Properties

Label 847.2.a.k.1.4
Level $847$
Weight $2$
Character 847.1
Self dual yes
Analytic conductor $6.763$
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] = [847,2,Mod(1,847)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(847, 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("847.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.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: \(6.76332905120\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.2525.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 4x^{2} + 5x + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.46673\) of defining polynomial
Character \(\chi\) \(=\) 847.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.46673 q^{2} -1.61803 q^{3} +0.151302 q^{4} +0.466732 q^{5} -2.37322 q^{6} +1.00000 q^{7} -2.71154 q^{8} -0.381966 q^{9} +O(q^{10})\) \(q+1.46673 q^{2} -1.61803 q^{3} +0.151302 q^{4} +0.466732 q^{5} -2.37322 q^{6} +1.00000 q^{7} -2.71154 q^{8} -0.381966 q^{9} +0.684570 q^{10} -0.244812 q^{12} +1.58232 q^{13} +1.46673 q^{14} -0.755188 q^{15} -4.27971 q^{16} -5.22732 q^{17} -0.560242 q^{18} -4.22192 q^{19} +0.0706175 q^{20} -1.61803 q^{21} -1.80505 q^{23} +4.38737 q^{24} -4.78216 q^{25} +2.32083 q^{26} +5.47214 q^{27} +0.151302 q^{28} -2.71947 q^{29} -1.10766 q^{30} +1.29386 q^{31} -0.854102 q^{32} -7.66708 q^{34} +0.466732 q^{35} -0.0577923 q^{36} -1.94221 q^{37} -6.19242 q^{38} -2.56024 q^{39} -1.26556 q^{40} +1.04112 q^{41} -2.37322 q^{42} -8.70820 q^{43} -0.178276 q^{45} -2.64753 q^{46} -6.39530 q^{47} +6.92472 q^{48} +1.00000 q^{49} -7.01415 q^{50} +8.45799 q^{51} +0.239408 q^{52} -13.2044 q^{53} +8.02616 q^{54} -2.71154 q^{56} +6.83121 q^{57} -3.98873 q^{58} -8.60389 q^{59} -0.114262 q^{60} +15.2401 q^{61} +1.89775 q^{62} -0.381966 q^{63} +7.30669 q^{64} +0.738517 q^{65} -4.67583 q^{67} -0.790906 q^{68} +2.92064 q^{69} +0.684570 q^{70} +9.74310 q^{71} +1.03572 q^{72} +13.3200 q^{73} -2.84870 q^{74} +7.73770 q^{75} -0.638786 q^{76} -3.75519 q^{78} -3.58232 q^{79} -1.99748 q^{80} -7.70820 q^{81} +1.52705 q^{82} +17.2589 q^{83} -0.244812 q^{84} -2.43976 q^{85} -12.7726 q^{86} +4.40020 q^{87} -8.91982 q^{89} -0.261483 q^{90} +1.58232 q^{91} -0.273109 q^{92} -2.09351 q^{93} -9.38018 q^{94} -1.97050 q^{95} +1.38197 q^{96} -2.70362 q^{97} +1.46673 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 6 q^{5} + q^{6} + 4 q^{7} - 9 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 6 q^{5} + q^{6} + 4 q^{7} - 9 q^{8} - 6 q^{9} + 14 q^{10} - 7 q^{12} - 2 q^{14} + 3 q^{15} - 4 q^{16} + 3 q^{17} + 3 q^{18} - 3 q^{19} - 17 q^{20} - 2 q^{21} - 8 q^{23} + 12 q^{24} - 12 q^{26} + 4 q^{27} + 4 q^{28} - 3 q^{29} - 12 q^{30} - 3 q^{31} + 10 q^{32} - 12 q^{34} - 6 q^{35} - q^{36} - 7 q^{37} - 20 q^{38} - 5 q^{39} + 13 q^{40} - 4 q^{41} + q^{42} - 8 q^{43} + 9 q^{45} + 3 q^{46} - 14 q^{47} - 3 q^{48} + 4 q^{49} - 33 q^{50} + 11 q^{51} + 17 q^{52} - 9 q^{53} - 2 q^{54} - 9 q^{56} - 6 q^{57} + 3 q^{58} - 25 q^{59} + 21 q^{60} + 19 q^{61} - 10 q^{62} - 6 q^{63} + 3 q^{64} - 12 q^{65} - 15 q^{67} + q^{68} + 14 q^{69} + 14 q^{70} - 7 q^{71} + 6 q^{72} + 11 q^{73} - 8 q^{74} - 5 q^{75} + 26 q^{76} - 9 q^{78} - 8 q^{79} - 4 q^{80} - 4 q^{81} + 3 q^{82} + q^{83} - 7 q^{84} - 15 q^{85} + 4 q^{86} - 6 q^{87} - 17 q^{89} - 16 q^{90} - 17 q^{92} - 11 q^{93} + 20 q^{94} - 17 q^{95} + 10 q^{96} - 15 q^{97} - 2 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.46673 1.03714 0.518568 0.855036i \(-0.326465\pi\)
0.518568 + 0.855036i \(0.326465\pi\)
\(3\) −1.61803 −0.934172 −0.467086 0.884212i \(-0.654696\pi\)
−0.467086 + 0.884212i \(0.654696\pi\)
\(4\) 0.151302 0.0756511
\(5\) 0.466732 0.208729 0.104364 0.994539i \(-0.466719\pi\)
0.104364 + 0.994539i \(0.466719\pi\)
\(6\) −2.37322 −0.968864
\(7\) 1.00000 0.377964
\(8\) −2.71154 −0.958676
\(9\) −0.381966 −0.127322
\(10\) 0.684570 0.216480
\(11\) 0 0
\(12\) −0.244812 −0.0706712
\(13\) 1.58232 0.438856 0.219428 0.975629i \(-0.429581\pi\)
0.219428 + 0.975629i \(0.429581\pi\)
\(14\) 1.46673 0.392001
\(15\) −0.755188 −0.194989
\(16\) −4.27971 −1.06993
\(17\) −5.22732 −1.26781 −0.633906 0.773410i \(-0.718549\pi\)
−0.633906 + 0.773410i \(0.718549\pi\)
\(18\) −0.560242 −0.132050
\(19\) −4.22192 −0.968575 −0.484287 0.874909i \(-0.660921\pi\)
−0.484287 + 0.874909i \(0.660921\pi\)
\(20\) 0.0706175 0.0157906
\(21\) −1.61803 −0.353084
\(22\) 0 0
\(23\) −1.80505 −0.376380 −0.188190 0.982133i \(-0.560262\pi\)
−0.188190 + 0.982133i \(0.560262\pi\)
\(24\) 4.38737 0.895568
\(25\) −4.78216 −0.956432
\(26\) 2.32083 0.455153
\(27\) 5.47214 1.05311
\(28\) 0.151302 0.0285934
\(29\) −2.71947 −0.504993 −0.252496 0.967598i \(-0.581252\pi\)
−0.252496 + 0.967598i \(0.581252\pi\)
\(30\) −1.10766 −0.202230
\(31\) 1.29386 0.232384 0.116192 0.993227i \(-0.462931\pi\)
0.116192 + 0.993227i \(0.462931\pi\)
\(32\) −0.854102 −0.150985
\(33\) 0 0
\(34\) −7.66708 −1.31489
\(35\) 0.466732 0.0788921
\(36\) −0.0577923 −0.00963205
\(37\) −1.94221 −0.319297 −0.159648 0.987174i \(-0.551036\pi\)
−0.159648 + 0.987174i \(0.551036\pi\)
\(38\) −6.19242 −1.00454
\(39\) −2.56024 −0.409967
\(40\) −1.26556 −0.200103
\(41\) 1.04112 0.162596 0.0812980 0.996690i \(-0.474093\pi\)
0.0812980 + 0.996690i \(0.474093\pi\)
\(42\) −2.37322 −0.366196
\(43\) −8.70820 −1.32799 −0.663994 0.747738i \(-0.731140\pi\)
−0.663994 + 0.747738i \(0.731140\pi\)
\(44\) 0 0
\(45\) −0.178276 −0.0265758
\(46\) −2.64753 −0.390357
\(47\) −6.39530 −0.932850 −0.466425 0.884561i \(-0.654458\pi\)
−0.466425 + 0.884561i \(0.654458\pi\)
\(48\) 6.92472 0.999497
\(49\) 1.00000 0.142857
\(50\) −7.01415 −0.991950
\(51\) 8.45799 1.18436
\(52\) 0.239408 0.0331999
\(53\) −13.2044 −1.81377 −0.906884 0.421380i \(-0.861546\pi\)
−0.906884 + 0.421380i \(0.861546\pi\)
\(54\) 8.02616 1.09222
\(55\) 0 0
\(56\) −2.71154 −0.362345
\(57\) 6.83121 0.904816
\(58\) −3.98873 −0.523746
\(59\) −8.60389 −1.12013 −0.560065 0.828448i \(-0.689224\pi\)
−0.560065 + 0.828448i \(0.689224\pi\)
\(60\) −0.114262 −0.0147511
\(61\) 15.2401 1.95130 0.975651 0.219331i \(-0.0703874\pi\)
0.975651 + 0.219331i \(0.0703874\pi\)
\(62\) 1.89775 0.241014
\(63\) −0.381966 −0.0481232
\(64\) 7.30669 0.913336
\(65\) 0.738517 0.0916018
\(66\) 0 0
\(67\) −4.67583 −0.571243 −0.285622 0.958342i \(-0.592200\pi\)
−0.285622 + 0.958342i \(0.592200\pi\)
\(68\) −0.790906 −0.0959114
\(69\) 2.92064 0.351604
\(70\) 0.684570 0.0818218
\(71\) 9.74310 1.15629 0.578147 0.815933i \(-0.303776\pi\)
0.578147 + 0.815933i \(0.303776\pi\)
\(72\) 1.03572 0.122060
\(73\) 13.3200 1.55899 0.779495 0.626408i \(-0.215476\pi\)
0.779495 + 0.626408i \(0.215476\pi\)
\(74\) −2.84870 −0.331154
\(75\) 7.73770 0.893473
\(76\) −0.638786 −0.0732737
\(77\) 0 0
\(78\) −3.75519 −0.425191
\(79\) −3.58232 −0.403042 −0.201521 0.979484i \(-0.564588\pi\)
−0.201521 + 0.979484i \(0.564588\pi\)
\(80\) −1.99748 −0.223325
\(81\) −7.70820 −0.856467
\(82\) 1.52705 0.168634
\(83\) 17.2589 1.89441 0.947204 0.320631i \(-0.103895\pi\)
0.947204 + 0.320631i \(0.103895\pi\)
\(84\) −0.244812 −0.0267112
\(85\) −2.43976 −0.264629
\(86\) −12.7726 −1.37730
\(87\) 4.40020 0.471750
\(88\) 0 0
\(89\) −8.91982 −0.945499 −0.472750 0.881197i \(-0.656738\pi\)
−0.472750 + 0.881197i \(0.656738\pi\)
\(90\) −0.261483 −0.0275627
\(91\) 1.58232 0.165872
\(92\) −0.273109 −0.0284735
\(93\) −2.09351 −0.217087
\(94\) −9.38018 −0.967492
\(95\) −1.97050 −0.202169
\(96\) 1.38197 0.141046
\(97\) −2.70362 −0.274511 −0.137255 0.990536i \(-0.543828\pi\)
−0.137255 + 0.990536i \(0.543828\pi\)
\(98\) 1.46673 0.148162
\(99\) 0 0
\(100\) −0.723551 −0.0723551
\(101\) −0.178781 −0.0177894 −0.00889469 0.999960i \(-0.502831\pi\)
−0.00889469 + 0.999960i \(0.502831\pi\)
\(102\) 12.4056 1.22834
\(103\) 16.8772 1.66296 0.831481 0.555553i \(-0.187493\pi\)
0.831481 + 0.555553i \(0.187493\pi\)
\(104\) −4.29052 −0.420720
\(105\) −0.755188 −0.0736988
\(106\) −19.3674 −1.88112
\(107\) 15.4762 1.49614 0.748071 0.663618i \(-0.230980\pi\)
0.748071 + 0.663618i \(0.230980\pi\)
\(108\) 0.827946 0.0796692
\(109\) −11.0349 −1.05695 −0.528476 0.848948i \(-0.677236\pi\)
−0.528476 + 0.848948i \(0.677236\pi\)
\(110\) 0 0
\(111\) 3.14256 0.298278
\(112\) −4.27971 −0.404395
\(113\) 1.77008 0.166515 0.0832574 0.996528i \(-0.473468\pi\)
0.0832574 + 0.996528i \(0.473468\pi\)
\(114\) 10.0196 0.938417
\(115\) −0.842476 −0.0785613
\(116\) −0.411462 −0.0382033
\(117\) −0.604391 −0.0558760
\(118\) −12.6196 −1.16173
\(119\) −5.22732 −0.479188
\(120\) 2.04773 0.186931
\(121\) 0 0
\(122\) 22.3532 2.02376
\(123\) −1.68457 −0.151893
\(124\) 0.195764 0.0175801
\(125\) −4.56565 −0.408364
\(126\) −0.560242 −0.0499103
\(127\) 8.54023 0.757823 0.378911 0.925433i \(-0.376298\pi\)
0.378911 + 0.925433i \(0.376298\pi\)
\(128\) 12.4252 1.09824
\(129\) 14.0902 1.24057
\(130\) 1.08321 0.0950035
\(131\) 9.66708 0.844617 0.422308 0.906452i \(-0.361220\pi\)
0.422308 + 0.906452i \(0.361220\pi\)
\(132\) 0 0
\(133\) −4.22192 −0.366087
\(134\) −6.85818 −0.592457
\(135\) 2.55402 0.219815
\(136\) 14.1741 1.21542
\(137\) −14.0108 −1.19702 −0.598512 0.801114i \(-0.704241\pi\)
−0.598512 + 0.801114i \(0.704241\pi\)
\(138\) 4.28379 0.364661
\(139\) −9.57765 −0.812366 −0.406183 0.913792i \(-0.633140\pi\)
−0.406183 + 0.913792i \(0.633140\pi\)
\(140\) 0.0706175 0.00596827
\(141\) 10.3478 0.871443
\(142\) 14.2905 1.19923
\(143\) 0 0
\(144\) 1.63470 0.136225
\(145\) −1.26926 −0.105407
\(146\) 19.5369 1.61688
\(147\) −1.61803 −0.133453
\(148\) −0.293860 −0.0241552
\(149\) −14.9625 −1.22578 −0.612888 0.790170i \(-0.709992\pi\)
−0.612888 + 0.790170i \(0.709992\pi\)
\(150\) 11.3491 0.926653
\(151\) 2.87233 0.233747 0.116874 0.993147i \(-0.462713\pi\)
0.116874 + 0.993147i \(0.462713\pi\)
\(152\) 11.4479 0.928549
\(153\) 1.99666 0.161420
\(154\) 0 0
\(155\) 0.603886 0.0485053
\(156\) −0.387370 −0.0310144
\(157\) −18.8823 −1.50697 −0.753487 0.657463i \(-0.771630\pi\)
−0.753487 + 0.657463i \(0.771630\pi\)
\(158\) −5.25430 −0.418009
\(159\) 21.3652 1.69437
\(160\) −0.398637 −0.0315150
\(161\) −1.80505 −0.142258
\(162\) −11.3059 −0.888273
\(163\) 11.5951 0.908202 0.454101 0.890950i \(-0.349961\pi\)
0.454101 + 0.890950i \(0.349961\pi\)
\(164\) 0.157524 0.0123006
\(165\) 0 0
\(166\) 25.3142 1.96476
\(167\) −6.32491 −0.489437 −0.244718 0.969594i \(-0.578695\pi\)
−0.244718 + 0.969594i \(0.578695\pi\)
\(168\) 4.38737 0.338493
\(169\) −10.4963 −0.807406
\(170\) −3.57847 −0.274456
\(171\) 1.61263 0.123321
\(172\) −1.31757 −0.100464
\(173\) −1.33906 −0.101807 −0.0509035 0.998704i \(-0.516210\pi\)
−0.0509035 + 0.998704i \(0.516210\pi\)
\(174\) 6.45391 0.489269
\(175\) −4.78216 −0.361497
\(176\) 0 0
\(177\) 13.9214 1.04639
\(178\) −13.0830 −0.980611
\(179\) 17.7888 1.32960 0.664799 0.747022i \(-0.268517\pi\)
0.664799 + 0.747022i \(0.268517\pi\)
\(180\) −0.0269735 −0.00201049
\(181\) −0.963777 −0.0716370 −0.0358185 0.999358i \(-0.511404\pi\)
−0.0358185 + 0.999358i \(0.511404\pi\)
\(182\) 2.32083 0.172032
\(183\) −24.6591 −1.82285
\(184\) 4.89448 0.360826
\(185\) −0.906490 −0.0666465
\(186\) −3.07062 −0.225149
\(187\) 0 0
\(188\) −0.967622 −0.0705711
\(189\) 5.47214 0.398039
\(190\) −2.89020 −0.209677
\(191\) −16.0888 −1.16415 −0.582074 0.813136i \(-0.697759\pi\)
−0.582074 + 0.813136i \(0.697759\pi\)
\(192\) −11.8225 −0.853213
\(193\) −12.1475 −0.874393 −0.437197 0.899366i \(-0.644029\pi\)
−0.437197 + 0.899366i \(0.644029\pi\)
\(194\) −3.96548 −0.284705
\(195\) −1.19495 −0.0855719
\(196\) 0.151302 0.0108073
\(197\) 2.30179 0.163996 0.0819978 0.996633i \(-0.473870\pi\)
0.0819978 + 0.996633i \(0.473870\pi\)
\(198\) 0 0
\(199\) 20.2797 1.43759 0.718795 0.695222i \(-0.244694\pi\)
0.718795 + 0.695222i \(0.244694\pi\)
\(200\) 12.9670 0.916908
\(201\) 7.56565 0.533640
\(202\) −0.262224 −0.0184500
\(203\) −2.71947 −0.190869
\(204\) 1.27971 0.0895978
\(205\) 0.485925 0.0339384
\(206\) 24.7544 1.72472
\(207\) 0.689469 0.0479214
\(208\) −6.77186 −0.469544
\(209\) 0 0
\(210\) −1.10766 −0.0764357
\(211\) −5.36530 −0.369362 −0.184681 0.982799i \(-0.559125\pi\)
−0.184681 + 0.982799i \(0.559125\pi\)
\(212\) −1.99786 −0.137214
\(213\) −15.7647 −1.08018
\(214\) 22.6995 1.55170
\(215\) −4.06440 −0.277189
\(216\) −14.8379 −1.00959
\(217\) 1.29386 0.0878330
\(218\) −16.1852 −1.09620
\(219\) −21.5522 −1.45637
\(220\) 0 0
\(221\) −8.27128 −0.556387
\(222\) 4.60929 0.309355
\(223\) −25.4230 −1.70245 −0.851225 0.524800i \(-0.824140\pi\)
−0.851225 + 0.524800i \(0.824140\pi\)
\(224\) −0.854102 −0.0570671
\(225\) 1.82662 0.121775
\(226\) 2.59623 0.172699
\(227\) 21.7098 1.44093 0.720464 0.693493i \(-0.243929\pi\)
0.720464 + 0.693493i \(0.243929\pi\)
\(228\) 1.03358 0.0684503
\(229\) 20.5307 1.35670 0.678352 0.734737i \(-0.262694\pi\)
0.678352 + 0.734737i \(0.262694\pi\)
\(230\) −1.23569 −0.0814787
\(231\) 0 0
\(232\) 7.37396 0.484124
\(233\) −0.694056 −0.0454691 −0.0227345 0.999742i \(-0.507237\pi\)
−0.0227345 + 0.999742i \(0.507237\pi\)
\(234\) −0.886480 −0.0579510
\(235\) −2.98489 −0.194713
\(236\) −1.30179 −0.0847391
\(237\) 5.79631 0.376511
\(238\) −7.66708 −0.496983
\(239\) −0.346561 −0.0224171 −0.0112086 0.999937i \(-0.503568\pi\)
−0.0112086 + 0.999937i \(0.503568\pi\)
\(240\) 3.23199 0.208624
\(241\) 10.4372 0.672317 0.336158 0.941806i \(-0.390872\pi\)
0.336158 + 0.941806i \(0.390872\pi\)
\(242\) 0 0
\(243\) −3.94427 −0.253025
\(244\) 2.30587 0.147618
\(245\) 0.466732 0.0298184
\(246\) −2.47081 −0.157533
\(247\) −6.68041 −0.425064
\(248\) −3.50836 −0.222781
\(249\) −27.9255 −1.76970
\(250\) −6.69658 −0.423529
\(251\) −6.99502 −0.441522 −0.220761 0.975328i \(-0.570854\pi\)
−0.220761 + 0.975328i \(0.570854\pi\)
\(252\) −0.0577923 −0.00364057
\(253\) 0 0
\(254\) 12.5262 0.785965
\(255\) 3.94761 0.247209
\(256\) 3.61099 0.225687
\(257\) −9.94843 −0.620566 −0.310283 0.950644i \(-0.600424\pi\)
−0.310283 + 0.950644i \(0.600424\pi\)
\(258\) 20.6665 1.28664
\(259\) −1.94221 −0.120683
\(260\) 0.111739 0.00692978
\(261\) 1.03875 0.0642967
\(262\) 14.1790 0.875983
\(263\) −14.1803 −0.874397 −0.437199 0.899365i \(-0.644029\pi\)
−0.437199 + 0.899365i \(0.644029\pi\)
\(264\) 0 0
\(265\) −6.16293 −0.378586
\(266\) −6.19242 −0.379682
\(267\) 14.4326 0.883259
\(268\) −0.707463 −0.0432152
\(269\) −18.4031 −1.12206 −0.561028 0.827797i \(-0.689594\pi\)
−0.561028 + 0.827797i \(0.689594\pi\)
\(270\) 3.74606 0.227978
\(271\) −0.730591 −0.0443802 −0.0221901 0.999754i \(-0.507064\pi\)
−0.0221901 + 0.999754i \(0.507064\pi\)
\(272\) 22.3714 1.35647
\(273\) −2.56024 −0.154953
\(274\) −20.5501 −1.24148
\(275\) 0 0
\(276\) 0.441899 0.0265992
\(277\) −15.0644 −0.905132 −0.452566 0.891731i \(-0.649491\pi\)
−0.452566 + 0.891731i \(0.649491\pi\)
\(278\) −14.0478 −0.842534
\(279\) −0.494211 −0.0295876
\(280\) −1.26556 −0.0756319
\(281\) 10.6961 0.638077 0.319039 0.947742i \(-0.396640\pi\)
0.319039 + 0.947742i \(0.396640\pi\)
\(282\) 15.1775 0.903804
\(283\) −9.10890 −0.541468 −0.270734 0.962654i \(-0.587266\pi\)
−0.270734 + 0.962654i \(0.587266\pi\)
\(284\) 1.47415 0.0874749
\(285\) 3.18834 0.188861
\(286\) 0 0
\(287\) 1.04112 0.0614555
\(288\) 0.326238 0.0192238
\(289\) 10.3249 0.607348
\(290\) −1.86167 −0.109321
\(291\) 4.37454 0.256440
\(292\) 2.01535 0.117939
\(293\) −11.8890 −0.694563 −0.347281 0.937761i \(-0.612895\pi\)
−0.347281 + 0.937761i \(0.612895\pi\)
\(294\) −2.37322 −0.138409
\(295\) −4.01571 −0.233804
\(296\) 5.26638 0.306102
\(297\) 0 0
\(298\) −21.9460 −1.27130
\(299\) −2.85617 −0.165176
\(300\) 1.17073 0.0675922
\(301\) −8.70820 −0.501933
\(302\) 4.21294 0.242427
\(303\) 0.289274 0.0166183
\(304\) 18.0686 1.03631
\(305\) 7.11306 0.407293
\(306\) 2.92856 0.167415
\(307\) −2.22072 −0.126743 −0.0633716 0.997990i \(-0.520185\pi\)
−0.0633716 + 0.997990i \(0.520185\pi\)
\(308\) 0 0
\(309\) −27.3079 −1.55349
\(310\) 0.885738 0.0503066
\(311\) −21.4126 −1.21420 −0.607098 0.794627i \(-0.707666\pi\)
−0.607098 + 0.794627i \(0.707666\pi\)
\(312\) 6.94221 0.393025
\(313\) −31.5548 −1.78358 −0.891790 0.452449i \(-0.850550\pi\)
−0.891790 + 0.452449i \(0.850550\pi\)
\(314\) −27.6953 −1.56294
\(315\) −0.178276 −0.0100447
\(316\) −0.542012 −0.0304906
\(317\) −13.2007 −0.741423 −0.370712 0.928748i \(-0.620886\pi\)
−0.370712 + 0.928748i \(0.620886\pi\)
\(318\) 31.3370 1.75729
\(319\) 0 0
\(320\) 3.41026 0.190639
\(321\) −25.0410 −1.39765
\(322\) −2.64753 −0.147541
\(323\) 22.0693 1.22797
\(324\) −1.16627 −0.0647927
\(325\) −7.56689 −0.419736
\(326\) 17.0070 0.941929
\(327\) 17.8548 0.987375
\(328\) −2.82305 −0.155877
\(329\) −6.39530 −0.352584
\(330\) 0 0
\(331\) 9.47653 0.520877 0.260439 0.965490i \(-0.416133\pi\)
0.260439 + 0.965490i \(0.416133\pi\)
\(332\) 2.61131 0.143314
\(333\) 0.741857 0.0406535
\(334\) −9.27695 −0.507612
\(335\) −2.18236 −0.119235
\(336\) 6.92472 0.377774
\(337\) 19.2011 1.04595 0.522975 0.852348i \(-0.324822\pi\)
0.522975 + 0.852348i \(0.324822\pi\)
\(338\) −15.3952 −0.837390
\(339\) −2.86405 −0.155554
\(340\) −0.369141 −0.0200195
\(341\) 0 0
\(342\) 2.36530 0.127901
\(343\) 1.00000 0.0539949
\(344\) 23.6127 1.27311
\(345\) 1.36315 0.0733898
\(346\) −1.96405 −0.105588
\(347\) −3.04831 −0.163642 −0.0818208 0.996647i \(-0.526074\pi\)
−0.0818208 + 0.996647i \(0.526074\pi\)
\(348\) 0.665759 0.0356884
\(349\) −19.3961 −1.03825 −0.519125 0.854698i \(-0.673742\pi\)
−0.519125 + 0.854698i \(0.673742\pi\)
\(350\) −7.01415 −0.374922
\(351\) 8.65865 0.462165
\(352\) 0 0
\(353\) −10.7585 −0.572619 −0.286309 0.958137i \(-0.592428\pi\)
−0.286309 + 0.958137i \(0.592428\pi\)
\(354\) 20.4189 1.08525
\(355\) 4.54742 0.241352
\(356\) −1.34959 −0.0715280
\(357\) 8.45799 0.447644
\(358\) 26.0914 1.37897
\(359\) −0.607226 −0.0320481 −0.0160241 0.999872i \(-0.505101\pi\)
−0.0160241 + 0.999872i \(0.505101\pi\)
\(360\) 0.483402 0.0254775
\(361\) −1.17539 −0.0618628
\(362\) −1.41360 −0.0742973
\(363\) 0 0
\(364\) 0.239408 0.0125484
\(365\) 6.21688 0.325406
\(366\) −36.1683 −1.89055
\(367\) 27.6628 1.44399 0.721994 0.691899i \(-0.243226\pi\)
0.721994 + 0.691899i \(0.243226\pi\)
\(368\) 7.72511 0.402699
\(369\) −0.397673 −0.0207020
\(370\) −1.32958 −0.0691215
\(371\) −13.2044 −0.685540
\(372\) −0.316753 −0.0164229
\(373\) 29.4513 1.52493 0.762465 0.647029i \(-0.223989\pi\)
0.762465 + 0.647029i \(0.223989\pi\)
\(374\) 0 0
\(375\) 7.38737 0.381482
\(376\) 17.3411 0.894300
\(377\) −4.30306 −0.221619
\(378\) 8.02616 0.412821
\(379\) 25.3436 1.30182 0.650908 0.759157i \(-0.274388\pi\)
0.650908 + 0.759157i \(0.274388\pi\)
\(380\) −0.298142 −0.0152943
\(381\) −13.8184 −0.707937
\(382\) −23.5980 −1.20738
\(383\) −31.9322 −1.63166 −0.815829 0.578293i \(-0.803719\pi\)
−0.815829 + 0.578293i \(0.803719\pi\)
\(384\) −20.1043 −1.02594
\(385\) 0 0
\(386\) −17.8171 −0.906865
\(387\) 3.32624 0.169082
\(388\) −0.409063 −0.0207670
\(389\) 17.7517 0.900047 0.450024 0.893017i \(-0.351416\pi\)
0.450024 + 0.893017i \(0.351416\pi\)
\(390\) −1.75267 −0.0887497
\(391\) 9.43560 0.477179
\(392\) −2.71154 −0.136954
\(393\) −15.6417 −0.789018
\(394\) 3.37610 0.170086
\(395\) −1.67198 −0.0841265
\(396\) 0 0
\(397\) −13.3047 −0.667742 −0.333871 0.942619i \(-0.608355\pi\)
−0.333871 + 0.942619i \(0.608355\pi\)
\(398\) 29.7449 1.49098
\(399\) 6.83121 0.341988
\(400\) 20.4663 1.02331
\(401\) −3.48962 −0.174264 −0.0871318 0.996197i \(-0.527770\pi\)
−0.0871318 + 0.996197i \(0.527770\pi\)
\(402\) 11.0968 0.553457
\(403\) 2.04730 0.101983
\(404\) −0.0270500 −0.00134579
\(405\) −3.59766 −0.178769
\(406\) −3.98873 −0.197958
\(407\) 0 0
\(408\) −22.9342 −1.13541
\(409\) 29.6255 1.46489 0.732443 0.680828i \(-0.238380\pi\)
0.732443 + 0.680828i \(0.238380\pi\)
\(410\) 0.712721 0.0351988
\(411\) 22.6700 1.11823
\(412\) 2.55356 0.125805
\(413\) −8.60389 −0.423370
\(414\) 1.01127 0.0497010
\(415\) 8.05527 0.395418
\(416\) −1.35146 −0.0662608
\(417\) 15.4970 0.758890
\(418\) 0 0
\(419\) −11.6452 −0.568907 −0.284454 0.958690i \(-0.591812\pi\)
−0.284454 + 0.958690i \(0.591812\pi\)
\(420\) −0.114262 −0.00557539
\(421\) 19.8848 0.969128 0.484564 0.874756i \(-0.338978\pi\)
0.484564 + 0.874756i \(0.338978\pi\)
\(422\) −7.86945 −0.383079
\(423\) 2.44279 0.118772
\(424\) 35.8044 1.73882
\(425\) 24.9979 1.21258
\(426\) −23.1225 −1.12029
\(427\) 15.2401 0.737523
\(428\) 2.34159 0.113185
\(429\) 0 0
\(430\) −5.96138 −0.287483
\(431\) −30.2464 −1.45692 −0.728458 0.685090i \(-0.759763\pi\)
−0.728458 + 0.685090i \(0.759763\pi\)
\(432\) −23.4192 −1.12676
\(433\) −5.70719 −0.274270 −0.137135 0.990552i \(-0.543789\pi\)
−0.137135 + 0.990552i \(0.543789\pi\)
\(434\) 1.89775 0.0910947
\(435\) 2.05371 0.0984679
\(436\) −1.66960 −0.0799596
\(437\) 7.62079 0.364552
\(438\) −31.6114 −1.51045
\(439\) −6.84875 −0.326873 −0.163436 0.986554i \(-0.552258\pi\)
−0.163436 + 0.986554i \(0.552258\pi\)
\(440\) 0 0
\(441\) −0.381966 −0.0181889
\(442\) −12.1317 −0.577048
\(443\) 0.100695 0.00478417 0.00239209 0.999997i \(-0.499239\pi\)
0.00239209 + 0.999997i \(0.499239\pi\)
\(444\) 0.475476 0.0225651
\(445\) −4.16316 −0.197353
\(446\) −37.2887 −1.76567
\(447\) 24.2098 1.14509
\(448\) 7.30669 0.345208
\(449\) −30.8047 −1.45377 −0.726883 0.686762i \(-0.759032\pi\)
−0.726883 + 0.686762i \(0.759032\pi\)
\(450\) 2.67917 0.126297
\(451\) 0 0
\(452\) 0.267817 0.0125970
\(453\) −4.64753 −0.218360
\(454\) 31.8424 1.49444
\(455\) 0.738517 0.0346222
\(456\) −18.5231 −0.867425
\(457\) −23.1356 −1.08224 −0.541118 0.840947i \(-0.681999\pi\)
−0.541118 + 0.840947i \(0.681999\pi\)
\(458\) 30.1130 1.40709
\(459\) −28.6046 −1.33515
\(460\) −0.127468 −0.00594325
\(461\) 2.77839 0.129403 0.0647013 0.997905i \(-0.479391\pi\)
0.0647013 + 0.997905i \(0.479391\pi\)
\(462\) 0 0
\(463\) −26.0950 −1.21274 −0.606369 0.795184i \(-0.707374\pi\)
−0.606369 + 0.795184i \(0.707374\pi\)
\(464\) 11.6385 0.540306
\(465\) −0.977108 −0.0453123
\(466\) −1.01799 −0.0471576
\(467\) −2.65829 −0.123011 −0.0615055 0.998107i \(-0.519590\pi\)
−0.0615055 + 0.998107i \(0.519590\pi\)
\(468\) −0.0914457 −0.00422708
\(469\) −4.67583 −0.215910
\(470\) −4.37803 −0.201943
\(471\) 30.5522 1.40777
\(472\) 23.3298 1.07384
\(473\) 0 0
\(474\) 8.50163 0.390493
\(475\) 20.1899 0.926376
\(476\) −0.790906 −0.0362511
\(477\) 5.04364 0.230933
\(478\) −0.508312 −0.0232496
\(479\) 8.28223 0.378425 0.189212 0.981936i \(-0.439407\pi\)
0.189212 + 0.981936i \(0.439407\pi\)
\(480\) 0.645007 0.0294404
\(481\) −3.07319 −0.140125
\(482\) 15.3085 0.697284
\(483\) 2.92064 0.132894
\(484\) 0 0
\(485\) −1.26186 −0.0572983
\(486\) −5.78519 −0.262422
\(487\) −19.5956 −0.887960 −0.443980 0.896037i \(-0.646434\pi\)
−0.443980 + 0.896037i \(0.646434\pi\)
\(488\) −41.3243 −1.87066
\(489\) −18.7613 −0.848417
\(490\) 0.684570 0.0309257
\(491\) 28.6817 1.29439 0.647193 0.762327i \(-0.275943\pi\)
0.647193 + 0.762327i \(0.275943\pi\)
\(492\) −0.254879 −0.0114908
\(493\) 14.2156 0.640236
\(494\) −9.79837 −0.440850
\(495\) 0 0
\(496\) −5.53735 −0.248634
\(497\) 9.74310 0.437038
\(498\) −40.9592 −1.83542
\(499\) 27.9499 1.25121 0.625605 0.780140i \(-0.284852\pi\)
0.625605 + 0.780140i \(0.284852\pi\)
\(500\) −0.690792 −0.0308932
\(501\) 10.2339 0.457218
\(502\) −10.2598 −0.457918
\(503\) 8.09736 0.361043 0.180522 0.983571i \(-0.442221\pi\)
0.180522 + 0.983571i \(0.442221\pi\)
\(504\) 1.03572 0.0461345
\(505\) −0.0834428 −0.00371316
\(506\) 0 0
\(507\) 16.9833 0.754256
\(508\) 1.29216 0.0573301
\(509\) 16.3002 0.722492 0.361246 0.932471i \(-0.382352\pi\)
0.361246 + 0.932471i \(0.382352\pi\)
\(510\) 5.79009 0.256389
\(511\) 13.3200 0.589243
\(512\) −19.5539 −0.864170
\(513\) −23.1029 −1.02002
\(514\) −14.5917 −0.643611
\(515\) 7.87714 0.347108
\(516\) 2.13187 0.0938505
\(517\) 0 0
\(518\) −2.84870 −0.125165
\(519\) 2.16665 0.0951054
\(520\) −2.00252 −0.0878164
\(521\) 7.68605 0.336732 0.168366 0.985725i \(-0.446151\pi\)
0.168366 + 0.985725i \(0.446151\pi\)
\(522\) 1.52356 0.0666844
\(523\) −26.1229 −1.14228 −0.571138 0.820854i \(-0.693498\pi\)
−0.571138 + 0.820854i \(0.693498\pi\)
\(524\) 1.46265 0.0638962
\(525\) 7.73770 0.337701
\(526\) −20.7988 −0.906869
\(527\) −6.76343 −0.294619
\(528\) 0 0
\(529\) −19.7418 −0.858338
\(530\) −9.03936 −0.392645
\(531\) 3.28639 0.142617
\(532\) −0.638786 −0.0276949
\(533\) 1.64738 0.0713561
\(534\) 21.1687 0.916060
\(535\) 7.22324 0.312288
\(536\) 12.6787 0.547637
\(537\) −28.7829 −1.24207
\(538\) −26.9924 −1.16372
\(539\) 0 0
\(540\) 0.386429 0.0166292
\(541\) 25.8777 1.11257 0.556284 0.830992i \(-0.312227\pi\)
0.556284 + 0.830992i \(0.312227\pi\)
\(542\) −1.07158 −0.0460283
\(543\) 1.55942 0.0669213
\(544\) 4.46467 0.191421
\(545\) −5.15034 −0.220616
\(546\) −3.75519 −0.160707
\(547\) 38.0968 1.62890 0.814451 0.580232i \(-0.197038\pi\)
0.814451 + 0.580232i \(0.197038\pi\)
\(548\) −2.11987 −0.0905562
\(549\) −5.82122 −0.248444
\(550\) 0 0
\(551\) 11.4814 0.489123
\(552\) −7.91944 −0.337074
\(553\) −3.58232 −0.152336
\(554\) −22.0954 −0.938745
\(555\) 1.46673 0.0622593
\(556\) −1.44912 −0.0614564
\(557\) −34.5422 −1.46360 −0.731799 0.681520i \(-0.761319\pi\)
−0.731799 + 0.681520i \(0.761319\pi\)
\(558\) −0.724874 −0.0306864
\(559\) −13.7791 −0.582795
\(560\) −1.99748 −0.0844088
\(561\) 0 0
\(562\) 15.6883 0.661773
\(563\) −19.4819 −0.821066 −0.410533 0.911846i \(-0.634657\pi\)
−0.410533 + 0.911846i \(0.634657\pi\)
\(564\) 1.56565 0.0659256
\(565\) 0.826151 0.0347565
\(566\) −13.3603 −0.561576
\(567\) −7.70820 −0.323714
\(568\) −26.4189 −1.10851
\(569\) −17.1288 −0.718075 −0.359038 0.933323i \(-0.616895\pi\)
−0.359038 + 0.933323i \(0.616895\pi\)
\(570\) 4.67644 0.195875
\(571\) 3.85581 0.161360 0.0806802 0.996740i \(-0.474291\pi\)
0.0806802 + 0.996740i \(0.474291\pi\)
\(572\) 0 0
\(573\) 26.0323 1.08751
\(574\) 1.52705 0.0637377
\(575\) 8.63206 0.359982
\(576\) −2.79091 −0.116288
\(577\) −9.78185 −0.407224 −0.203612 0.979052i \(-0.565268\pi\)
−0.203612 + 0.979052i \(0.565268\pi\)
\(578\) 15.1439 0.629902
\(579\) 19.6550 0.816834
\(580\) −0.192042 −0.00797412
\(581\) 17.2589 0.716019
\(582\) 6.41628 0.265964
\(583\) 0 0
\(584\) −36.1178 −1.49457
\(585\) −0.282089 −0.0116629
\(586\) −17.4380 −0.720356
\(587\) −6.09891 −0.251729 −0.125865 0.992047i \(-0.540170\pi\)
−0.125865 + 0.992047i \(0.540170\pi\)
\(588\) −0.244812 −0.0100959
\(589\) −5.46257 −0.225081
\(590\) −5.88997 −0.242486
\(591\) −3.72437 −0.153200
\(592\) 8.31209 0.341625
\(593\) −13.2330 −0.543413 −0.271706 0.962380i \(-0.587588\pi\)
−0.271706 + 0.962380i \(0.587588\pi\)
\(594\) 0 0
\(595\) −2.43976 −0.100020
\(596\) −2.26386 −0.0927313
\(597\) −32.8133 −1.34296
\(598\) −4.18923 −0.171310
\(599\) −5.92515 −0.242095 −0.121048 0.992647i \(-0.538625\pi\)
−0.121048 + 0.992647i \(0.538625\pi\)
\(600\) −20.9811 −0.856550
\(601\) −12.7408 −0.519708 −0.259854 0.965648i \(-0.583674\pi\)
−0.259854 + 0.965648i \(0.583674\pi\)
\(602\) −12.7726 −0.520572
\(603\) 1.78601 0.0727318
\(604\) 0.434590 0.0176832
\(605\) 0 0
\(606\) 0.424287 0.0172355
\(607\) −8.36141 −0.339379 −0.169690 0.985498i \(-0.554276\pi\)
−0.169690 + 0.985498i \(0.554276\pi\)
\(608\) 3.60595 0.146241
\(609\) 4.40020 0.178305
\(610\) 10.4330 0.422418
\(611\) −10.1194 −0.409386
\(612\) 0.302099 0.0122116
\(613\) 6.68294 0.269921 0.134961 0.990851i \(-0.456909\pi\)
0.134961 + 0.990851i \(0.456909\pi\)
\(614\) −3.25720 −0.131450
\(615\) −0.786243 −0.0317044
\(616\) 0 0
\(617\) 11.8669 0.477741 0.238871 0.971051i \(-0.423223\pi\)
0.238871 + 0.971051i \(0.423223\pi\)
\(618\) −40.0534 −1.61118
\(619\) 20.6206 0.828814 0.414407 0.910092i \(-0.363989\pi\)
0.414407 + 0.910092i \(0.363989\pi\)
\(620\) 0.0913692 0.00366948
\(621\) −9.87750 −0.396370
\(622\) −31.4065 −1.25929
\(623\) −8.91982 −0.357365
\(624\) 10.9571 0.438635
\(625\) 21.7799 0.871195
\(626\) −46.2824 −1.84982
\(627\) 0 0
\(628\) −2.85694 −0.114004
\(629\) 10.1525 0.404809
\(630\) −0.261483 −0.0104177
\(631\) −15.4795 −0.616228 −0.308114 0.951349i \(-0.599698\pi\)
−0.308114 + 0.951349i \(0.599698\pi\)
\(632\) 9.71361 0.386387
\(633\) 8.68123 0.345048
\(634\) −19.3618 −0.768957
\(635\) 3.98600 0.158179
\(636\) 3.23260 0.128181
\(637\) 1.58232 0.0626937
\(638\) 0 0
\(639\) −3.72153 −0.147222
\(640\) 5.79921 0.229234
\(641\) −23.5785 −0.931294 −0.465647 0.884971i \(-0.654178\pi\)
−0.465647 + 0.884971i \(0.654178\pi\)
\(642\) −36.7285 −1.44956
\(643\) 28.6806 1.13105 0.565527 0.824730i \(-0.308673\pi\)
0.565527 + 0.824730i \(0.308673\pi\)
\(644\) −0.273109 −0.0107620
\(645\) 6.57633 0.258943
\(646\) 32.3698 1.27357
\(647\) 5.42763 0.213382 0.106691 0.994292i \(-0.465974\pi\)
0.106691 + 0.994292i \(0.465974\pi\)
\(648\) 20.9011 0.821074
\(649\) 0 0
\(650\) −11.0986 −0.435323
\(651\) −2.09351 −0.0820511
\(652\) 1.75437 0.0687064
\(653\) −12.3421 −0.482983 −0.241492 0.970403i \(-0.577637\pi\)
−0.241492 + 0.970403i \(0.577637\pi\)
\(654\) 26.1883 1.02404
\(655\) 4.51193 0.176296
\(656\) −4.45570 −0.173966
\(657\) −5.08779 −0.198494
\(658\) −9.38018 −0.365678
\(659\) −16.2115 −0.631512 −0.315756 0.948840i \(-0.602258\pi\)
−0.315756 + 0.948840i \(0.602258\pi\)
\(660\) 0 0
\(661\) 43.7050 1.69993 0.849964 0.526840i \(-0.176623\pi\)
0.849964 + 0.526840i \(0.176623\pi\)
\(662\) 13.8995 0.540220
\(663\) 13.3832 0.519761
\(664\) −46.7982 −1.81612
\(665\) −1.97050 −0.0764129
\(666\) 1.08811 0.0421632
\(667\) 4.90879 0.190069
\(668\) −0.956973 −0.0370264
\(669\) 41.1353 1.59038
\(670\) −3.20093 −0.123663
\(671\) 0 0
\(672\) 1.38197 0.0533105
\(673\) 5.86102 0.225926 0.112963 0.993599i \(-0.463966\pi\)
0.112963 + 0.993599i \(0.463966\pi\)
\(674\) 28.1629 1.08479
\(675\) −26.1686 −1.00723
\(676\) −1.58811 −0.0610811
\(677\) 20.5279 0.788952 0.394476 0.918906i \(-0.370926\pi\)
0.394476 + 0.918906i \(0.370926\pi\)
\(678\) −4.20079 −0.161330
\(679\) −2.70362 −0.103755
\(680\) 6.61551 0.253693
\(681\) −35.1271 −1.34607
\(682\) 0 0
\(683\) 38.7055 1.48103 0.740513 0.672042i \(-0.234583\pi\)
0.740513 + 0.672042i \(0.234583\pi\)
\(684\) 0.243994 0.00932936
\(685\) −6.53929 −0.249853
\(686\) 1.46673 0.0560001
\(687\) −33.2193 −1.26740
\(688\) 37.2686 1.42085
\(689\) −20.8936 −0.795982
\(690\) 1.99938 0.0761152
\(691\) −23.1300 −0.879907 −0.439954 0.898020i \(-0.645005\pi\)
−0.439954 + 0.898020i \(0.645005\pi\)
\(692\) −0.202603 −0.00770182
\(693\) 0 0
\(694\) −4.47105 −0.169719
\(695\) −4.47020 −0.169564
\(696\) −11.9313 −0.452256
\(697\) −5.44228 −0.206141
\(698\) −28.4489 −1.07681
\(699\) 1.12301 0.0424760
\(700\) −0.723551 −0.0273477
\(701\) −35.5107 −1.34122 −0.670610 0.741810i \(-0.733968\pi\)
−0.670610 + 0.741810i \(0.733968\pi\)
\(702\) 12.6999 0.479328
\(703\) 8.19985 0.309263
\(704\) 0 0
\(705\) 4.82965 0.181895
\(706\) −15.7799 −0.593883
\(707\) −0.178781 −0.00672375
\(708\) 2.10634 0.0791609
\(709\) 43.8045 1.64511 0.822556 0.568684i \(-0.192547\pi\)
0.822556 + 0.568684i \(0.192547\pi\)
\(710\) 6.66984 0.250315
\(711\) 1.36832 0.0513161
\(712\) 24.1865 0.906427
\(713\) −2.33549 −0.0874647
\(714\) 12.4056 0.464268
\(715\) 0 0
\(716\) 2.69149 0.100586
\(717\) 0.560747 0.0209415
\(718\) −0.890637 −0.0332383
\(719\) 15.8605 0.591496 0.295748 0.955266i \(-0.404431\pi\)
0.295748 + 0.955266i \(0.404431\pi\)
\(720\) 0.762969 0.0284342
\(721\) 16.8772 0.628541
\(722\) −1.72399 −0.0641602
\(723\) −16.8877 −0.628060
\(724\) −0.145822 −0.00541942
\(725\) 13.0049 0.482992
\(726\) 0 0
\(727\) 13.7719 0.510770 0.255385 0.966839i \(-0.417798\pi\)
0.255385 + 0.966839i \(0.417798\pi\)
\(728\) −4.29052 −0.159017
\(729\) 29.5066 1.09284
\(730\) 9.11849 0.337490
\(731\) 45.5206 1.68364
\(732\) −3.73097 −0.137901
\(733\) 19.5677 0.722750 0.361375 0.932421i \(-0.382307\pi\)
0.361375 + 0.932421i \(0.382307\pi\)
\(734\) 40.5740 1.49761
\(735\) −0.755188 −0.0278555
\(736\) 1.54170 0.0568278
\(737\) 0 0
\(738\) −0.583280 −0.0214708
\(739\) 11.1542 0.410313 0.205157 0.978729i \(-0.434230\pi\)
0.205157 + 0.978729i \(0.434230\pi\)
\(740\) −0.137154 −0.00504188
\(741\) 10.8091 0.397083
\(742\) −19.3674 −0.710998
\(743\) −20.6550 −0.757757 −0.378878 0.925446i \(-0.623690\pi\)
−0.378878 + 0.925446i \(0.623690\pi\)
\(744\) 5.67664 0.208116
\(745\) −6.98348 −0.255855
\(746\) 43.1972 1.58156
\(747\) −6.59231 −0.241200
\(748\) 0 0
\(749\) 15.4762 0.565489
\(750\) 10.8353 0.395649
\(751\) 26.5991 0.970614 0.485307 0.874344i \(-0.338708\pi\)
0.485307 + 0.874344i \(0.338708\pi\)
\(752\) 27.3700 0.998082
\(753\) 11.3182 0.412458
\(754\) −6.31144 −0.229849
\(755\) 1.34061 0.0487897
\(756\) 0.827946 0.0301121
\(757\) 21.0999 0.766890 0.383445 0.923564i \(-0.374738\pi\)
0.383445 + 0.923564i \(0.374738\pi\)
\(758\) 37.1723 1.35016
\(759\) 0 0
\(760\) 5.34311 0.193815
\(761\) 7.99743 0.289907 0.144953 0.989438i \(-0.453697\pi\)
0.144953 + 0.989438i \(0.453697\pi\)
\(762\) −20.2679 −0.734227
\(763\) −11.0349 −0.399490
\(764\) −2.43428 −0.0880691
\(765\) 0.931905 0.0336931
\(766\) −46.8360 −1.69225
\(767\) −13.6141 −0.491576
\(768\) −5.84271 −0.210831
\(769\) 52.0476 1.87689 0.938443 0.345435i \(-0.112269\pi\)
0.938443 + 0.345435i \(0.112269\pi\)
\(770\) 0 0
\(771\) 16.0969 0.579716
\(772\) −1.83794 −0.0661488
\(773\) −1.58099 −0.0568644 −0.0284322 0.999596i \(-0.509051\pi\)
−0.0284322 + 0.999596i \(0.509051\pi\)
\(774\) 4.87870 0.175361
\(775\) −6.18745 −0.222260
\(776\) 7.33098 0.263167
\(777\) 3.14256 0.112739
\(778\) 26.0370 0.933471
\(779\) −4.39553 −0.157486
\(780\) −0.180798 −0.00647361
\(781\) 0 0
\(782\) 13.8395 0.494899
\(783\) −14.8813 −0.531815
\(784\) −4.27971 −0.152847
\(785\) −8.81298 −0.314549
\(786\) −22.9421 −0.818319
\(787\) −23.3907 −0.833789 −0.416894 0.908955i \(-0.636882\pi\)
−0.416894 + 0.908955i \(0.636882\pi\)
\(788\) 0.348265 0.0124064
\(789\) 22.9443 0.816838
\(790\) −2.45235 −0.0872506
\(791\) 1.77008 0.0629367
\(792\) 0 0
\(793\) 24.1147 0.856339
\(794\) −19.5144 −0.692539
\(795\) 9.97183 0.353664
\(796\) 3.06836 0.108755
\(797\) −46.1518 −1.63478 −0.817391 0.576084i \(-0.804580\pi\)
−0.817391 + 0.576084i \(0.804580\pi\)
\(798\) 10.0196 0.354688
\(799\) 33.4303 1.18268
\(800\) 4.08445 0.144407
\(801\) 3.40707 0.120383
\(802\) −5.11834 −0.180735
\(803\) 0 0
\(804\) 1.14470 0.0403704
\(805\) −0.842476 −0.0296934
\(806\) 3.00283 0.105770
\(807\) 29.7768 1.04819
\(808\) 0.484773 0.0170542
\(809\) −33.7501 −1.18659 −0.593295 0.804985i \(-0.702173\pi\)
−0.593295 + 0.804985i \(0.702173\pi\)
\(810\) −5.27681 −0.185408
\(811\) 30.7650 1.08030 0.540152 0.841567i \(-0.318367\pi\)
0.540152 + 0.841567i \(0.318367\pi\)
\(812\) −0.411462 −0.0144395
\(813\) 1.18212 0.0414588
\(814\) 0 0
\(815\) 5.41182 0.189568
\(816\) −36.1978 −1.26717
\(817\) 36.7653 1.28626
\(818\) 43.4527 1.51929
\(819\) −0.604391 −0.0211191
\(820\) 0.0735215 0.00256748
\(821\) −12.0784 −0.421538 −0.210769 0.977536i \(-0.567597\pi\)
−0.210769 + 0.977536i \(0.567597\pi\)
\(822\) 33.2508 1.15975
\(823\) −24.8187 −0.865124 −0.432562 0.901604i \(-0.642390\pi\)
−0.432562 + 0.901604i \(0.642390\pi\)
\(824\) −45.7634 −1.59424
\(825\) 0 0
\(826\) −12.6196 −0.439092
\(827\) −5.17330 −0.179893 −0.0899466 0.995947i \(-0.528670\pi\)
−0.0899466 + 0.995947i \(0.528670\pi\)
\(828\) 0.104318 0.00362531
\(829\) 31.5535 1.09590 0.547949 0.836511i \(-0.315409\pi\)
0.547949 + 0.836511i \(0.315409\pi\)
\(830\) 11.8149 0.410102
\(831\) 24.3747 0.845549
\(832\) 11.5615 0.400822
\(833\) −5.22732 −0.181116
\(834\) 22.7299 0.787072
\(835\) −2.95204 −0.102160
\(836\) 0 0
\(837\) 7.08018 0.244727
\(838\) −17.0804 −0.590034
\(839\) −5.83642 −0.201496 −0.100748 0.994912i \(-0.532124\pi\)
−0.100748 + 0.994912i \(0.532124\pi\)
\(840\) 2.04773 0.0706532
\(841\) −21.6045 −0.744982
\(842\) 29.1657 1.00512
\(843\) −17.3067 −0.596074
\(844\) −0.811781 −0.0279427
\(845\) −4.89895 −0.168529
\(846\) 3.58291 0.123183
\(847\) 0 0
\(848\) 56.5112 1.94060
\(849\) 14.7385 0.505825
\(850\) 36.6652 1.25761
\(851\) 3.50579 0.120177
\(852\) −2.38523 −0.0817166
\(853\) 20.3462 0.696640 0.348320 0.937376i \(-0.386752\pi\)
0.348320 + 0.937376i \(0.386752\pi\)
\(854\) 22.3532 0.764911
\(855\) 0.752666 0.0257406
\(856\) −41.9644 −1.43432
\(857\) 15.1087 0.516104 0.258052 0.966131i \(-0.416919\pi\)
0.258052 + 0.966131i \(0.416919\pi\)
\(858\) 0 0
\(859\) −33.9641 −1.15884 −0.579420 0.815029i \(-0.696721\pi\)
−0.579420 + 0.815029i \(0.696721\pi\)
\(860\) −0.614952 −0.0209697
\(861\) −1.68457 −0.0574100
\(862\) −44.3633 −1.51102
\(863\) −2.77734 −0.0945417 −0.0472709 0.998882i \(-0.515052\pi\)
−0.0472709 + 0.998882i \(0.515052\pi\)
\(864\) −4.67376 −0.159005
\(865\) −0.624983 −0.0212501
\(866\) −8.37092 −0.284456
\(867\) −16.7061 −0.567368
\(868\) 0.195764 0.00664466
\(869\) 0 0
\(870\) 3.01224 0.102125
\(871\) −7.39864 −0.250693
\(872\) 29.9216 1.01327
\(873\) 1.03269 0.0349513
\(874\) 11.1777 0.378090
\(875\) −4.56565 −0.154347
\(876\) −3.26090 −0.110176
\(877\) −49.6783 −1.67752 −0.838759 0.544503i \(-0.816718\pi\)
−0.838759 + 0.544503i \(0.816718\pi\)
\(878\) −10.0453 −0.339012
\(879\) 19.2368 0.648841
\(880\) 0 0
\(881\) −27.3064 −0.919975 −0.459988 0.887925i \(-0.652146\pi\)
−0.459988 + 0.887925i \(0.652146\pi\)
\(882\) −0.560242 −0.0188643
\(883\) −17.8109 −0.599386 −0.299693 0.954036i \(-0.596884\pi\)
−0.299693 + 0.954036i \(0.596884\pi\)
\(884\) −1.25146 −0.0420912
\(885\) 6.49755 0.218413
\(886\) 0.147693 0.00496184
\(887\) 16.4729 0.553105 0.276553 0.960999i \(-0.410808\pi\)
0.276553 + 0.960999i \(0.410808\pi\)
\(888\) −8.52118 −0.285952
\(889\) 8.54023 0.286430
\(890\) −6.10625 −0.204682
\(891\) 0 0
\(892\) −3.84656 −0.128792
\(893\) 27.0004 0.903535
\(894\) 35.5093 1.18761
\(895\) 8.30260 0.277525
\(896\) 12.4252 0.415095
\(897\) 4.62137 0.154303
\(898\) −45.1823 −1.50775
\(899\) −3.51861 −0.117352
\(900\) 0.276372 0.00921240
\(901\) 69.0238 2.29952
\(902\) 0 0
\(903\) 14.0902 0.468891
\(904\) −4.79964 −0.159634
\(905\) −0.449825 −0.0149527
\(906\) −6.81668 −0.226469
\(907\) 28.4877 0.945918 0.472959 0.881084i \(-0.343186\pi\)
0.472959 + 0.881084i \(0.343186\pi\)
\(908\) 3.28473 0.109008
\(909\) 0.0682883 0.00226498
\(910\) 1.08321 0.0359080
\(911\) −11.2353 −0.372242 −0.186121 0.982527i \(-0.559592\pi\)
−0.186121 + 0.982527i \(0.559592\pi\)
\(912\) −29.2356 −0.968088
\(913\) 0 0
\(914\) −33.9337 −1.12243
\(915\) −11.5092 −0.380482
\(916\) 3.10634 0.102636
\(917\) 9.66708 0.319235
\(918\) −41.9553 −1.38473
\(919\) −0.780457 −0.0257449 −0.0128724 0.999917i \(-0.504098\pi\)
−0.0128724 + 0.999917i \(0.504098\pi\)
\(920\) 2.28441 0.0753148
\(921\) 3.59320 0.118400
\(922\) 4.07516 0.134208
\(923\) 15.4167 0.507446
\(924\) 0 0
\(925\) 9.28795 0.305386
\(926\) −38.2744 −1.25777
\(927\) −6.44653 −0.211732
\(928\) 2.32270 0.0762465
\(929\) 26.5963 0.872597 0.436298 0.899802i \(-0.356289\pi\)
0.436298 + 0.899802i \(0.356289\pi\)
\(930\) −1.43315 −0.0469950
\(931\) −4.22192 −0.138368
\(932\) −0.105012 −0.00343979
\(933\) 34.6463 1.13427
\(934\) −3.89900 −0.127579
\(935\) 0 0
\(936\) 1.63883 0.0535669
\(937\) −41.9697 −1.37109 −0.685544 0.728031i \(-0.740436\pi\)
−0.685544 + 0.728031i \(0.740436\pi\)
\(938\) −6.85818 −0.223928
\(939\) 51.0567 1.66617
\(940\) −0.451620 −0.0147302
\(941\) 49.0330 1.59843 0.799215 0.601046i \(-0.205249\pi\)
0.799215 + 0.601046i \(0.205249\pi\)
\(942\) 44.8119 1.46005
\(943\) −1.87928 −0.0611978
\(944\) 36.8222 1.19846
\(945\) 2.55402 0.0830823
\(946\) 0 0
\(947\) −27.2953 −0.886978 −0.443489 0.896280i \(-0.646259\pi\)
−0.443489 + 0.896280i \(0.646259\pi\)
\(948\) 0.876994 0.0284835
\(949\) 21.0765 0.684171
\(950\) 29.6132 0.960778
\(951\) 21.3591 0.692617
\(952\) 14.1741 0.459386
\(953\) −19.7408 −0.639466 −0.319733 0.947508i \(-0.603593\pi\)
−0.319733 + 0.947508i \(0.603593\pi\)
\(954\) 7.39767 0.239509
\(955\) −7.50918 −0.242991
\(956\) −0.0524354 −0.00169588
\(957\) 0 0
\(958\) 12.1478 0.392478
\(959\) −14.0108 −0.452433
\(960\) −5.51792 −0.178090
\(961\) −29.3259 −0.945998
\(962\) −4.50754 −0.145329
\(963\) −5.91139 −0.190492
\(964\) 1.57917 0.0508615
\(965\) −5.66960 −0.182511
\(966\) 4.28379 0.137829
\(967\) 12.6734 0.407551 0.203775 0.979018i \(-0.434679\pi\)
0.203775 + 0.979018i \(0.434679\pi\)
\(968\) 0 0
\(969\) −35.7089 −1.14714
\(970\) −1.85082 −0.0594261
\(971\) 16.7036 0.536045 0.268022 0.963413i \(-0.413630\pi\)
0.268022 + 0.963413i \(0.413630\pi\)
\(972\) −0.596777 −0.0191416
\(973\) −9.57765 −0.307045
\(974\) −28.7414 −0.920935
\(975\) 12.2435 0.392105
\(976\) −65.2234 −2.08775
\(977\) 27.3452 0.874851 0.437425 0.899255i \(-0.355890\pi\)
0.437425 + 0.899255i \(0.355890\pi\)
\(978\) −27.5178 −0.879924
\(979\) 0 0
\(980\) 0.0706175 0.00225579
\(981\) 4.21496 0.134573
\(982\) 42.0683 1.34245
\(983\) −55.0065 −1.75443 −0.877217 0.480093i \(-0.840603\pi\)
−0.877217 + 0.480093i \(0.840603\pi\)
\(984\) 4.56779 0.145616
\(985\) 1.07432 0.0342306
\(986\) 20.8504 0.664012
\(987\) 10.3478 0.329374
\(988\) −1.01076 −0.0321566
\(989\) 15.7188 0.499828
\(990\) 0 0
\(991\) 53.2327 1.69099 0.845497 0.533980i \(-0.179304\pi\)
0.845497 + 0.533980i \(0.179304\pi\)
\(992\) −1.10509 −0.0350866
\(993\) −15.3333 −0.486589
\(994\) 14.2905 0.453268
\(995\) 9.46519 0.300067
\(996\) −4.22518 −0.133880
\(997\) −32.7546 −1.03735 −0.518674 0.854972i \(-0.673574\pi\)
−0.518674 + 0.854972i \(0.673574\pi\)
\(998\) 40.9950 1.29767
\(999\) −10.6280 −0.336256
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 847.2.a.k.1.4 4
3.2 odd 2 7623.2.a.co.1.1 4
7.6 odd 2 5929.2.a.bb.1.4 4
11.2 odd 10 77.2.f.a.15.2 8
11.3 even 5 847.2.f.s.372.2 8
11.4 even 5 847.2.f.s.148.2 8
11.5 even 5 847.2.f.q.729.1 8
11.6 odd 10 77.2.f.a.36.2 yes 8
11.7 odd 10 847.2.f.p.148.1 8
11.8 odd 10 847.2.f.p.372.1 8
11.9 even 5 847.2.f.q.323.1 8
11.10 odd 2 847.2.a.l.1.1 4
33.2 even 10 693.2.m.g.631.1 8
33.17 even 10 693.2.m.g.190.1 8
33.32 even 2 7623.2.a.ch.1.4 4
77.2 odd 30 539.2.q.c.312.2 16
77.6 even 10 539.2.f.d.344.2 8
77.13 even 10 539.2.f.d.246.2 8
77.17 even 30 539.2.q.b.520.2 16
77.24 even 30 539.2.q.b.422.1 16
77.39 odd 30 539.2.q.c.520.2 16
77.46 odd 30 539.2.q.c.422.1 16
77.61 even 30 539.2.q.b.410.1 16
77.68 even 30 539.2.q.b.312.2 16
77.72 odd 30 539.2.q.c.410.1 16
77.76 even 2 5929.2.a.bi.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.f.a.15.2 8 11.2 odd 10
77.2.f.a.36.2 yes 8 11.6 odd 10
539.2.f.d.246.2 8 77.13 even 10
539.2.f.d.344.2 8 77.6 even 10
539.2.q.b.312.2 16 77.68 even 30
539.2.q.b.410.1 16 77.61 even 30
539.2.q.b.422.1 16 77.24 even 30
539.2.q.b.520.2 16 77.17 even 30
539.2.q.c.312.2 16 77.2 odd 30
539.2.q.c.410.1 16 77.72 odd 30
539.2.q.c.422.1 16 77.46 odd 30
539.2.q.c.520.2 16 77.39 odd 30
693.2.m.g.190.1 8 33.17 even 10
693.2.m.g.631.1 8 33.2 even 10
847.2.a.k.1.4 4 1.1 even 1 trivial
847.2.a.l.1.1 4 11.10 odd 2
847.2.f.p.148.1 8 11.7 odd 10
847.2.f.p.372.1 8 11.8 odd 10
847.2.f.q.323.1 8 11.9 even 5
847.2.f.q.729.1 8 11.5 even 5
847.2.f.s.148.2 8 11.4 even 5
847.2.f.s.372.2 8 11.3 even 5
5929.2.a.bb.1.4 4 7.6 odd 2
5929.2.a.bi.1.1 4 77.76 even 2
7623.2.a.ch.1.4 4 33.32 even 2
7623.2.a.co.1.1 4 3.2 odd 2