.332597 0.0783939
\(19\) −4.20573 −0.964860 −0.482430 0.875935i \(-0.660246\pi\)
−0.482430 + 0.875935i \(0.660246\pi\)
\(20\) 0 0
\(21\) −8.02655 −1.75154
\(22\) 0.629015 0.134106
\(23\) 4.24412 0.884959 0.442480 0.896779i \(-0.354099\pi\)
0.442480 + 0.896779i \(0.354099\pi\)
\(24\) −4.00000 −0.816497
\(25\) 0 0
\(26\) −2.70579 −0.530649
\(27\) −4.58297 −0.881993
\(28\) 7.07665 1.33736
\(29\) 6.02655 1.11910 0.559552 0.828796i \(-0.310973\pi\)
0.559552 + 0.828796i \(0.310973\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 5.44984 0.963405
\(33\) 2.06273 0.359076
\(34\) −1.61510 −0.276987
\(35\) 0 0
\(36\) −0.961612 −0.160269
\(37\) −6.53428 −1.07423 −0.537114 0.843510i \(-0.680486\pi\)
−0.537114 + 0.843510i \(0.680486\pi\)
\(38\) −2.42550 −0.393467
\(39\) −8.87313 −1.42084
\(40\) 0 0
\(41\) −7.42189 −1.15910 −0.579552 0.814935i \(-0.696772\pi\)
−0.579552 + 0.814935i \(0.696772\pi\)
\(42\) −4.62901 −0.714272
\(43\) −8.58297 −1.30889 −0.654445 0.756109i \(-0.727098\pi\)
−0.654445 + 0.756109i \(0.727098\pi\)
\(44\) −1.81862 −0.274167
\(45\) 0 0
\(46\) 2.44763 0.360884
\(47\) 10.0893 1.47167 0.735837 0.677159i \(-0.236789\pi\)
0.735837 + 0.677159i \(0.236789\pi\)
\(48\) 4.00000 0.577350
\(49\) 11.0125 1.57322
\(50\) 0 0
\(51\) −5.29642 −0.741647
\(52\) 7.82304 1.08486
\(53\) 7.45610 1.02417 0.512087 0.858934i \(-0.328873\pi\)
0.512087 + 0.858934i \(0.328873\pi\)
\(54\) −2.64306 −0.359674
\(55\) 0 0
\(56\) 8.97646 1.19953
\(57\) −7.95395 −1.05353
\(58\) 3.47559 0.456367
\(59\) 0.716096 0.0932278 0.0466139 0.998913i \(-0.485157\pi\)
0.0466139 + 0.998913i \(0.485157\pi\)
\(60\) 0 0
\(61\) 9.78244 1.25251 0.626257 0.779617i \(-0.284586\pi\)
0.626257 + 0.779617i \(0.284586\pi\)
\(62\) −0.576713 −0.0732426
\(63\) −2.44763 −0.308373
\(64\) −1.08708 −0.135885
\(65\) 0 0
\(66\) 1.18961 0.146430
\(67\) −3.27207 −0.399747 −0.199874 0.979822i \(-0.564053\pi\)
−0.199874 + 0.979822i \(0.564053\pi\)
\(68\) 4.66961 0.566273
\(69\) 8.02655 0.966284
\(70\) 0 0
\(71\) −4.42329 −0.524948 −0.262474 0.964939i \(-0.584538\pi\)
−0.262474 + 0.964939i \(0.584538\pi\)
\(72\) −1.21977 −0.143751
\(73\) −2.50772 −0.293507 −0.146753 0.989173i \(-0.546882\pi\)
−0.146753 + 0.989173i \(0.546882\pi\)
\(74\) −3.76840 −0.438067
\(75\) 0 0
\(76\) 7.01264 0.804405
\(77\) −4.62901 −0.527525
\(78\) −5.11725 −0.579414
\(79\) −17.1925 −1.93431 −0.967153 0.254194i \(-0.918190\pi\)
−0.967153 + 0.254194i \(0.918190\pi\)
\(80\) 0 0
\(81\) −10.3975 −1.15528
\(82\) −4.28030 −0.472680
\(83\) 5.55501 0.609742 0.304871 0.952394i \(-0.401387\pi\)
0.304871 + 0.952394i \(0.401387\pi\)
\(84\) 13.3835 1.46026
\(85\) 0 0
\(86\) −4.94991 −0.533762
\(87\) 11.3975 1.22194
\(88\) −2.30685 −0.245911
\(89\) 4.55097 0.482401 0.241201 0.970475i \(-0.422459\pi\)
0.241201 + 0.970475i \(0.422459\pi\)
\(90\) 0 0
\(91\) 19.9123 2.08738
\(92\) −7.07665 −0.737792
\(93\) −1.89122 −0.196110
\(94\) 5.81862 0.600145
\(95\) 0 0
\(96\) 10.3069 1.05194
\(97\) −3.42690 −0.347949 −0.173974 0.984750i \(-0.555661\pi\)
−0.173974 + 0.984750i \(0.555661\pi\)
\(98\) 6.35105 0.641553
\(99\) 0.629015 0.0632184
\(100\) 0 0
\(101\) 3.93226 0.391274 0.195637 0.980676i \(-0.437323\pi\)
0.195637 + 0.980676i \(0.437323\pi\)
\(102\) −3.05451 −0.302442
\(103\) −2.44763 −0.241173 −0.120586 0.992703i \(-0.538477\pi\)
−0.120586 + 0.992703i \(0.538477\pi\)
\(104\) 9.92322 0.973052
\(105\) 0 0
\(106\) 4.30003 0.417655
\(107\) 20.1452 1.94751 0.973755 0.227599i \(-0.0730874\pi\)
0.973755 + 0.227599i \(0.0730874\pi\)
\(108\) 7.64166 0.735319
\(109\) −13.0495 −1.24992 −0.624958 0.780659i \(-0.714884\pi\)
−0.624958 + 0.780659i \(0.714884\pi\)
\(110\) 0 0
\(111\) −12.3578 −1.17295
\(112\) −8.97646 −0.848196
\(113\) −18.5148 −1.74172 −0.870862 0.491527i \(-0.836439\pi\)
−0.870862 + 0.491527i \(0.836439\pi\)
\(114\) −4.58715 −0.429626
\(115\) 0 0
\(116\) −10.0487 −0.932998
\(117\) −2.70579 −0.250150
\(118\) 0.412982 0.0380181
\(119\) 11.8858 1.08957
\(120\) 0 0
\(121\) −9.81039 −0.891854
\(122\) 5.64166 0.510771
\(123\) −14.0364 −1.26562
\(124\) 1.66740 0.149737
\(125\) 0 0
\(126\) −1.41158 −0.125754
\(127\) 11.9779 1.06286 0.531432 0.847101i \(-0.321654\pi\)
0.531432 + 0.847101i \(0.321654\pi\)
\(128\) −11.5266 −1.01882
\(129\) −16.2323 −1.42917
\(130\) 0 0
\(131\) −2.19321 −0.191622 −0.0958110 0.995400i \(-0.530544\pi\)
−0.0958110 + 0.995400i \(0.530544\pi\)
\(132\) −3.43941 −0.299362
\(133\) 17.8496 1.54776
\(134\) −1.88704 −0.163016
\(135\) 0 0
\(136\) 5.92322 0.507912
\(137\) 9.10056 0.777513 0.388756 0.921341i \(-0.372905\pi\)
0.388756 + 0.921341i \(0.372905\pi\)
\(138\) 4.62901 0.394048
\(139\) 4.11865 0.349339 0.174669 0.984627i \(-0.444114\pi\)
0.174669 + 0.984627i \(0.444114\pi\)
\(140\) 0 0
\(141\) 19.0811 1.60692
\(142\) −2.55097 −0.214072
\(143\) −5.11725 −0.427926
\(144\) 1.21977 0.101647
\(145\) 0 0
\(146\) −1.44623 −0.119691
\(147\) 20.8271 1.71779
\(148\) 10.8953 0.895586
\(149\) −20.4137 −1.67235 −0.836176 0.548461i \(-0.815214\pi\)
−0.836176 + 0.548461i \(0.815214\pi\)
\(150\) 0 0
\(151\) −9.79648 −0.797226 −0.398613 0.917119i \(-0.630508\pi\)
−0.398613 + 0.917119i \(0.630508\pi\)
\(152\) 8.89527 0.721502
\(153\) −1.61510 −0.130573
\(154\) −2.66961 −0.215123
\(155\) 0 0
\(156\) 14.7951 1.18455
\(157\) 6.82444 0.544649 0.272325 0.962205i \(-0.412208\pi\)
0.272325 + 0.962205i \(0.412208\pi\)
\(158\) −9.91513 −0.788805
\(159\) 14.1011 1.11829
\(160\) 0 0
\(161\) −18.0125 −1.41958
\(162\) −5.99639 −0.471121
\(163\) −21.2958 −1.66802 −0.834009 0.551751i \(-0.813960\pi\)
−0.834009 + 0.551751i \(0.813960\pi\)
\(164\) 12.3753 0.966347
\(165\) 0 0
\(166\) 3.20365 0.248651
\(167\) −11.3780 −0.880460 −0.440230 0.897885i \(-0.645103\pi\)
−0.440230 + 0.897885i \(0.645103\pi\)
\(168\) 16.9765 1.30976
\(169\) 9.01251 0.693270
\(170\) 0 0
\(171\) −2.42550 −0.185482
\(172\) 14.3113 1.09122
\(173\) −9.56641 −0.727320 −0.363660 0.931532i \(-0.618473\pi\)
−0.363660 + 0.931532i \(0.618473\pi\)
\(174\) 6.57310 0.498306
\(175\) 0 0
\(176\) 2.30685 0.173885
\(177\) 1.35430 0.101795
\(178\) 2.62460 0.196722
\(179\) −3.44094 −0.257188 −0.128594 0.991697i \(-0.541046\pi\)
−0.128594 + 0.991697i \(0.541046\pi\)
\(180\) 0 0
\(181\) −12.9890 −0.965463 −0.482731 0.875768i \(-0.660355\pi\)
−0.482731 + 0.875768i \(0.660355\pi\)
\(182\) 11.4837 0.851228
\(183\) 18.5007 1.36761
\(184\) −8.97646 −0.661753
\(185\) 0 0
\(186\) −1.09069 −0.0799733
\(187\) −3.05451 −0.223368
\(188\) −16.8229 −1.22694
\(189\) 19.4506 1.41483
\(190\) 0 0
\(191\) 26.5539 1.92137 0.960685 0.277642i \(-0.0895528\pi\)
0.960685 + 0.277642i \(0.0895528\pi\)
\(192\) −2.05591 −0.148373
\(193\) 10.8510 0.781071 0.390536 0.920588i \(-0.372290\pi\)
0.390536 + 0.920588i \(0.372290\pi\)
\(194\) −1.97633 −0.141892
\(195\) 0 0
\(196\) −18.3623 −1.31159
\(197\) 15.0517 1.07239 0.536195 0.844094i \(-0.319861\pi\)
0.536195 + 0.844094i \(0.319861\pi\)
\(198\) 0.362761 0.0257803
\(199\) 2.64306 0.187361 0.0936806 0.995602i \(-0.470137\pi\)
0.0936806 + 0.995602i \(0.470137\pi\)
\(200\) 0 0
\(201\) −6.18820 −0.436482
\(202\) 2.26778 0.159561
\(203\) −25.5774 −1.79518
\(204\) 8.83126 0.618312
\(205\) 0 0
\(206\) −1.41158 −0.0983495
\(207\) 2.44763 0.170122
\(208\) −9.92322 −0.688052
\(209\) −4.58715 −0.317299
\(210\) 0 0
\(211\) 6.93586 0.477485 0.238742 0.971083i \(-0.423265\pi\)
0.238742 + 0.971083i \(0.423265\pi\)
\(212\) −12.4323 −0.853855
\(213\) −8.36541 −0.573188
\(214\) 11.6180 0.794189
\(215\) 0 0
\(216\) 9.69315 0.659535
\(217\) 4.24412 0.288109
\(218\) −7.52581 −0.509712
\(219\) −4.74265 −0.320479
\(220\) 0 0
\(221\) 13.1394 0.883851
\(222\) −7.12687 −0.478324
\(223\) 19.0616 1.27646 0.638229 0.769847i \(-0.279667\pi\)
0.638229 + 0.769847i \(0.279667\pi\)
\(224\) −23.1298 −1.54542
\(225\) 0 0
\(226\) −10.6777 −0.710271
\(227\) −17.8983 −1.18795 −0.593975 0.804483i \(-0.702442\pi\)
−0.593975 + 0.804483i \(0.702442\pi\)
\(228\) 13.2624 0.878327
\(229\) 28.3334 1.87232 0.936162 0.351569i \(-0.114352\pi\)
0.936162 + 0.351569i \(0.114352\pi\)
\(230\) 0 0
\(231\) −8.75448 −0.576003
\(232\) −12.7464 −0.836841
\(233\) −12.4742 −0.817211 −0.408606 0.912711i \(-0.633985\pi\)
−0.408606 + 0.912711i \(0.633985\pi\)
\(234\) −1.56046 −0.102011
\(235\) 0 0
\(236\) −1.19402 −0.0777242
\(237\) −32.5148 −2.11206
\(238\) 6.85467 0.444322
\(239\) 12.4395 0.804647 0.402323 0.915498i \(-0.368203\pi\)
0.402323 + 0.915498i \(0.368203\pi\)
\(240\) 0 0
\(241\) −12.5229 −0.806670 −0.403335 0.915052i \(-0.632149\pi\)
−0.403335 + 0.915052i \(0.632149\pi\)
\(242\) −5.65778 −0.363696
\(243\) −5.91513 −0.379456
\(244\) −16.3113 −1.04422
\(245\) 0 0
\(246\) −8.09498 −0.516117
\(247\) 19.7322 1.25553
\(248\) 2.11504 0.134305
\(249\) 10.5058 0.665775
\(250\) 0 0
\(251\) −20.2273 −1.27673 −0.638367 0.769732i \(-0.720390\pi\)
−0.638367 + 0.769732i \(0.720390\pi\)
\(252\) 4.08119 0.257091
\(253\) 4.62901 0.291024
\(254\) 6.90778 0.433433
\(255\) 0 0
\(256\) −4.47338 −0.279586
\(257\) 9.58702 0.598022 0.299011 0.954250i \(-0.403343\pi\)
0.299011 + 0.954250i \(0.403343\pi\)
\(258\) −9.36136 −0.582813
\(259\) 27.7322 1.72320
\(260\) 0 0
\(261\) 3.47559 0.215134
\(262\) −1.26485 −0.0781429
\(263\) 5.18985 0.320020 0.160010 0.987115i \(-0.448847\pi\)
0.160010 + 0.987115i \(0.448847\pi\)
\(264\) −4.36276 −0.268509
\(265\) 0 0
\(266\) 10.2941 0.631171
\(267\) 8.60688 0.526732
\(268\) 5.45586 0.333270
\(269\) 8.03907 0.490151 0.245075 0.969504i \(-0.421187\pi\)
0.245075 + 0.969504i \(0.421187\pi\)
\(270\) 0 0
\(271\) 21.8215 1.32556 0.662781 0.748813i \(-0.269376\pi\)
0.662781 + 0.748813i \(0.269376\pi\)
\(272\) −5.92322 −0.359148
\(273\) 37.6586 2.27920
\(274\) 5.24840 0.317068
\(275\) 0 0
\(276\) −13.3835 −0.805592
\(277\) −20.7378 −1.24601 −0.623007 0.782217i \(-0.714089\pi\)
−0.623007 + 0.782217i \(0.714089\pi\)
\(278\) 2.37527 0.142459
\(279\) −0.576713 −0.0345269
\(280\) 0 0
\(281\) −24.7986 −1.47936 −0.739679 0.672960i \(-0.765023\pi\)
−0.739679 + 0.672960i \(0.765023\pi\)
\(282\) 11.0043 0.655296
\(283\) −7.78244 −0.462618 −0.231309 0.972880i \(-0.574301\pi\)
−0.231309 + 0.972880i \(0.574301\pi\)
\(284\) 7.37540 0.437650
\(285\) 0 0
\(286\) −2.95118 −0.174507
\(287\) 31.4993 1.85935
\(288\) 3.14299 0.185203
\(289\) −9.15703 −0.538649
\(290\) 0 0
\(291\) −6.48101 −0.379924
\(292\) 4.18138 0.244697
\(293\) 8.25663 0.482357 0.241179 0.970481i \(-0.422466\pi\)
0.241179 + 0.970481i \(0.422466\pi\)
\(294\) 12.0112 0.700510
\(295\) 0 0
\(296\) 13.8202 0.803285
\(297\) −4.99860 −0.290048
\(298\) −11.7728 −0.681981
\(299\) −19.9123 −1.15156
\(300\) 0 0
\(301\) 36.4271 2.09962
\(302\) −5.64975 −0.325107
\(303\) 7.43676 0.427231
\(304\) −8.89527 −0.510179
\(305\) 0 0
\(306\) −0.931449 −0.0532474
\(307\) −4.21603 −0.240622 −0.120311 0.992736i \(-0.538389\pi\)
−0.120311 + 0.992736i \(0.538389\pi\)
\(308\) 7.71843 0.439798
\(309\) −4.62901 −0.263335
\(310\) 0 0
\(311\) 10.3452 0.586625 0.293312 0.956017i \(-0.405242\pi\)
0.293312 + 0.956017i \(0.405242\pi\)
\(312\) 18.7670 1.06247
\(313\) 10.9679 0.619941 0.309970 0.950746i \(-0.399681\pi\)
0.309970 + 0.950746i \(0.399681\pi\)
\(314\) 3.93574 0.222107
\(315\) 0 0
\(316\) 28.6668 1.61263
\(317\) 0.214760 0.0120621 0.00603105 0.999982i \(-0.498080\pi\)
0.00603105 + 0.999982i \(0.498080\pi\)
\(318\) 8.13229 0.456036
\(319\) 6.57310 0.368023
\(320\) 0 0
\(321\) 38.0990 2.12648
\(322\) −10.3880 −0.578903
\(323\) 11.7783 0.655360
\(324\) 17.3369 0.963160
\(325\) 0 0
\(326\) −12.2816 −0.680213
\(327\) −24.6795 −1.36478
\(328\) 15.6976 0.866753
\(329\) −42.8201 −2.36075
\(330\) 0 0
\(331\) −10.7130 −0.588840 −0.294420 0.955676i \(-0.595126\pi\)
−0.294420 + 0.955676i \(0.595126\pi\)
\(332\) −9.26245 −0.508343
\(333\) −3.76840 −0.206507
\(334\) −6.56186 −0.359049
\(335\) 0 0
\(336\) −16.9765 −0.926142
\(337\) 9.54692 0.520054 0.260027 0.965601i \(-0.416269\pi\)
0.260027 + 0.965601i \(0.416269\pi\)
\(338\) 5.19763 0.282714
\(339\) −35.0155 −1.90178
\(340\) 0 0
\(341\) −1.09069 −0.0590642
\(342\) −1.39881 −0.0756392
\(343\) −17.0296 −0.919510
\(344\) 18.1533 0.978760
\(345\) 0 0
\(346\) −5.51707 −0.296599
\(347\) −9.01391 −0.483892 −0.241946 0.970290i \(-0.577786\pi\)
−0.241946 + 0.970290i \(0.577786\pi\)
\(348\) −19.0043 −1.01874
\(349\) −34.1933 −1.83033 −0.915163 0.403085i \(-0.867938\pi\)
−0.915163 + 0.403085i \(0.867938\pi\)
\(350\) 0 0
\(351\) 21.5021 1.14770
\(352\) 5.94409 0.316821
\(353\) −12.1312 −0.645676 −0.322838 0.946454i \(-0.604637\pi\)
−0.322838 + 0.946454i \(0.604637\pi\)
\(354\) 0.781039 0.0415118
\(355\) 0 0
\(356\) −7.58829 −0.402179
\(357\) 22.4786 1.18969
\(358\) −1.98443 −0.104880
\(359\) −28.2317 −1.49001 −0.745006 0.667058i \(-0.767553\pi\)
−0.745006 + 0.667058i \(0.767553\pi\)
\(360\) 0 0
\(361\) −1.31186 −0.0690452
\(362\) −7.49090 −0.393713
\(363\) −18.5536 −0.973812
\(364\) −33.2019 −1.74025
\(365\) 0 0
\(366\) 10.6696 0.557709
\(367\) 1.11989 0.0584580 0.0292290 0.999573i \(-0.490695\pi\)
0.0292290 + 0.999573i \(0.490695\pi\)
\(368\) 8.97646 0.467930
\(369\) −4.28030 −0.222823
\(370\) 0 0
\(371\) −31.6445 −1.64290
\(372\) 3.15343 0.163497
\(373\) −1.97786 −0.102410 −0.0512049 0.998688i \(-0.516306\pi\)
−0.0512049 + 0.998688i \(0.516306\pi\)
\(374\) −1.76157 −0.0910888
\(375\) 0 0
\(376\) −21.3392 −1.10049
\(377\) −28.2751 −1.45624
\(378\) 11.2174 0.576963
\(379\) −30.4957 −1.56646 −0.783230 0.621732i \(-0.786429\pi\)
−0.783230 + 0.621732i \(0.786429\pi\)
\(380\) 0 0
\(381\) 22.6528 1.16054
\(382\) 15.3139 0.783529
\(383\) −26.7575 −1.36725 −0.683623 0.729835i \(-0.739597\pi\)
−0.683623 + 0.729835i \(0.739597\pi\)
\(384\) −21.7994 −1.11244
\(385\) 0 0
\(386\) 6.25790 0.318519
\(387\) −4.94991 −0.251618
\(388\) 5.71402 0.290085
\(389\) −26.7102 −1.35426 −0.677131 0.735863i \(-0.736777\pi\)
−0.677131 + 0.735863i \(0.736777\pi\)
\(390\) 0 0
\(391\) −11.8858 −0.601089
\(392\) −23.2919 −1.17642
\(393\) −4.14785 −0.209231
\(394\) 8.68051 0.437318
\(395\) 0 0
\(396\) −1.04882 −0.0527052
\(397\) 5.25523 0.263752 0.131876 0.991266i \(-0.457900\pi\)
0.131876 + 0.991266i \(0.457900\pi\)
\(398\) 1.52428 0.0764054
\(399\) 33.7575 1.68999
\(400\) 0 0
\(401\) 0.256629 0.0128154 0.00640772 0.999979i \(-0.497960\pi\)
0.00640772 + 0.999979i \(0.497960\pi\)
\(402\) −3.56882 −0.177996
\(403\) 4.69175 0.233713
\(404\) −6.55665 −0.326206
\(405\) 0 0
\(406\) −14.7508 −0.732070
\(407\) −7.12687 −0.353266
\(408\) 11.2021 0.554588
\(409\) 23.1285 1.14363 0.571815 0.820383i \(-0.306240\pi\)
0.571815 + 0.820383i \(0.306240\pi\)
\(410\) 0 0
\(411\) 17.2112 0.848963
\(412\) 4.08119 0.201066
\(413\) −3.03920 −0.149549
\(414\) 1.41158 0.0693754
\(415\) 0 0
\(416\) −25.5693 −1.25364
\(417\) 7.78926 0.381442
\(418\) −2.64546 −0.129394
\(419\) 25.7315 1.25707 0.628534 0.777782i \(-0.283655\pi\)
0.628534 + 0.777782i \(0.283655\pi\)
\(420\) 0 0
\(421\) −13.8459 −0.674806 −0.337403 0.941360i \(-0.609548\pi\)
−0.337403 + 0.941360i \(0.609548\pi\)
\(422\) 4.00000 0.194717
\(423\) 5.81862 0.282911
\(424\) −15.7699 −0.765855
\(425\) 0 0
\(426\) −4.82444 −0.233745
\(427\) −41.5178 −2.00919
\(428\) −33.5902 −1.62364
\(429\) −9.67784 −0.467250
\(430\) 0 0
\(431\) −19.1263 −0.921280 −0.460640 0.887587i \(-0.652380\pi\)
−0.460640 + 0.887587i \(0.652380\pi\)
\(432\) −9.69315 −0.466362
\(433\) 9.00140 0.432580 0.216290 0.976329i \(-0.430604\pi\)
0.216290 + 0.976329i \(0.430604\pi\)
\(434\) 2.44763 0.117490
\(435\) 0 0
\(436\) 21.7588 1.04206
\(437\) −17.8496 −0.853862
\(438\) −2.73515 −0.130690
\(439\) 3.44972 0.164646 0.0823230 0.996606i \(-0.473766\pi\)
0.0823230 + 0.996606i \(0.473766\pi\)
\(440\) 0 0
\(441\) 6.35105 0.302431
\(442\) 7.57765 0.360432
\(443\) 16.6053 0.788944 0.394472 0.918908i \(-0.370928\pi\)
0.394472 + 0.918908i \(0.370928\pi\)
\(444\) 20.6053 0.977887
\(445\) 0 0
\(446\) 10.9930 0.520536
\(447\) −38.6067 −1.82604
\(448\) 4.61370 0.217977
\(449\) −4.15635 −0.196150 −0.0980752 0.995179i \(-0.531269\pi\)
−0.0980752 + 0.995179i \(0.531269\pi\)
\(450\) 0 0
\(451\) −8.09498 −0.381178
\(452\) 30.8716 1.45208
\(453\) −18.5273 −0.870488
\(454\) −10.3222 −0.484443
\(455\) 0 0
\(456\) 16.8229 0.787805
\(457\) −31.0517 −1.45254 −0.726269 0.687411i \(-0.758747\pi\)
−0.726269 + 0.687411i \(0.758747\pi\)
\(458\) 16.3402 0.763529
\(459\) 12.8347 0.599074
\(460\) 0 0
\(461\) 0.666722 0.0310523 0.0155262 0.999879i \(-0.495058\pi\)
0.0155262 + 0.999879i \(0.495058\pi\)
\(462\) −5.04882 −0.234892
\(463\) −2.06273 −0.0958634 −0.0479317 0.998851i \(-0.515263\pi\)
−0.0479317 + 0.998851i \(0.515263\pi\)
\(464\) 12.7464 0.591736
\(465\) 0 0
\(466\) −7.19402 −0.333257
\(467\) −40.8173 −1.88880 −0.944400 0.328799i \(-0.893356\pi\)
−0.944400 + 0.328799i \(0.893356\pi\)
\(468\) 4.51164 0.208551
\(469\) 13.8870 0.641244
\(470\) 0 0
\(471\) 12.9065 0.594701
\(472\) −1.51457 −0.0697138
\(473\) −9.36136 −0.430436
\(474\) −18.7517 −0.861293
\(475\) 0 0
\(476\) −19.8184 −0.908373
\(477\) 4.30003 0.196885
\(478\) 7.17404 0.328133
\(479\) 27.4772 1.25547 0.627733 0.778429i \(-0.283983\pi\)
0.627733 + 0.778429i \(0.283983\pi\)
\(480\) 0 0
\(481\) 30.6572 1.39785
\(482\) −7.22210 −0.328958
\(483\) −34.0656 −1.55004
\(484\) 16.3579 0.743540
\(485\) 0 0
\(486\) −3.41133 −0.154741
\(487\) 37.8676 1.71594 0.857971 0.513698i \(-0.171725\pi\)
0.857971 + 0.513698i \(0.171725\pi\)
\(488\) −20.6902 −0.936602
\(489\) −40.2751 −1.82130
\(490\) 0 0
\(491\) 6.12827 0.276565 0.138282 0.990393i \(-0.455842\pi\)
0.138282 + 0.990393i \(0.455842\pi\)
\(492\) 23.4044 1.05515
\(493\) −16.8775 −0.760126
\(494\) 11.3798 0.512002
\(495\) 0 0
\(496\) −2.11504 −0.0949680
\(497\) 18.7729 0.842082
\(498\) 6.05880 0.271501
\(499\) −13.3767 −0.598822 −0.299411 0.954124i \(-0.596790\pi\)
−0.299411 + 0.954124i \(0.596790\pi\)
\(500\) 0 0
\(501\) −21.5184 −0.961371
\(502\) −11.6653 −0.520649
\(503\) 9.11877 0.406586 0.203293 0.979118i \(-0.434836\pi\)
0.203293 + 0.979118i \(0.434836\pi\)
\(504\) 5.17684 0.230595
\(505\) 0 0
\(506\) 2.66961 0.118679
\(507\) 17.0446 0.756979
\(508\) −19.9719 −0.886111
\(509\) 23.4853 1.04097 0.520484 0.853872i \(-0.325752\pi\)
0.520484 + 0.853872i \(0.325752\pi\)
\(510\) 0 0
\(511\) 10.6431 0.470821
\(512\) 20.4734 0.904804
\(513\) 19.2747 0.851000
\(514\) 5.52895 0.243872
\(515\) 0 0
\(516\) 27.0658 1.19150
\(517\) 11.0043 0.483968
\(518\) 15.9935 0.702715
\(519\) −18.0922 −0.794158
\(520\) 0 0
\(521\) −29.0776 −1.27391 −0.636956 0.770900i \(-0.719807\pi\)
−0.636956 + 0.770900i \(0.719807\pi\)
\(522\) 2.00442 0.0877309
\(523\) 22.7741 0.995840 0.497920 0.867223i \(-0.334097\pi\)
0.497920 + 0.867223i \(0.334097\pi\)
\(524\) 3.65697 0.159755
\(525\) 0 0
\(526\) 2.99305 0.130503
\(527\) 2.80053 0.121993
\(528\) 4.36276 0.189865
\(529\) −4.98749 −0.216847
\(530\) 0 0
\(531\) 0.412982 0.0179219
\(532\) −29.7625 −1.29037
\(533\) 34.8216 1.50829
\(534\) 4.96369 0.214800
\(535\) 0 0
\(536\) 6.92055 0.298922
\(537\) −6.50757 −0.280822
\(538\) 4.63623 0.199882
\(539\) 12.0112 0.517361
\(540\) 0 0
\(541\) −12.8201 −0.551179 −0.275590 0.961275i \(-0.588873\pi\)
−0.275590 + 0.961275i \(0.588873\pi\)
\(542\) 12.5847 0.540561
\(543\) −24.5650 −1.05419
\(544\) −15.2624 −0.654372
\(545\) 0 0
\(546\) 21.7182 0.929452
\(547\) 29.9317 1.27979 0.639893 0.768464i \(-0.278979\pi\)
0.639893 + 0.768464i \(0.278979\pi\)
\(548\) −15.1743 −0.648214
\(549\) 5.64166 0.240780
\(550\) 0 0
\(551\) −25.3460 −1.07978
\(552\) −16.9765 −0.722566
\(553\) 72.9669 3.10287
\(554\) −11.9597 −0.508121
\(555\) 0 0
\(556\) −6.86744 −0.291244
\(557\) −40.3195 −1.70839 −0.854195 0.519953i \(-0.825950\pi\)
−0.854195 + 0.519953i \(0.825950\pi\)
\(558\) −0.332597 −0.0140800
\(559\) 40.2691 1.70320
\(560\) 0 0
\(561\) −5.77675 −0.243895
\(562\) −14.3016 −0.603278
\(563\) 26.0825 1.09924 0.549622 0.835413i \(-0.314772\pi\)
0.549622 + 0.835413i \(0.314772\pi\)
\(564\) −31.8158 −1.33969
\(565\) 0 0
\(566\) −4.48823 −0.188654
\(567\) 44.1284 1.85322
\(568\) 9.35542 0.392545
\(569\) −29.1351 −1.22141 −0.610703 0.791860i \(-0.709113\pi\)
−0.610703 + 0.791860i \(0.709113\pi\)
\(570\) 0 0
\(571\) 8.58185 0.359139 0.179570 0.983745i \(-0.442529\pi\)
0.179570 + 0.983745i \(0.442529\pi\)
\(572\) 8.53251 0.356762
\(573\) 50.2192 2.09794
\(574\) 18.1661 0.758237
\(575\) 0 0
\(576\) −0.626934 −0.0261222
\(577\) −0.522883 −0.0217679 −0.0108840 0.999941i \(-0.503465\pi\)
−0.0108840 + 0.999941i \(0.503465\pi\)
\(578\) −5.28098 −0.219660
\(579\) 20.5216 0.852849
\(580\) 0 0
\(581\) −23.5761 −0.978102
\(582\) −3.73768 −0.154932
\(583\) 8.13229 0.336805
\(584\) 5.30392 0.219478
\(585\) 0 0
\(586\) 4.76170 0.196704
\(587\) 0.829733 0.0342467 0.0171234 0.999853i \(-0.494549\pi\)
0.0171234 + 0.999853i \(0.494549\pi\)
\(588\) −34.7271 −1.43212
\(589\) 4.20573 0.173294
\(590\) 0 0
\(591\) 28.4661 1.17094
\(592\) −13.8202 −0.568008
\(593\) 11.7950 0.484361 0.242180 0.970231i \(-0.422137\pi\)
0.242180 + 0.970231i \(0.422137\pi\)
\(594\) −2.88275 −0.118281
\(595\) 0 0
\(596\) 34.0378 1.39424
\(597\) 4.99860 0.204579
\(598\) −11.4837 −0.469603
\(599\) 33.7841 1.38038 0.690190 0.723629i \(-0.257527\pi\)
0.690190 + 0.723629i \(0.257527\pi\)
\(600\) 0 0
\(601\) −2.98329 −0.121691 −0.0608454 0.998147i \(-0.519380\pi\)
−0.0608454 + 0.998147i \(0.519380\pi\)
\(602\) 21.0080 0.856221
\(603\) −1.88704 −0.0768464
\(604\) 16.3347 0.664649
\(605\) 0 0
\(606\) 4.28887 0.174224
\(607\) −5.99567 −0.243357 −0.121678 0.992570i \(-0.538828\pi\)
−0.121678 + 0.992570i \(0.538828\pi\)
\(608\) −22.9206 −0.929551
\(609\) −48.3725 −1.96015
\(610\) 0 0
\(611\) −47.3364 −1.91503
\(612\) 2.69302 0.108859
\(613\) −22.7615 −0.919330 −0.459665 0.888092i \(-0.652031\pi\)
−0.459665 + 0.888092i \(0.652031\pi\)
\(614\) −2.43144 −0.0981249
\(615\) 0 0
\(616\) 9.79054 0.394472
\(617\) 9.98316 0.401907 0.200953 0.979601i \(-0.435596\pi\)
0.200953 + 0.979601i \(0.435596\pi\)
\(618\) −2.66961 −0.107387
\(619\) 19.1144 0.768274 0.384137 0.923276i \(-0.374499\pi\)
0.384137 + 0.923276i \(0.374499\pi\)
\(620\) 0 0
\(621\) −19.4506 −0.780528
\(622\) 5.96623 0.239224
\(623\) −19.3148 −0.773832
\(624\) −18.7670 −0.751281
\(625\) 0 0
\(626\) 6.32531 0.252810
\(627\) −8.67530 −0.346458
\(628\) −11.3791 −0.454075
\(629\) 18.2994 0.729646
\(630\) 0 0
\(631\) 5.19932 0.206982 0.103491 0.994630i \(-0.466999\pi\)
0.103491 + 0.994630i \(0.466999\pi\)
\(632\) 36.3628 1.44643
\(633\) 13.1172 0.521364
\(634\) 0.123854 0.00491889
\(635\) 0 0
\(636\) −23.5122 −0.932321
\(637\) −51.6680 −2.04716
\(638\) 3.79079 0.150079
\(639\) −2.55097 −0.100915
\(640\) 0 0
\(641\) 8.96924 0.354264 0.177132 0.984187i \(-0.443318\pi\)
0.177132 + 0.984187i \(0.443318\pi\)
\(642\) 21.9722 0.867172
\(643\) −9.69580 −0.382365 −0.191182 0.981555i \(-0.561232\pi\)
−0.191182 + 0.981555i \(0.561232\pi\)
\(644\) 30.0341 1.18351
\(645\) 0 0
\(646\) 6.79267 0.267254
\(647\) −0.631663 −0.0248332 −0.0124166 0.999923i \(-0.503952\pi\)
−0.0124166 + 0.999923i \(0.503952\pi\)
\(648\) 21.9912 0.863895
\(649\) 0.781039 0.0306585
\(650\) 0 0
\(651\) 8.02655 0.314585
\(652\) 35.5087 1.39063
\(653\) 10.1074 0.395533 0.197767 0.980249i \(-0.436631\pi\)
0.197767 + 0.980249i \(0.436631\pi\)
\(654\) −14.2330 −0.556553
\(655\) 0 0
\(656\) −15.6976 −0.612887
\(657\) −1.44623 −0.0564230
\(658\) −24.6949 −0.962707
\(659\) 36.3066 1.41431 0.707153 0.707060i \(-0.249979\pi\)
0.707153 + 0.707060i \(0.249979\pi\)
\(660\) 0 0
\(661\) −13.4456 −0.522975 −0.261487 0.965207i \(-0.584213\pi\)
−0.261487 + 0.965207i \(0.584213\pi\)
\(662\) −6.17833 −0.240127
\(663\) 24.8495 0.965073
\(664\) −11.7491 −0.455952
\(665\) 0 0
\(666\) −2.17328 −0.0842130
\(667\) 25.5774 0.990361
\(668\) 18.9718 0.734041
\(669\) 36.0496 1.39376
\(670\) 0 0
\(671\) 10.6696 0.411896
\(672\) −43.7435 −1.68744
\(673\) −21.1118 −0.813800 −0.406900 0.913473i \(-0.633390\pi\)
−0.406900 + 0.913473i \(0.633390\pi\)
\(674\) 5.50583 0.212077
\(675\) 0 0
\(676\) −15.0275 −0.577980
\(677\) −19.7726 −0.759922 −0.379961 0.925003i \(-0.624063\pi\)
−0.379961 + 0.925003i \(0.624063\pi\)
\(678\) −20.1939 −0.775542
\(679\) 14.5441 0.558153
\(680\) 0 0
\(681\) −33.8496 −1.29712
\(682\) −0.629015 −0.0240862
\(683\) 5.64293 0.215921 0.107960 0.994155i \(-0.465568\pi\)
0.107960 + 0.994155i \(0.465568\pi\)
\(684\) 4.04428 0.154637
\(685\) 0 0
\(686\) −9.82117 −0.374974
\(687\) 53.5847 2.04438
\(688\) −18.1533 −0.692088
\(689\) −34.9822 −1.33271
\(690\) 0 0
\(691\) 9.97718 0.379550 0.189775 0.981828i \(-0.439224\pi\)
0.189775 + 0.981828i \(0.439224\pi\)
\(692\) 15.9511 0.606368
\(693\) −2.66961 −0.101410
\(694\) −5.19844 −0.197330
\(695\) 0 0
\(696\) −24.1062 −0.913744
\(697\) 20.7852 0.787296
\(698\) −19.7197 −0.746402
\(699\) −23.5914 −0.892310
\(700\) 0 0
\(701\) 32.9600 1.24488 0.622441 0.782667i \(-0.286141\pi\)
0.622441 + 0.782667i \(0.286141\pi\)
\(702\) 12.4006 0.468029
\(703\) 27.4814 1.03648
\(704\) −1.18567 −0.0446866
\(705\) 0 0
\(706\) −6.99619 −0.263305
\(707\) −16.6889 −0.627653
\(708\) −2.25816 −0.0848667
\(709\) 26.2501 0.985842 0.492921 0.870074i \(-0.335929\pi\)
0.492921 + 0.870074i \(0.335929\pi\)
\(710\) 0 0
\(711\) −9.91513 −0.371846
\(712\) −9.62546 −0.360729
\(713\) −4.24412 −0.158943
\(714\) 12.9637 0.485154
\(715\) 0 0
\(716\) 5.73743 0.214418
\(717\) 23.5259 0.878591
\(718\) −16.2816 −0.607623
\(719\) −10.2635 −0.382762 −0.191381 0.981516i \(-0.561297\pi\)
−0.191381 + 0.981516i \(0.561297\pi\)
\(720\) 0 0
\(721\) 10.3880 0.386871
\(722\) −0.756565 −0.0281564
\(723\) −23.6835 −0.880800
\(724\) 21.6579 0.804908
\(725\) 0 0
\(726\) −10.7001 −0.397118
\(727\) −11.7588 −0.436109 −0.218054 0.975937i \(-0.569971\pi\)
−0.218054 + 0.975937i \(0.569971\pi\)
\(728\) −42.1153 −1.56090
\(729\) 20.0058 0.740956
\(730\) 0 0
\(731\) 24.0369 0.889035
\(732\) −30.8482 −1.14018
\(733\) −15.7669 −0.582363 −0.291181 0.956668i \(-0.594048\pi\)
−0.291181 + 0.956668i \(0.594048\pi\)
\(734\) 0.645857 0.0238390
\(735\) 0 0
\(736\) 23.1298 0.852574
\(737\) −3.56882 −0.131459
\(738\) −2.46850 −0.0908667
\(739\) −6.39806 −0.235356 −0.117678 0.993052i \(-0.537545\pi\)
−0.117678 + 0.993052i \(0.537545\pi\)
\(740\) 0 0
\(741\) 37.3180 1.37091
\(742\) −18.2498 −0.669971
\(743\) 4.17152 0.153038 0.0765190 0.997068i \(-0.475619\pi\)
0.0765190 + 0.997068i \(0.475619\pi\)
\(744\) 4.00000 0.146647
\(745\) 0 0
\(746\) −1.14066 −0.0417624
\(747\) 3.20365 0.117215
\(748\) 5.09310 0.186222
\(749\) −85.4986 −3.12405
\(750\) 0 0
\(751\) 24.9734 0.911293 0.455646 0.890161i \(-0.349408\pi\)
0.455646 + 0.890161i \(0.349408\pi\)
\(752\) 21.3392 0.778161
\(753\) −38.2542 −1.39406
\(754\) −16.3066 −0.593851
\(755\) 0 0
\(756\) −32.4321 −1.17954
\(757\) −51.6338 −1.87666 −0.938331 0.345738i \(-0.887629\pi\)
−0.938331 + 0.345738i \(0.887629\pi\)
\(758\) −17.5873 −0.638798
\(759\) 8.75448 0.317768
\(760\) 0 0
\(761\) −26.3238 −0.954236 −0.477118 0.878839i \(-0.658319\pi\)
−0.477118 + 0.878839i \(0.658319\pi\)
\(762\) 13.0641 0.473264
\(763\) 55.3836 2.00502
\(764\) −44.2760 −1.60185
\(765\) 0 0
\(766\) −15.4314 −0.557559
\(767\) −3.35974 −0.121313
\(768\) −8.46015 −0.305279
\(769\) 51.5996 1.86073 0.930364 0.366636i \(-0.119491\pi\)
0.930364 + 0.366636i \(0.119491\pi\)
\(770\) 0 0
\(771\) 18.1312 0.652978
\(772\) −18.0930 −0.651180
\(773\) 12.0627 0.433866 0.216933 0.976186i \(-0.430395\pi\)
0.216933 + 0.976186i \(0.430395\pi\)
\(774\) −2.85467 −0.102609
\(775\) 0 0
\(776\) 7.24801 0.260188
\(777\) 52.4477 1.88155
\(778\) −15.4041 −0.552264
\(779\) 31.2144 1.11837
\(780\) 0 0
\(781\) −4.82444 −0.172632
\(782\) −6.85467 −0.245123
\(783\) −27.6195 −0.987041
\(784\) 23.2919 0.831853
\(785\) 0 0
\(786\) −2.39212 −0.0853240
\(787\) 33.6557 1.19970 0.599848 0.800114i \(-0.295228\pi\)
0.599848 + 0.800114i \(0.295228\pi\)
\(788\) −25.0973 −0.894053
\(789\) 9.81514 0.349428
\(790\) 0 0
\(791\) 78.5789 2.79394
\(792\) −1.33039 −0.0472733
\(793\) −45.8968 −1.62984
\(794\) 3.03076 0.107558
\(795\) 0 0
\(796\) −4.40704 −0.156203
\(797\) 44.8414 1.58836 0.794181 0.607681i \(-0.207900\pi\)
0.794181 + 0.607681i \(0.207900\pi\)
\(798\) 19.4684 0.689173
\(799\) −28.2554 −0.999603
\(800\) 0 0
\(801\) 2.62460 0.0927356
\(802\) 0.148001 0.00522610
\(803\) −2.73515 −0.0965212
\(804\) 10.3182 0.363896
\(805\) 0 0
\(806\) 2.70579 0.0953074
\(807\) 15.2036 0.535194
\(808\) −8.31687 −0.292586
\(809\) 3.60700 0.126815 0.0634077 0.997988i \(-0.479803\pi\)
0.0634077 + 0.997988i \(0.479803\pi\)
\(810\) 0 0
\(811\) 43.5156 1.52804 0.764020 0.645193i \(-0.223223\pi\)
0.764020 + 0.645193i \(0.223223\pi\)
\(812\) 42.6478 1.49664
\(813\) 41.2693 1.44738
\(814\) −4.11016 −0.144061
\(815\) 0 0
\(816\) −11.2021 −0.392153
\(817\) 36.0976 1.26290
\(818\) 13.3385 0.466369
\(819\) 11.4837 0.401273
\(820\) 0 0
\(821\) 26.7855 0.934819 0.467409 0.884041i \(-0.345187\pi\)
0.467409 + 0.884041i \(0.345187\pi\)
\(822\) 9.92589 0.346205
\(823\) 1.00125 0.0349013 0.0174507 0.999848i \(-0.494445\pi\)
0.0174507 + 0.999848i \(0.494445\pi\)
\(824\) 5.17684 0.180344
\(825\) 0 0
\(826\) −1.75274 −0.0609857
\(827\) −4.95816 −0.172412 −0.0862060 0.996277i \(-0.527474\pi\)
−0.0862060 + 0.996277i \(0.527474\pi\)
\(828\) −4.08119 −0.141831
\(829\) 42.6668 1.48188 0.740940 0.671571i \(-0.234380\pi\)
0.740940 + 0.671571i \(0.234380\pi\)
\(830\) 0 0
\(831\) −39.2197 −1.36052
\(832\) 5.10032 0.176822
\(833\) −30.8409 −1.06857
\(834\) 4.49217 0.155551
\(835\) 0 0
\(836\) 7.64862 0.264533
\(837\) 4.58297 0.158411
\(838\) 14.8397 0.512629
\(839\) 56.4093 1.94747 0.973733 0.227692i \(-0.0731179\pi\)
0.973733 + 0.227692i \(0.0731179\pi\)
\(840\) 0 0
\(841\) 7.31936 0.252392
\(842\) −7.98508 −0.275184
\(843\) −46.8995 −1.61531
\(844\) −11.5649 −0.398080
\(845\) 0 0
\(846\) 3.35567 0.115370
\(847\) 41.6364 1.43065
\(848\) 15.7699 0.541541
\(849\) −14.7183 −0.505131
\(850\) 0 0
\(851\) −27.7322 −0.950648
\(852\) 13.9485 0.477868
\(853\) 1.39072 0.0476172 0.0238086 0.999717i \(-0.492421\pi\)
0.0238086 + 0.999717i \(0.492421\pi\)
\(854\) −23.9438 −0.819341
\(855\) 0 0
\(856\) −42.6078 −1.45631
\(857\) −44.5988 −1.52347 −0.761733 0.647892i \(-0.775651\pi\)
−0.761733 + 0.647892i \(0.775651\pi\)
\(858\) −5.58133 −0.190543
\(859\) 4.63054 0.157992 0.0789960 0.996875i \(-0.474829\pi\)
0.0789960 + 0.996875i \(0.474829\pi\)
\(860\) 0 0
\(861\) 59.5722 2.03021
\(862\) −11.0304 −0.375696
\(863\) −5.29847 −0.180362 −0.0901811 0.995925i \(-0.528745\pi\)
−0.0901811 + 0.995925i \(0.528745\pi\)
\(864\) −24.9765 −0.849716
\(865\) 0 0
\(866\) 5.19122 0.176405
\(867\) −17.3180 −0.588149
\(868\) −7.07665 −0.240197
\(869\) −18.7517 −0.636107
\(870\) 0 0
\(871\) 15.3517 0.520174
\(872\) 27.6002 0.934660
\(873\) −1.97633 −0.0668888
\(874\) −10.2941 −0.348203
\(875\) 0 0
\(876\) 7.90791 0.267183
\(877\) −29.4379 −0.994047 −0.497023 0.867737i \(-0.665574\pi\)
−0.497023 + 0.867737i \(0.665574\pi\)
\(878\) 1.98949 0.0671422
\(879\) 15.6151 0.526684
\(880\) 0 0
\(881\) −12.2360 −0.412242 −0.206121 0.978527i \(-0.566084\pi\)
−0.206121 + 0.978527i \(0.566084\pi\)
\(882\) 3.66273 0.123331
\(883\) 13.0166 0.438042 0.219021 0.975720i \(-0.429714\pi\)
0.219021 + 0.975720i \(0.429714\pi\)
\(884\) −21.9086 −0.736867
\(885\) 0 0
\(886\) 9.57651 0.321729
\(887\) 42.6931 1.43349 0.716747 0.697333i \(-0.245630\pi\)
0.716747 + 0.697333i \(0.245630\pi\)
\(888\) 26.1371 0.877104
\(889\) −50.8354 −1.70497
\(890\) 0 0
\(891\) −11.3405 −0.379921
\(892\) −31.7833 −1.06418
\(893\) −42.4328 −1.41996
\(894\) −22.2650 −0.744652
\(895\) 0 0
\(896\) 48.9203 1.63431
\(897\) −37.6586 −1.25738
\(898\) −2.39702 −0.0799896
\(899\) −6.02655 −0.200997
\(900\) 0 0
\(901\) −20.8810 −0.695648
\(902\) −4.66848 −0.155443
\(903\) 68.8917 2.29257
\(904\) 39.1595 1.30242
\(905\) 0 0
\(906\) −10.6849 −0.354983
\(907\) −2.96089 −0.0983148 −0.0491574 0.998791i \(-0.515654\pi\)
−0.0491574 + 0.998791i \(0.515654\pi\)
\(908\) 29.8436 0.990396
\(909\) 2.26778 0.0752176
\(910\) 0 0
\(911\) 50.6094 1.67676 0.838382 0.545084i \(-0.183502\pi\)
0.838382 + 0.545084i \(0.183502\pi\)
\(912\) −16.8229 −0.557062
\(913\) 6.05880 0.200517
\(914\) −17.9079 −0.592341
\(915\) 0 0
\(916\) −47.2432 −1.56096
\(917\) 9.30825 0.307386
\(918\) 7.40196 0.244301
\(919\) −53.7822 −1.77411 −0.887056 0.461661i \(-0.847253\pi\)
−0.887056 + 0.461661i \(0.847253\pi\)
\(920\) 0 0
\(921\) −7.97345 −0.262734
\(922\) 0.384507 0.0126631
\(923\) 20.7530 0.683092
\(924\) 14.5973 0.480214
\(925\) 0 0
\(926\) −1.18961 −0.0390928
\(927\) −1.41158 −0.0463624
\(928\) 32.8438 1.07815
\(929\) −23.7699 −0.779866 −0.389933 0.920843i \(-0.627502\pi\)
−0.389933 + 0.920843i \(0.627502\pi\)
\(930\) 0 0
\(931\) −46.3156 −1.51793
\(932\) 20.7995 0.681310
\(933\) 19.5651 0.640533
\(934\) −23.5399 −0.770247
\(935\) 0 0
\(936\) 5.72285 0.187057
\(937\) 15.4351 0.504242 0.252121 0.967696i \(-0.418872\pi\)
0.252121 + 0.967696i \(0.418872\pi\)
\(938\) 8.00883 0.261498
\(939\) 20.7427 0.676911
\(940\) 0 0
\(941\) −11.8314 −0.385692 −0.192846 0.981229i \(-0.561772\pi\)
−0.192846 + 0.981229i \(0.561772\pi\)
\(942\) 7.44335 0.242517
\(943\) −31.4993 −1.02576
\(944\) 1.51457 0.0492951
\(945\) 0 0
\(946\) −5.39881 −0.175531
\(947\) −25.9237 −0.842408 −0.421204 0.906966i \(-0.638392\pi\)
−0.421204 + 0.906966i \(0.638392\pi\)
\(948\) 54.2152 1.76083
\(949\) 11.7656 0.381927
\(950\) 0 0
\(951\) 0.406157 0.0131706
\(952\) −25.1388 −0.814755
\(953\) −33.0212 −1.06966 −0.534831 0.844959i \(-0.679625\pi\)
−0.534831 + 0.844959i \(0.679625\pi\)
\(954\) 2.47988 0.0802890
\(955\) 0 0
\(956\) −20.7417 −0.670835
\(957\) 12.4312 0.401843
\(958\) 15.8464 0.511975
\(959\) −38.6238 −1.24723
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 17.6804 0.570038
\(963\) 11.6180 0.374384
\(964\) 20.8807 0.672522
\(965\) 0 0
\(966\) −19.6461 −0.632102
\(967\) 7.17998 0.230893 0.115446 0.993314i \(-0.463170\pi\)
0.115446 + 0.993314i \(0.463170\pi\)
\(968\) 20.7493 0.666909
\(969\) 22.2753 0.715585
\(970\) 0 0
\(971\) 10.1377 0.325334 0.162667 0.986681i \(-0.447990\pi\)
0.162667 + 0.986681i \(0.447990\pi\)
\(972\) 9.86290 0.316353
\(973\) −17.4800 −0.560383
\(974\) 21.8387 0.699757
\(975\) 0 0
\(976\) 20.6902 0.662278
\(977\) −2.04187 −0.0653252 −0.0326626 0.999466i \(-0.510399\pi\)
−0.0326626 + 0.999466i \(0.510399\pi\)
\(978\) −23.2271 −0.742722
\(979\) 4.96369 0.158640
\(980\) 0 0
\(981\) −7.52581 −0.240281
\(982\) 3.53425 0.112782
\(983\) 38.5229 1.22869 0.614345 0.789038i \(-0.289420\pi\)
0.614345 + 0.789038i \(0.289420\pi\)
\(984\) 29.6875 0.946404
\(985\) 0 0
\(986\) −9.73349 −0.309978
\(987\) −80.9822 −2.57769
\(988\) −32.9016 −1.04674
\(989\) −36.4271 −1.15831
\(990\) 0 0
\(991\) −43.9389 −1.39576 −0.697882 0.716212i \(-0.745874\pi\)
−0.697882 + 0.716212i \(0.745874\pi\)
\(992\) −5.44984 −0.173033
\(993\) −20.2607 −0.642952
\(994\) 10.8266 0.343399
\(995\) 0 0
\(996\) −17.5173 −0.555058
\(997\) −58.9278 −1.86626 −0.933131 0.359538i \(-0.882934\pi\)
−0.933131 + 0.359538i \(0.882934\pi\)
\(998\) −7.71450 −0.244198
\(999\) 29.9464 0.947462
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 775.2.a.e.1.3 4
3.2 odd 2 6975.2.a.bn.1.2 4
5.2 odd 4 775.2.b.f.249.5 8
5.3 odd 4 775.2.b.f.249.4 8
5.4 even 2 155.2.a.e.1.2 4
15.14 odd 2 1395.2.a.l.1.3 4
20.19 odd 2 2480.2.a.x.1.4 4
35.34 odd 2 7595.2.a.s.1.2 4
40.19 odd 2 9920.2.a.cg.1.1 4
40.29 even 2 9920.2.a.cb.1.4 4
155.154 odd 2 4805.2.a.n.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
155.2.a.e.1.2 4 5.4 even 2
775.2.a.e.1.3 4 1.1 even 1 trivial
775.2.b.f.249.4 8 5.3 odd 4
775.2.b.f.249.5 8 5.2 odd 4
1395.2.a.l.1.3 4 15.14 odd 2
2480.2.a.x.1.4 4 20.19 odd 2
4805.2.a.n.1.2 4 155.154 odd 2
6975.2.a.bn.1.2 4 3.2 odd 2
7595.2.a.s.1.2 4 35.34 odd 2
9920.2.a.cb.1.4 4 40.29 even 2
9920.2.a.cg.1.1 4 40.19 odd 2