Properties

Label 1339.2.a.e.1.17
Level $1339$
Weight $2$
Character 1339.1
Self dual yes
Analytic conductor $10.692$
Analytic rank $1$
Dimension $21$
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] = [1339,2,Mod(1,1339)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1339, 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("1339.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1339 = 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1339.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: \(10.6919688306\)
Analytic rank: \(1\)
Dimension: \(21\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.17
Character \(\chi\) \(=\) 1339.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.85697 q^{2} -2.04069 q^{3} +1.44833 q^{4} +0.337566 q^{5} -3.78949 q^{6} -0.868388 q^{7} -1.02443 q^{8} +1.16441 q^{9} +O(q^{10})\) \(q+1.85697 q^{2} -2.04069 q^{3} +1.44833 q^{4} +0.337566 q^{5} -3.78949 q^{6} -0.868388 q^{7} -1.02443 q^{8} +1.16441 q^{9} +0.626849 q^{10} +4.02651 q^{11} -2.95559 q^{12} -1.00000 q^{13} -1.61257 q^{14} -0.688867 q^{15} -4.79900 q^{16} +4.17665 q^{17} +2.16226 q^{18} -6.30143 q^{19} +0.488907 q^{20} +1.77211 q^{21} +7.47711 q^{22} -5.15149 q^{23} +2.09055 q^{24} -4.88605 q^{25} -1.85697 q^{26} +3.74587 q^{27} -1.25771 q^{28} -6.31422 q^{29} -1.27920 q^{30} +0.244159 q^{31} -6.86272 q^{32} -8.21686 q^{33} +7.75590 q^{34} -0.293138 q^{35} +1.68644 q^{36} -6.75531 q^{37} -11.7016 q^{38} +2.04069 q^{39} -0.345814 q^{40} -1.95590 q^{41} +3.29075 q^{42} +0.258091 q^{43} +5.83172 q^{44} +0.393064 q^{45} -9.56614 q^{46} -4.80034 q^{47} +9.79326 q^{48} -6.24590 q^{49} -9.07324 q^{50} -8.52323 q^{51} -1.44833 q^{52} +4.90729 q^{53} +6.95597 q^{54} +1.35921 q^{55} +0.889607 q^{56} +12.8593 q^{57} -11.7253 q^{58} +0.669589 q^{59} -0.997706 q^{60} -8.72309 q^{61} +0.453396 q^{62} -1.01116 q^{63} -3.14585 q^{64} -0.337566 q^{65} -15.2584 q^{66} +4.50319 q^{67} +6.04916 q^{68} +10.5126 q^{69} -0.544349 q^{70} +11.1310 q^{71} -1.19286 q^{72} -13.1290 q^{73} -12.5444 q^{74} +9.97090 q^{75} -9.12655 q^{76} -3.49658 q^{77} +3.78949 q^{78} +0.454458 q^{79} -1.61998 q^{80} -11.1374 q^{81} -3.63205 q^{82} +1.72510 q^{83} +2.56660 q^{84} +1.40989 q^{85} +0.479266 q^{86} +12.8853 q^{87} -4.12490 q^{88} +4.72748 q^{89} +0.729907 q^{90} +0.868388 q^{91} -7.46105 q^{92} -0.498253 q^{93} -8.91408 q^{94} -2.12715 q^{95} +14.0047 q^{96} -7.91621 q^{97} -11.5984 q^{98} +4.68850 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 21 q - q^{2} - 6 q^{3} + 15 q^{4} - 18 q^{5} - 8 q^{6} - 2 q^{7} + 6 q^{8} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 21 q - q^{2} - 6 q^{3} + 15 q^{4} - 18 q^{5} - 8 q^{6} - 2 q^{7} + 6 q^{8} + 11 q^{9} - 10 q^{10} - 5 q^{11} - 22 q^{12} - 21 q^{13} - 24 q^{14} - q^{15} - 5 q^{16} - 18 q^{17} + 10 q^{18} - 6 q^{19} - 30 q^{20} - 17 q^{21} - 8 q^{22} - 14 q^{23} - 13 q^{24} + 11 q^{25} + q^{26} - 33 q^{27} - 2 q^{28} - 36 q^{29} + 2 q^{30} + q^{31} - 9 q^{32} - 11 q^{33} - 25 q^{34} - 37 q^{35} + 3 q^{36} + 5 q^{37} - 13 q^{38} + 6 q^{39} - 2 q^{40} - 32 q^{41} + 7 q^{42} + 2 q^{43} + 4 q^{44} - 32 q^{45} - 5 q^{46} - 12 q^{47} - 18 q^{48} + 5 q^{49} - 2 q^{50} - 27 q^{51} - 15 q^{52} - 49 q^{53} - 31 q^{54} - 9 q^{55} - 53 q^{56} + 14 q^{57} - 2 q^{58} - 34 q^{59} + 25 q^{60} - 27 q^{61} - 12 q^{62} + 41 q^{63} - 70 q^{64} + 18 q^{65} - 15 q^{66} - 12 q^{67} - 20 q^{68} - 70 q^{69} + 70 q^{70} - 26 q^{71} - 14 q^{72} + 23 q^{73} + 20 q^{74} - 9 q^{75} - 36 q^{76} - 62 q^{77} + 8 q^{78} - 7 q^{79} - 41 q^{80} - 3 q^{81} - 33 q^{82} - q^{83} - 48 q^{84} + 10 q^{85} - 23 q^{86} - 7 q^{87} + 9 q^{88} - 16 q^{89} - 30 q^{90} + 2 q^{91} - 27 q^{92} + 2 q^{93} - 37 q^{94} - 35 q^{95} + 55 q^{96} - 30 q^{97} - 43 q^{98} - 42 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.85697 1.31307 0.656537 0.754294i \(-0.272020\pi\)
0.656537 + 0.754294i \(0.272020\pi\)
\(3\) −2.04069 −1.17819 −0.589096 0.808063i \(-0.700516\pi\)
−0.589096 + 0.808063i \(0.700516\pi\)
\(4\) 1.44833 0.724165
\(5\) 0.337566 0.150964 0.0754821 0.997147i \(-0.475950\pi\)
0.0754821 + 0.997147i \(0.475950\pi\)
\(6\) −3.78949 −1.54705
\(7\) −0.868388 −0.328220 −0.164110 0.986442i \(-0.552475\pi\)
−0.164110 + 0.986442i \(0.552475\pi\)
\(8\) −1.02443 −0.362192
\(9\) 1.16441 0.388135
\(10\) 0.626849 0.198227
\(11\) 4.02651 1.21404 0.607020 0.794687i \(-0.292365\pi\)
0.607020 + 0.794687i \(0.292365\pi\)
\(12\) −2.95559 −0.853205
\(13\) −1.00000 −0.277350
\(14\) −1.61257 −0.430977
\(15\) −0.688867 −0.177865
\(16\) −4.79900 −1.19975
\(17\) 4.17665 1.01299 0.506493 0.862244i \(-0.330942\pi\)
0.506493 + 0.862244i \(0.330942\pi\)
\(18\) 2.16226 0.509651
\(19\) −6.30143 −1.44565 −0.722824 0.691032i \(-0.757156\pi\)
−0.722824 + 0.691032i \(0.757156\pi\)
\(20\) 0.488907 0.109323
\(21\) 1.77211 0.386706
\(22\) 7.47711 1.59412
\(23\) −5.15149 −1.07416 −0.537080 0.843532i \(-0.680473\pi\)
−0.537080 + 0.843532i \(0.680473\pi\)
\(24\) 2.09055 0.426732
\(25\) −4.88605 −0.977210
\(26\) −1.85697 −0.364181
\(27\) 3.74587 0.720894
\(28\) −1.25771 −0.237685
\(29\) −6.31422 −1.17252 −0.586260 0.810123i \(-0.699400\pi\)
−0.586260 + 0.810123i \(0.699400\pi\)
\(30\) −1.27920 −0.233550
\(31\) 0.244159 0.0438523 0.0219261 0.999760i \(-0.493020\pi\)
0.0219261 + 0.999760i \(0.493020\pi\)
\(32\) −6.86272 −1.21317
\(33\) −8.21686 −1.43037
\(34\) 7.75590 1.33013
\(35\) −0.293138 −0.0495494
\(36\) 1.68644 0.281074
\(37\) −6.75531 −1.11057 −0.555283 0.831661i \(-0.687390\pi\)
−0.555283 + 0.831661i \(0.687390\pi\)
\(38\) −11.7016 −1.89824
\(39\) 2.04069 0.326772
\(40\) −0.345814 −0.0546781
\(41\) −1.95590 −0.305461 −0.152730 0.988268i \(-0.548807\pi\)
−0.152730 + 0.988268i \(0.548807\pi\)
\(42\) 3.29075 0.507774
\(43\) 0.258091 0.0393584 0.0196792 0.999806i \(-0.493736\pi\)
0.0196792 + 0.999806i \(0.493736\pi\)
\(44\) 5.83172 0.879165
\(45\) 0.393064 0.0585945
\(46\) −9.56614 −1.41045
\(47\) −4.80034 −0.700202 −0.350101 0.936712i \(-0.613853\pi\)
−0.350101 + 0.936712i \(0.613853\pi\)
\(48\) 9.79326 1.41354
\(49\) −6.24590 −0.892272
\(50\) −9.07324 −1.28315
\(51\) −8.52323 −1.19349
\(52\) −1.44833 −0.200847
\(53\) 4.90729 0.674068 0.337034 0.941492i \(-0.390576\pi\)
0.337034 + 0.941492i \(0.390576\pi\)
\(54\) 6.95597 0.946587
\(55\) 1.35921 0.183276
\(56\) 0.889607 0.118879
\(57\) 12.8593 1.70325
\(58\) −11.7253 −1.53961
\(59\) 0.669589 0.0871731 0.0435865 0.999050i \(-0.486122\pi\)
0.0435865 + 0.999050i \(0.486122\pi\)
\(60\) −0.997706 −0.128803
\(61\) −8.72309 −1.11688 −0.558439 0.829546i \(-0.688599\pi\)
−0.558439 + 0.829546i \(0.688599\pi\)
\(62\) 0.453396 0.0575813
\(63\) −1.01116 −0.127394
\(64\) −3.14585 −0.393231
\(65\) −0.337566 −0.0418699
\(66\) −15.2584 −1.87818
\(67\) 4.50319 0.550153 0.275076 0.961422i \(-0.411297\pi\)
0.275076 + 0.961422i \(0.411297\pi\)
\(68\) 6.04916 0.733568
\(69\) 10.5126 1.26557
\(70\) −0.544349 −0.0650621
\(71\) 11.1310 1.32101 0.660503 0.750823i \(-0.270343\pi\)
0.660503 + 0.750823i \(0.270343\pi\)
\(72\) −1.19286 −0.140580
\(73\) −13.1290 −1.53663 −0.768316 0.640071i \(-0.778905\pi\)
−0.768316 + 0.640071i \(0.778905\pi\)
\(74\) −12.5444 −1.45826
\(75\) 9.97090 1.15134
\(76\) −9.12655 −1.04689
\(77\) −3.49658 −0.398472
\(78\) 3.78949 0.429075
\(79\) 0.454458 0.0511305 0.0255652 0.999673i \(-0.491861\pi\)
0.0255652 + 0.999673i \(0.491861\pi\)
\(80\) −1.61998 −0.181119
\(81\) −11.1374 −1.23749
\(82\) −3.63205 −0.401093
\(83\) 1.72510 0.189354 0.0946771 0.995508i \(-0.469818\pi\)
0.0946771 + 0.995508i \(0.469818\pi\)
\(84\) 2.56660 0.280039
\(85\) 1.40989 0.152924
\(86\) 0.479266 0.0516806
\(87\) 12.8853 1.38145
\(88\) −4.12490 −0.439716
\(89\) 4.72748 0.501112 0.250556 0.968102i \(-0.419387\pi\)
0.250556 + 0.968102i \(0.419387\pi\)
\(90\) 0.729907 0.0769390
\(91\) 0.868388 0.0910318
\(92\) −7.46105 −0.777868
\(93\) −0.498253 −0.0516664
\(94\) −8.91408 −0.919417
\(95\) −2.12715 −0.218241
\(96\) 14.0047 1.42935
\(97\) −7.91621 −0.803769 −0.401885 0.915690i \(-0.631645\pi\)
−0.401885 + 0.915690i \(0.631645\pi\)
\(98\) −11.5984 −1.17162
\(99\) 4.68850 0.471212
\(100\) −7.07661 −0.707661
\(101\) 8.42660 0.838478 0.419239 0.907876i \(-0.362297\pi\)
0.419239 + 0.907876i \(0.362297\pi\)
\(102\) −15.8274 −1.56714
\(103\) −1.00000 −0.0985329
\(104\) 1.02443 0.100454
\(105\) 0.598204 0.0583787
\(106\) 9.11268 0.885102
\(107\) 9.57014 0.925180 0.462590 0.886572i \(-0.346920\pi\)
0.462590 + 0.886572i \(0.346920\pi\)
\(108\) 5.42526 0.522046
\(109\) −3.36167 −0.321989 −0.160995 0.986955i \(-0.551470\pi\)
−0.160995 + 0.986955i \(0.551470\pi\)
\(110\) 2.52402 0.240656
\(111\) 13.7855 1.30846
\(112\) 4.16740 0.393782
\(113\) 8.19635 0.771048 0.385524 0.922698i \(-0.374021\pi\)
0.385524 + 0.922698i \(0.374021\pi\)
\(114\) 23.8792 2.23649
\(115\) −1.73897 −0.162159
\(116\) −9.14506 −0.849098
\(117\) −1.16441 −0.107649
\(118\) 1.24341 0.114465
\(119\) −3.62695 −0.332482
\(120\) 0.705699 0.0644212
\(121\) 5.21282 0.473893
\(122\) −16.1985 −1.46654
\(123\) 3.99139 0.359891
\(124\) 0.353623 0.0317563
\(125\) −3.33719 −0.298488
\(126\) −1.87769 −0.167278
\(127\) 12.8228 1.13784 0.568920 0.822393i \(-0.307362\pi\)
0.568920 + 0.822393i \(0.307362\pi\)
\(128\) 7.88370 0.696827
\(129\) −0.526682 −0.0463718
\(130\) −0.626849 −0.0549783
\(131\) −9.14472 −0.798978 −0.399489 0.916738i \(-0.630812\pi\)
−0.399489 + 0.916738i \(0.630812\pi\)
\(132\) −11.9007 −1.03582
\(133\) 5.47209 0.474490
\(134\) 8.36229 0.722392
\(135\) 1.26448 0.108829
\(136\) −4.27870 −0.366896
\(137\) 14.4798 1.23710 0.618548 0.785747i \(-0.287721\pi\)
0.618548 + 0.785747i \(0.287721\pi\)
\(138\) 19.5215 1.66178
\(139\) 7.34503 0.622997 0.311499 0.950247i \(-0.399169\pi\)
0.311499 + 0.950247i \(0.399169\pi\)
\(140\) −0.424561 −0.0358820
\(141\) 9.79600 0.824972
\(142\) 20.6699 1.73458
\(143\) −4.02651 −0.336714
\(144\) −5.58799 −0.465666
\(145\) −2.13147 −0.177009
\(146\) −24.3801 −2.01771
\(147\) 12.7459 1.05127
\(148\) −9.78392 −0.804233
\(149\) 8.17502 0.669724 0.334862 0.942267i \(-0.391310\pi\)
0.334862 + 0.942267i \(0.391310\pi\)
\(150\) 18.5156 1.51180
\(151\) 0.616426 0.0501641 0.0250820 0.999685i \(-0.492015\pi\)
0.0250820 + 0.999685i \(0.492015\pi\)
\(152\) 6.45540 0.523602
\(153\) 4.86331 0.393175
\(154\) −6.49303 −0.523224
\(155\) 0.0824198 0.00662012
\(156\) 2.95559 0.236636
\(157\) −11.1067 −0.886414 −0.443207 0.896419i \(-0.646159\pi\)
−0.443207 + 0.896419i \(0.646159\pi\)
\(158\) 0.843913 0.0671381
\(159\) −10.0142 −0.794182
\(160\) −2.31662 −0.183145
\(161\) 4.47349 0.352560
\(162\) −20.6818 −1.62491
\(163\) 7.10016 0.556128 0.278064 0.960563i \(-0.410307\pi\)
0.278064 + 0.960563i \(0.410307\pi\)
\(164\) −2.83279 −0.221204
\(165\) −2.77373 −0.215935
\(166\) 3.20345 0.248636
\(167\) −14.4379 −1.11724 −0.558618 0.829425i \(-0.688668\pi\)
−0.558618 + 0.829425i \(0.688668\pi\)
\(168\) −1.81541 −0.140062
\(169\) 1.00000 0.0769231
\(170\) 2.61813 0.200801
\(171\) −7.33743 −0.561107
\(172\) 0.373800 0.0285020
\(173\) −20.0730 −1.52612 −0.763060 0.646328i \(-0.776304\pi\)
−0.763060 + 0.646328i \(0.776304\pi\)
\(174\) 23.9277 1.81395
\(175\) 4.24299 0.320740
\(176\) −19.3232 −1.45654
\(177\) −1.36642 −0.102707
\(178\) 8.77878 0.657997
\(179\) −10.0573 −0.751718 −0.375859 0.926677i \(-0.622652\pi\)
−0.375859 + 0.926677i \(0.622652\pi\)
\(180\) 0.569286 0.0424321
\(181\) 1.03931 0.0772513 0.0386256 0.999254i \(-0.487702\pi\)
0.0386256 + 0.999254i \(0.487702\pi\)
\(182\) 1.61257 0.119532
\(183\) 17.8011 1.31590
\(184\) 5.27736 0.389052
\(185\) −2.28036 −0.167656
\(186\) −0.925239 −0.0678418
\(187\) 16.8173 1.22980
\(188\) −6.95248 −0.507062
\(189\) −3.25287 −0.236612
\(190\) −3.95005 −0.286567
\(191\) 6.31200 0.456720 0.228360 0.973577i \(-0.426664\pi\)
0.228360 + 0.973577i \(0.426664\pi\)
\(192\) 6.41970 0.463302
\(193\) 12.7685 0.919093 0.459547 0.888154i \(-0.348012\pi\)
0.459547 + 0.888154i \(0.348012\pi\)
\(194\) −14.7001 −1.05541
\(195\) 0.688867 0.0493308
\(196\) −9.04612 −0.646152
\(197\) −8.86713 −0.631757 −0.315878 0.948800i \(-0.602299\pi\)
−0.315878 + 0.948800i \(0.602299\pi\)
\(198\) 8.70639 0.618736
\(199\) −1.52396 −0.108030 −0.0540152 0.998540i \(-0.517202\pi\)
−0.0540152 + 0.998540i \(0.517202\pi\)
\(200\) 5.00544 0.353938
\(201\) −9.18961 −0.648185
\(202\) 15.6479 1.10098
\(203\) 5.48319 0.384845
\(204\) −12.3444 −0.864284
\(205\) −0.660247 −0.0461136
\(206\) −1.85697 −0.129381
\(207\) −5.99842 −0.416919
\(208\) 4.79900 0.332751
\(209\) −25.3728 −1.75507
\(210\) 1.11085 0.0766556
\(211\) −8.62201 −0.593564 −0.296782 0.954945i \(-0.595913\pi\)
−0.296782 + 0.954945i \(0.595913\pi\)
\(212\) 7.10737 0.488136
\(213\) −22.7149 −1.55640
\(214\) 17.7714 1.21483
\(215\) 0.0871226 0.00594171
\(216\) −3.83740 −0.261102
\(217\) −0.212025 −0.0143932
\(218\) −6.24251 −0.422796
\(219\) 26.7922 1.81045
\(220\) 1.96859 0.132722
\(221\) −4.17665 −0.280952
\(222\) 25.5992 1.71811
\(223\) −16.7390 −1.12092 −0.560462 0.828180i \(-0.689377\pi\)
−0.560462 + 0.828180i \(0.689377\pi\)
\(224\) 5.95951 0.398186
\(225\) −5.68935 −0.379290
\(226\) 15.2204 1.01244
\(227\) −8.60827 −0.571351 −0.285676 0.958326i \(-0.592218\pi\)
−0.285676 + 0.958326i \(0.592218\pi\)
\(228\) 18.6244 1.23343
\(229\) −19.8035 −1.30865 −0.654327 0.756211i \(-0.727048\pi\)
−0.654327 + 0.756211i \(0.727048\pi\)
\(230\) −3.22921 −0.212927
\(231\) 7.13543 0.469477
\(232\) 6.46850 0.424678
\(233\) 0.298831 0.0195771 0.00978853 0.999952i \(-0.496884\pi\)
0.00978853 + 0.999952i \(0.496884\pi\)
\(234\) −2.16226 −0.141352
\(235\) −1.62043 −0.105705
\(236\) 0.969786 0.0631277
\(237\) −0.927406 −0.0602415
\(238\) −6.73513 −0.436574
\(239\) 27.4369 1.77475 0.887373 0.461052i \(-0.152528\pi\)
0.887373 + 0.461052i \(0.152528\pi\)
\(240\) 3.30587 0.213393
\(241\) 13.8656 0.893161 0.446581 0.894743i \(-0.352642\pi\)
0.446581 + 0.894743i \(0.352642\pi\)
\(242\) 9.68004 0.622256
\(243\) 11.4903 0.737102
\(244\) −12.6339 −0.808803
\(245\) −2.10840 −0.134701
\(246\) 7.41188 0.472564
\(247\) 6.30143 0.400950
\(248\) −0.250125 −0.0158830
\(249\) −3.52039 −0.223095
\(250\) −6.19706 −0.391937
\(251\) 16.9991 1.07297 0.536485 0.843910i \(-0.319752\pi\)
0.536485 + 0.843910i \(0.319752\pi\)
\(252\) −1.46449 −0.0922541
\(253\) −20.7425 −1.30407
\(254\) 23.8115 1.49407
\(255\) −2.87715 −0.180174
\(256\) 20.9315 1.30822
\(257\) 9.17004 0.572011 0.286006 0.958228i \(-0.407672\pi\)
0.286006 + 0.958228i \(0.407672\pi\)
\(258\) −0.978032 −0.0608896
\(259\) 5.86623 0.364510
\(260\) −0.488907 −0.0303207
\(261\) −7.35231 −0.455097
\(262\) −16.9815 −1.04912
\(263\) −14.0834 −0.868420 −0.434210 0.900812i \(-0.642972\pi\)
−0.434210 + 0.900812i \(0.642972\pi\)
\(264\) 8.41763 0.518070
\(265\) 1.65653 0.101760
\(266\) 10.1615 0.623041
\(267\) −9.64731 −0.590406
\(268\) 6.52211 0.398401
\(269\) −13.8474 −0.844292 −0.422146 0.906528i \(-0.638723\pi\)
−0.422146 + 0.906528i \(0.638723\pi\)
\(270\) 2.34810 0.142901
\(271\) −15.1319 −0.919195 −0.459597 0.888127i \(-0.652006\pi\)
−0.459597 + 0.888127i \(0.652006\pi\)
\(272\) −20.0437 −1.21533
\(273\) −1.77211 −0.107253
\(274\) 26.8886 1.62440
\(275\) −19.6737 −1.18637
\(276\) 15.2257 0.916478
\(277\) 31.5583 1.89616 0.948078 0.318037i \(-0.103023\pi\)
0.948078 + 0.318037i \(0.103023\pi\)
\(278\) 13.6395 0.818042
\(279\) 0.284300 0.0170206
\(280\) 0.300301 0.0179464
\(281\) 6.64836 0.396608 0.198304 0.980141i \(-0.436457\pi\)
0.198304 + 0.980141i \(0.436457\pi\)
\(282\) 18.1909 1.08325
\(283\) −18.8468 −1.12033 −0.560163 0.828383i \(-0.689261\pi\)
−0.560163 + 0.828383i \(0.689261\pi\)
\(284\) 16.1213 0.956626
\(285\) 4.34085 0.257130
\(286\) −7.47711 −0.442131
\(287\) 1.69848 0.100258
\(288\) −7.99100 −0.470874
\(289\) 0.444366 0.0261392
\(290\) −3.95806 −0.232425
\(291\) 16.1545 0.946994
\(292\) −19.0151 −1.11277
\(293\) 19.3011 1.12758 0.563791 0.825917i \(-0.309342\pi\)
0.563791 + 0.825917i \(0.309342\pi\)
\(294\) 23.6688 1.38039
\(295\) 0.226031 0.0131600
\(296\) 6.92037 0.402239
\(297\) 15.0828 0.875194
\(298\) 15.1808 0.879397
\(299\) 5.15149 0.297918
\(300\) 14.4411 0.833760
\(301\) −0.224123 −0.0129182
\(302\) 1.14468 0.0658692
\(303\) −17.1961 −0.987888
\(304\) 30.2406 1.73442
\(305\) −2.94462 −0.168608
\(306\) 9.03101 0.516269
\(307\) 20.0124 1.14217 0.571085 0.820891i \(-0.306523\pi\)
0.571085 + 0.820891i \(0.306523\pi\)
\(308\) −5.06420 −0.288559
\(309\) 2.04069 0.116091
\(310\) 0.153051 0.00869271
\(311\) 10.3303 0.585780 0.292890 0.956146i \(-0.405383\pi\)
0.292890 + 0.956146i \(0.405383\pi\)
\(312\) −2.09055 −0.118354
\(313\) 0.429358 0.0242687 0.0121344 0.999926i \(-0.496137\pi\)
0.0121344 + 0.999926i \(0.496137\pi\)
\(314\) −20.6249 −1.16393
\(315\) −0.341332 −0.0192319
\(316\) 0.658204 0.0370269
\(317\) 16.8546 0.946647 0.473324 0.880889i \(-0.343054\pi\)
0.473324 + 0.880889i \(0.343054\pi\)
\(318\) −18.5961 −1.04282
\(319\) −25.4243 −1.42349
\(320\) −1.06193 −0.0593638
\(321\) −19.5297 −1.09004
\(322\) 8.30713 0.462938
\(323\) −26.3188 −1.46442
\(324\) −16.1306 −0.896144
\(325\) 4.88605 0.271029
\(326\) 13.1848 0.730237
\(327\) 6.86011 0.379365
\(328\) 2.00370 0.110636
\(329\) 4.16856 0.229820
\(330\) −5.15073 −0.283538
\(331\) −15.2855 −0.840167 −0.420083 0.907485i \(-0.637999\pi\)
−0.420083 + 0.907485i \(0.637999\pi\)
\(332\) 2.49851 0.137124
\(333\) −7.86593 −0.431050
\(334\) −26.8107 −1.46702
\(335\) 1.52013 0.0830533
\(336\) −8.50436 −0.463951
\(337\) 22.0961 1.20365 0.601826 0.798627i \(-0.294440\pi\)
0.601826 + 0.798627i \(0.294440\pi\)
\(338\) 1.85697 0.101006
\(339\) −16.7262 −0.908442
\(340\) 2.04199 0.110742
\(341\) 0.983110 0.0532384
\(342\) −13.6254 −0.736775
\(343\) 11.5026 0.621081
\(344\) −0.264397 −0.0142553
\(345\) 3.54869 0.191055
\(346\) −37.2748 −2.00391
\(347\) 34.8154 1.86899 0.934494 0.355979i \(-0.115853\pi\)
0.934494 + 0.355979i \(0.115853\pi\)
\(348\) 18.6622 1.00040
\(349\) −23.3164 −1.24810 −0.624049 0.781386i \(-0.714513\pi\)
−0.624049 + 0.781386i \(0.714513\pi\)
\(350\) 7.87909 0.421155
\(351\) −3.74587 −0.199940
\(352\) −27.6328 −1.47284
\(353\) −20.4173 −1.08670 −0.543351 0.839506i \(-0.682845\pi\)
−0.543351 + 0.839506i \(0.682845\pi\)
\(354\) −2.53740 −0.134861
\(355\) 3.75745 0.199424
\(356\) 6.84695 0.362888
\(357\) 7.40147 0.391727
\(358\) −18.6761 −0.987062
\(359\) 9.98075 0.526764 0.263382 0.964692i \(-0.415162\pi\)
0.263382 + 0.964692i \(0.415162\pi\)
\(360\) −0.402668 −0.0212225
\(361\) 20.7080 1.08990
\(362\) 1.92996 0.101437
\(363\) −10.6377 −0.558336
\(364\) 1.25771 0.0659220
\(365\) −4.43190 −0.231976
\(366\) 33.0561 1.72787
\(367\) 18.4927 0.965310 0.482655 0.875810i \(-0.339672\pi\)
0.482655 + 0.875810i \(0.339672\pi\)
\(368\) 24.7220 1.28872
\(369\) −2.27747 −0.118560
\(370\) −4.23456 −0.220144
\(371\) −4.26143 −0.221243
\(372\) −0.721634 −0.0374150
\(373\) −38.3755 −1.98701 −0.993505 0.113790i \(-0.963701\pi\)
−0.993505 + 0.113790i \(0.963701\pi\)
\(374\) 31.2292 1.61482
\(375\) 6.81017 0.351676
\(376\) 4.91764 0.253608
\(377\) 6.31422 0.325199
\(378\) −6.04048 −0.310689
\(379\) −13.7059 −0.704026 −0.352013 0.935995i \(-0.614503\pi\)
−0.352013 + 0.935995i \(0.614503\pi\)
\(380\) −3.08081 −0.158042
\(381\) −26.1673 −1.34059
\(382\) 11.7212 0.599708
\(383\) 12.0858 0.617555 0.308778 0.951134i \(-0.400080\pi\)
0.308778 + 0.951134i \(0.400080\pi\)
\(384\) −16.0882 −0.820996
\(385\) −1.18033 −0.0601550
\(386\) 23.7106 1.20684
\(387\) 0.300522 0.0152764
\(388\) −11.4653 −0.582061
\(389\) −30.2130 −1.53186 −0.765930 0.642924i \(-0.777721\pi\)
−0.765930 + 0.642924i \(0.777721\pi\)
\(390\) 1.27920 0.0647750
\(391\) −21.5159 −1.08811
\(392\) 6.39852 0.323174
\(393\) 18.6615 0.941349
\(394\) −16.4660 −0.829544
\(395\) 0.153409 0.00771887
\(396\) 6.79049 0.341235
\(397\) 22.2128 1.11483 0.557415 0.830234i \(-0.311793\pi\)
0.557415 + 0.830234i \(0.311793\pi\)
\(398\) −2.82994 −0.141852
\(399\) −11.1668 −0.559041
\(400\) 23.4482 1.17241
\(401\) −19.3717 −0.967376 −0.483688 0.875240i \(-0.660703\pi\)
−0.483688 + 0.875240i \(0.660703\pi\)
\(402\) −17.0648 −0.851116
\(403\) −0.244159 −0.0121624
\(404\) 12.2045 0.607196
\(405\) −3.75960 −0.186816
\(406\) 10.1821 0.505330
\(407\) −27.2004 −1.34827
\(408\) 8.73149 0.432273
\(409\) −11.2165 −0.554621 −0.277310 0.960780i \(-0.589443\pi\)
−0.277310 + 0.960780i \(0.589443\pi\)
\(410\) −1.22606 −0.0605506
\(411\) −29.5488 −1.45754
\(412\) −1.44833 −0.0713541
\(413\) −0.581464 −0.0286120
\(414\) −11.1389 −0.547446
\(415\) 0.582335 0.0285857
\(416\) 6.86272 0.336473
\(417\) −14.9889 −0.734010
\(418\) −47.1165 −2.30454
\(419\) −29.1424 −1.42370 −0.711849 0.702333i \(-0.752142\pi\)
−0.711849 + 0.702333i \(0.752142\pi\)
\(420\) 0.866396 0.0422758
\(421\) −7.82838 −0.381532 −0.190766 0.981636i \(-0.561097\pi\)
−0.190766 + 0.981636i \(0.561097\pi\)
\(422\) −16.0108 −0.779394
\(423\) −5.58955 −0.271773
\(424\) −5.02720 −0.244142
\(425\) −20.4073 −0.989899
\(426\) −42.1808 −2.04367
\(427\) 7.57503 0.366581
\(428\) 13.8607 0.669983
\(429\) 8.21686 0.396714
\(430\) 0.161784 0.00780191
\(431\) −20.1683 −0.971474 −0.485737 0.874105i \(-0.661449\pi\)
−0.485737 + 0.874105i \(0.661449\pi\)
\(432\) −17.9764 −0.864892
\(433\) 13.0824 0.628700 0.314350 0.949307i \(-0.398213\pi\)
0.314350 + 0.949307i \(0.398213\pi\)
\(434\) −0.393724 −0.0188993
\(435\) 4.34965 0.208550
\(436\) −4.86880 −0.233173
\(437\) 32.4617 1.55286
\(438\) 49.7522 2.37725
\(439\) −7.66674 −0.365913 −0.182957 0.983121i \(-0.558567\pi\)
−0.182957 + 0.983121i \(0.558567\pi\)
\(440\) −1.39243 −0.0663813
\(441\) −7.27277 −0.346322
\(442\) −7.75590 −0.368910
\(443\) 14.6959 0.698223 0.349111 0.937081i \(-0.386483\pi\)
0.349111 + 0.937081i \(0.386483\pi\)
\(444\) 19.9659 0.947540
\(445\) 1.59584 0.0756499
\(446\) −31.0837 −1.47186
\(447\) −16.6827 −0.789063
\(448\) 2.73182 0.129066
\(449\) −31.0829 −1.46689 −0.733447 0.679746i \(-0.762090\pi\)
−0.733447 + 0.679746i \(0.762090\pi\)
\(450\) −10.5649 −0.498036
\(451\) −7.87547 −0.370842
\(452\) 11.8710 0.558365
\(453\) −1.25793 −0.0591029
\(454\) −15.9853 −0.750226
\(455\) 0.293138 0.0137425
\(456\) −13.1735 −0.616904
\(457\) −16.9808 −0.794330 −0.397165 0.917747i \(-0.630006\pi\)
−0.397165 + 0.917747i \(0.630006\pi\)
\(458\) −36.7745 −1.71836
\(459\) 15.6452 0.730255
\(460\) −2.51860 −0.117430
\(461\) −40.1807 −1.87140 −0.935700 0.352797i \(-0.885231\pi\)
−0.935700 + 0.352797i \(0.885231\pi\)
\(462\) 13.2503 0.616458
\(463\) 7.71807 0.358689 0.179345 0.983786i \(-0.442602\pi\)
0.179345 + 0.983786i \(0.442602\pi\)
\(464\) 30.3019 1.40673
\(465\) −0.168193 −0.00779977
\(466\) 0.554919 0.0257061
\(467\) 16.5007 0.763560 0.381780 0.924253i \(-0.375311\pi\)
0.381780 + 0.924253i \(0.375311\pi\)
\(468\) −1.68644 −0.0779559
\(469\) −3.91052 −0.180571
\(470\) −3.00909 −0.138799
\(471\) 22.6654 1.04437
\(472\) −0.685950 −0.0315734
\(473\) 1.03921 0.0477827
\(474\) −1.72216 −0.0791016
\(475\) 30.7891 1.41270
\(476\) −5.25302 −0.240772
\(477\) 5.71408 0.261630
\(478\) 50.9495 2.33037
\(479\) 29.5562 1.35046 0.675229 0.737608i \(-0.264045\pi\)
0.675229 + 0.737608i \(0.264045\pi\)
\(480\) 4.72750 0.215780
\(481\) 6.75531 0.308016
\(482\) 25.7480 1.17279
\(483\) −9.12900 −0.415384
\(484\) 7.54988 0.343176
\(485\) −2.67224 −0.121340
\(486\) 21.3371 0.967870
\(487\) −24.8359 −1.12542 −0.562712 0.826653i \(-0.690242\pi\)
−0.562712 + 0.826653i \(0.690242\pi\)
\(488\) 8.93624 0.404524
\(489\) −14.4892 −0.655225
\(490\) −3.91524 −0.176872
\(491\) −23.1591 −1.04515 −0.522577 0.852592i \(-0.675029\pi\)
−0.522577 + 0.852592i \(0.675029\pi\)
\(492\) 5.78085 0.260621
\(493\) −26.3722 −1.18775
\(494\) 11.7016 0.526478
\(495\) 1.58268 0.0711361
\(496\) −1.17172 −0.0526118
\(497\) −9.66603 −0.433580
\(498\) −6.53724 −0.292941
\(499\) −22.8739 −1.02397 −0.511987 0.858993i \(-0.671091\pi\)
−0.511987 + 0.858993i \(0.671091\pi\)
\(500\) −4.83336 −0.216154
\(501\) 29.4632 1.31632
\(502\) 31.5667 1.40889
\(503\) −18.4696 −0.823517 −0.411759 0.911293i \(-0.635085\pi\)
−0.411759 + 0.911293i \(0.635085\pi\)
\(504\) 1.03586 0.0461411
\(505\) 2.84453 0.126580
\(506\) −38.5182 −1.71234
\(507\) −2.04069 −0.0906301
\(508\) 18.5716 0.823983
\(509\) −2.68795 −0.119141 −0.0595706 0.998224i \(-0.518973\pi\)
−0.0595706 + 0.998224i \(0.518973\pi\)
\(510\) −5.34278 −0.236582
\(511\) 11.4011 0.504353
\(512\) 23.1017 1.02096
\(513\) −23.6044 −1.04216
\(514\) 17.0285 0.751094
\(515\) −0.337566 −0.0148749
\(516\) −0.762809 −0.0335808
\(517\) −19.3287 −0.850073
\(518\) 10.8934 0.478629
\(519\) 40.9627 1.79806
\(520\) 0.345814 0.0151650
\(521\) −44.7668 −1.96127 −0.980635 0.195843i \(-0.937256\pi\)
−0.980635 + 0.195843i \(0.937256\pi\)
\(522\) −13.6530 −0.597576
\(523\) −2.66249 −0.116422 −0.0582112 0.998304i \(-0.518540\pi\)
−0.0582112 + 0.998304i \(0.518540\pi\)
\(524\) −13.2446 −0.578592
\(525\) −8.65861 −0.377893
\(526\) −26.1524 −1.14030
\(527\) 1.01977 0.0444217
\(528\) 39.4327 1.71609
\(529\) 3.53781 0.153818
\(530\) 3.07613 0.133619
\(531\) 0.779674 0.0338350
\(532\) 7.92539 0.343609
\(533\) 1.95590 0.0847196
\(534\) −17.9148 −0.775247
\(535\) 3.23055 0.139669
\(536\) −4.61323 −0.199261
\(537\) 20.5238 0.885668
\(538\) −25.7142 −1.10862
\(539\) −25.1492 −1.08325
\(540\) 1.83138 0.0788102
\(541\) −17.0596 −0.733450 −0.366725 0.930329i \(-0.619521\pi\)
−0.366725 + 0.930329i \(0.619521\pi\)
\(542\) −28.0994 −1.20697
\(543\) −2.12091 −0.0910168
\(544\) −28.6631 −1.22892
\(545\) −1.13478 −0.0486088
\(546\) −3.29075 −0.140831
\(547\) 39.9654 1.70880 0.854398 0.519619i \(-0.173926\pi\)
0.854398 + 0.519619i \(0.173926\pi\)
\(548\) 20.9716 0.895862
\(549\) −10.1572 −0.433500
\(550\) −36.5335 −1.55779
\(551\) 39.7886 1.69505
\(552\) −10.7694 −0.458378
\(553\) −0.394646 −0.0167820
\(554\) 58.6028 2.48980
\(555\) 4.65351 0.197531
\(556\) 10.6380 0.451153
\(557\) −44.0020 −1.86442 −0.932212 0.361913i \(-0.882124\pi\)
−0.932212 + 0.361913i \(0.882124\pi\)
\(558\) 0.527937 0.0223493
\(559\) −0.258091 −0.0109161
\(560\) 1.40677 0.0594470
\(561\) −34.3189 −1.44895
\(562\) 12.3458 0.520776
\(563\) 44.7639 1.88657 0.943286 0.331980i \(-0.107717\pi\)
0.943286 + 0.331980i \(0.107717\pi\)
\(564\) 14.1878 0.597416
\(565\) 2.76681 0.116401
\(566\) −34.9979 −1.47107
\(567\) 9.67157 0.406168
\(568\) −11.4030 −0.478458
\(569\) −17.4692 −0.732348 −0.366174 0.930546i \(-0.619332\pi\)
−0.366174 + 0.930546i \(0.619332\pi\)
\(570\) 8.06081 0.337630
\(571\) 12.0268 0.503304 0.251652 0.967818i \(-0.419026\pi\)
0.251652 + 0.967818i \(0.419026\pi\)
\(572\) −5.83172 −0.243836
\(573\) −12.8808 −0.538104
\(574\) 3.15403 0.131647
\(575\) 25.1704 1.04968
\(576\) −3.66305 −0.152627
\(577\) 15.0401 0.626127 0.313063 0.949732i \(-0.398645\pi\)
0.313063 + 0.949732i \(0.398645\pi\)
\(578\) 0.825173 0.0343227
\(579\) −26.0564 −1.08287
\(580\) −3.08706 −0.128183
\(581\) −1.49806 −0.0621498
\(582\) 29.9984 1.24347
\(583\) 19.7593 0.818346
\(584\) 13.4498 0.556556
\(585\) −0.393064 −0.0162512
\(586\) 35.8415 1.48060
\(587\) −24.0985 −0.994653 −0.497327 0.867563i \(-0.665685\pi\)
−0.497327 + 0.867563i \(0.665685\pi\)
\(588\) 18.4603 0.761290
\(589\) −1.53855 −0.0633949
\(590\) 0.419732 0.0172801
\(591\) 18.0950 0.744331
\(592\) 32.4187 1.33240
\(593\) −30.9840 −1.27236 −0.636180 0.771541i \(-0.719486\pi\)
−0.636180 + 0.771541i \(0.719486\pi\)
\(594\) 28.0083 1.14919
\(595\) −1.22434 −0.0501929
\(596\) 11.8401 0.484990
\(597\) 3.10992 0.127281
\(598\) 9.56614 0.391189
\(599\) 23.7689 0.971172 0.485586 0.874189i \(-0.338606\pi\)
0.485586 + 0.874189i \(0.338606\pi\)
\(600\) −10.2145 −0.417007
\(601\) −11.2370 −0.458368 −0.229184 0.973383i \(-0.573606\pi\)
−0.229184 + 0.973383i \(0.573606\pi\)
\(602\) −0.416189 −0.0169626
\(603\) 5.24355 0.213534
\(604\) 0.892789 0.0363271
\(605\) 1.75967 0.0715408
\(606\) −31.9325 −1.29717
\(607\) −16.3757 −0.664668 −0.332334 0.943162i \(-0.607836\pi\)
−0.332334 + 0.943162i \(0.607836\pi\)
\(608\) 43.2450 1.75381
\(609\) −11.1895 −0.453421
\(610\) −5.46806 −0.221395
\(611\) 4.80034 0.194201
\(612\) 7.04368 0.284724
\(613\) −14.6355 −0.591122 −0.295561 0.955324i \(-0.595506\pi\)
−0.295561 + 0.955324i \(0.595506\pi\)
\(614\) 37.1625 1.49976
\(615\) 1.34736 0.0543307
\(616\) 3.58202 0.144324
\(617\) 12.9127 0.519845 0.259923 0.965629i \(-0.416303\pi\)
0.259923 + 0.965629i \(0.416303\pi\)
\(618\) 3.78949 0.152436
\(619\) −6.77381 −0.272262 −0.136131 0.990691i \(-0.543467\pi\)
−0.136131 + 0.990691i \(0.543467\pi\)
\(620\) 0.119371 0.00479406
\(621\) −19.2968 −0.774354
\(622\) 19.1831 0.769173
\(623\) −4.10529 −0.164475
\(624\) −9.79326 −0.392044
\(625\) 23.3037 0.932149
\(626\) 0.797304 0.0318667
\(627\) 51.7780 2.06781
\(628\) −16.0862 −0.641910
\(629\) −28.2145 −1.12499
\(630\) −0.633843 −0.0252529
\(631\) −37.5702 −1.49564 −0.747822 0.663899i \(-0.768900\pi\)
−0.747822 + 0.663899i \(0.768900\pi\)
\(632\) −0.465562 −0.0185191
\(633\) 17.5948 0.699332
\(634\) 31.2984 1.24302
\(635\) 4.32854 0.171773
\(636\) −14.5039 −0.575118
\(637\) 6.24590 0.247472
\(638\) −47.2121 −1.86914
\(639\) 12.9610 0.512729
\(640\) 2.66127 0.105196
\(641\) 5.76346 0.227643 0.113822 0.993501i \(-0.463691\pi\)
0.113822 + 0.993501i \(0.463691\pi\)
\(642\) −36.2660 −1.43130
\(643\) 45.6320 1.79955 0.899776 0.436352i \(-0.143730\pi\)
0.899776 + 0.436352i \(0.143730\pi\)
\(644\) 6.47909 0.255312
\(645\) −0.177790 −0.00700048
\(646\) −48.8732 −1.92289
\(647\) −20.8063 −0.817980 −0.408990 0.912539i \(-0.634119\pi\)
−0.408990 + 0.912539i \(0.634119\pi\)
\(648\) 11.4095 0.448208
\(649\) 2.69611 0.105832
\(650\) 9.07324 0.355882
\(651\) 0.432677 0.0169579
\(652\) 10.2834 0.402728
\(653\) −20.2172 −0.791161 −0.395581 0.918431i \(-0.629457\pi\)
−0.395581 + 0.918431i \(0.629457\pi\)
\(654\) 12.7390 0.498135
\(655\) −3.08695 −0.120617
\(656\) 9.38638 0.366477
\(657\) −15.2875 −0.596421
\(658\) 7.74089 0.301771
\(659\) −1.16763 −0.0454845 −0.0227423 0.999741i \(-0.507240\pi\)
−0.0227423 + 0.999741i \(0.507240\pi\)
\(660\) −4.01728 −0.156372
\(661\) 13.1516 0.511540 0.255770 0.966738i \(-0.417671\pi\)
0.255770 + 0.966738i \(0.417671\pi\)
\(662\) −28.3847 −1.10320
\(663\) 8.52323 0.331015
\(664\) −1.76725 −0.0685826
\(665\) 1.84719 0.0716310
\(666\) −14.6068 −0.566001
\(667\) 32.5276 1.25947
\(668\) −20.9108 −0.809063
\(669\) 34.1590 1.32066
\(670\) 2.82282 0.109055
\(671\) −35.1237 −1.35593
\(672\) −12.1615 −0.469140
\(673\) −3.04069 −0.117210 −0.0586050 0.998281i \(-0.518665\pi\)
−0.0586050 + 0.998281i \(0.518665\pi\)
\(674\) 41.0318 1.58048
\(675\) −18.3025 −0.704464
\(676\) 1.44833 0.0557050
\(677\) −8.26113 −0.317501 −0.158750 0.987319i \(-0.550747\pi\)
−0.158750 + 0.987319i \(0.550747\pi\)
\(678\) −31.0600 −1.19285
\(679\) 6.87434 0.263813
\(680\) −1.44434 −0.0553881
\(681\) 17.5668 0.673161
\(682\) 1.82560 0.0699060
\(683\) 22.9900 0.879686 0.439843 0.898075i \(-0.355034\pi\)
0.439843 + 0.898075i \(0.355034\pi\)
\(684\) −10.6270 −0.406334
\(685\) 4.88790 0.186757
\(686\) 21.3599 0.815526
\(687\) 40.4128 1.54185
\(688\) −1.23858 −0.0472203
\(689\) −4.90729 −0.186953
\(690\) 6.58980 0.250869
\(691\) 8.44262 0.321173 0.160586 0.987022i \(-0.448662\pi\)
0.160586 + 0.987022i \(0.448662\pi\)
\(692\) −29.0723 −1.10516
\(693\) −4.07144 −0.154661
\(694\) 64.6510 2.45412
\(695\) 2.47943 0.0940503
\(696\) −13.2002 −0.500352
\(697\) −8.16912 −0.309427
\(698\) −43.2978 −1.63884
\(699\) −0.609820 −0.0230655
\(700\) 6.14524 0.232268
\(701\) 27.1532 1.02556 0.512781 0.858519i \(-0.328615\pi\)
0.512781 + 0.858519i \(0.328615\pi\)
\(702\) −6.95597 −0.262536
\(703\) 42.5681 1.60549
\(704\) −12.6668 −0.477398
\(705\) 3.30680 0.124541
\(706\) −37.9142 −1.42692
\(707\) −7.31756 −0.275205
\(708\) −1.97903 −0.0743765
\(709\) 14.0594 0.528014 0.264007 0.964521i \(-0.414956\pi\)
0.264007 + 0.964521i \(0.414956\pi\)
\(710\) 6.97746 0.261859
\(711\) 0.529173 0.0198456
\(712\) −4.84300 −0.181499
\(713\) −1.25778 −0.0471043
\(714\) 13.7443 0.514367
\(715\) −1.35921 −0.0508317
\(716\) −14.5663 −0.544368
\(717\) −55.9902 −2.09099
\(718\) 18.5339 0.691680
\(719\) 30.8032 1.14876 0.574382 0.818587i \(-0.305242\pi\)
0.574382 + 0.818587i \(0.305242\pi\)
\(720\) −1.88631 −0.0702988
\(721\) 0.868388 0.0323405
\(722\) 38.4541 1.43112
\(723\) −28.2953 −1.05231
\(724\) 1.50526 0.0559426
\(725\) 30.8516 1.14580
\(726\) −19.7539 −0.733137
\(727\) 19.3629 0.718131 0.359066 0.933312i \(-0.383095\pi\)
0.359066 + 0.933312i \(0.383095\pi\)
\(728\) −0.889607 −0.0329710
\(729\) 9.96404 0.369039
\(730\) −8.22989 −0.304602
\(731\) 1.07795 0.0398695
\(732\) 25.7819 0.952925
\(733\) −47.4521 −1.75269 −0.876343 0.481689i \(-0.840024\pi\)
−0.876343 + 0.481689i \(0.840024\pi\)
\(734\) 34.3403 1.26752
\(735\) 4.30259 0.158704
\(736\) 35.3532 1.30314
\(737\) 18.1322 0.667907
\(738\) −4.22918 −0.155678
\(739\) 25.8525 0.950999 0.475499 0.879716i \(-0.342267\pi\)
0.475499 + 0.879716i \(0.342267\pi\)
\(740\) −3.30272 −0.121410
\(741\) −12.8593 −0.472396
\(742\) −7.91335 −0.290508
\(743\) 17.0589 0.625829 0.312915 0.949781i \(-0.398695\pi\)
0.312915 + 0.949781i \(0.398695\pi\)
\(744\) 0.510427 0.0187132
\(745\) 2.75961 0.101104
\(746\) −71.2621 −2.60909
\(747\) 2.00872 0.0734950
\(748\) 24.3570 0.890581
\(749\) −8.31060 −0.303663
\(750\) 12.6463 0.461776
\(751\) 18.1348 0.661749 0.330875 0.943675i \(-0.392656\pi\)
0.330875 + 0.943675i \(0.392656\pi\)
\(752\) 23.0368 0.840067
\(753\) −34.6898 −1.26417
\(754\) 11.7253 0.427010
\(755\) 0.208085 0.00757298
\(756\) −4.71123 −0.171346
\(757\) 0.447506 0.0162649 0.00813244 0.999967i \(-0.497411\pi\)
0.00813244 + 0.999967i \(0.497411\pi\)
\(758\) −25.4515 −0.924438
\(759\) 42.3290 1.53645
\(760\) 2.17913 0.0790452
\(761\) 39.2873 1.42416 0.712081 0.702097i \(-0.247753\pi\)
0.712081 + 0.702097i \(0.247753\pi\)
\(762\) −48.5919 −1.76030
\(763\) 2.91923 0.105683
\(764\) 9.14186 0.330741
\(765\) 1.64169 0.0593554
\(766\) 22.4429 0.810896
\(767\) −0.669589 −0.0241775
\(768\) −42.7146 −1.54133
\(769\) 24.3862 0.879388 0.439694 0.898148i \(-0.355087\pi\)
0.439694 + 0.898148i \(0.355087\pi\)
\(770\) −2.19183 −0.0789880
\(771\) −18.7132 −0.673939
\(772\) 18.4929 0.665575
\(773\) 33.8399 1.21714 0.608568 0.793502i \(-0.291744\pi\)
0.608568 + 0.793502i \(0.291744\pi\)
\(774\) 0.558060 0.0200591
\(775\) −1.19297 −0.0428529
\(776\) 8.10964 0.291119
\(777\) −11.9712 −0.429463
\(778\) −56.1045 −2.01145
\(779\) 12.3250 0.441589
\(780\) 0.997706 0.0357236
\(781\) 44.8191 1.60375
\(782\) −39.9544 −1.42877
\(783\) −23.6523 −0.845263
\(784\) 29.9741 1.07050
\(785\) −3.74926 −0.133817
\(786\) 34.6538 1.23606
\(787\) −5.07402 −0.180869 −0.0904346 0.995902i \(-0.528826\pi\)
−0.0904346 + 0.995902i \(0.528826\pi\)
\(788\) −12.8425 −0.457496
\(789\) 28.7398 1.02316
\(790\) 0.284876 0.0101355
\(791\) −7.11761 −0.253073
\(792\) −4.80306 −0.170669
\(793\) 8.72309 0.309766
\(794\) 41.2485 1.46385
\(795\) −3.38047 −0.119893
\(796\) −2.20719 −0.0782319
\(797\) −40.4586 −1.43312 −0.716558 0.697527i \(-0.754284\pi\)
−0.716558 + 0.697527i \(0.754284\pi\)
\(798\) −20.7364 −0.734062
\(799\) −20.0493 −0.709294
\(800\) 33.5316 1.18552
\(801\) 5.50471 0.194499
\(802\) −35.9726 −1.27024
\(803\) −52.8640 −1.86553
\(804\) −13.3096 −0.469393
\(805\) 1.51010 0.0532240
\(806\) −0.453396 −0.0159702
\(807\) 28.2582 0.994737
\(808\) −8.63250 −0.303690
\(809\) 33.6236 1.18214 0.591072 0.806619i \(-0.298705\pi\)
0.591072 + 0.806619i \(0.298705\pi\)
\(810\) −6.98146 −0.245303
\(811\) −31.9444 −1.12172 −0.560860 0.827911i \(-0.689529\pi\)
−0.560860 + 0.827911i \(0.689529\pi\)
\(812\) 7.94147 0.278691
\(813\) 30.8794 1.08299
\(814\) −50.5102 −1.77038
\(815\) 2.39677 0.0839553
\(816\) 40.9030 1.43189
\(817\) −1.62634 −0.0568984
\(818\) −20.8287 −0.728258
\(819\) 1.01116 0.0353327
\(820\) −0.956255 −0.0333939
\(821\) −52.2211 −1.82253 −0.911264 0.411824i \(-0.864892\pi\)
−0.911264 + 0.411824i \(0.864892\pi\)
\(822\) −54.8712 −1.91385
\(823\) 47.2992 1.64875 0.824374 0.566046i \(-0.191527\pi\)
0.824374 + 0.566046i \(0.191527\pi\)
\(824\) 1.02443 0.0356879
\(825\) 40.1480 1.39777
\(826\) −1.07976 −0.0375696
\(827\) −28.6641 −0.996747 −0.498374 0.866962i \(-0.666069\pi\)
−0.498374 + 0.866962i \(0.666069\pi\)
\(828\) −8.68769 −0.301918
\(829\) 11.7300 0.407399 0.203700 0.979033i \(-0.434703\pi\)
0.203700 + 0.979033i \(0.434703\pi\)
\(830\) 1.08138 0.0375351
\(831\) −64.4007 −2.23404
\(832\) 3.14585 0.109063
\(833\) −26.0869 −0.903858
\(834\) −27.8339 −0.963810
\(835\) −4.87374 −0.168663
\(836\) −36.7482 −1.27096
\(837\) 0.914589 0.0316128
\(838\) −54.1164 −1.86942
\(839\) −36.8104 −1.27084 −0.635418 0.772168i \(-0.719172\pi\)
−0.635418 + 0.772168i \(0.719172\pi\)
\(840\) −0.612821 −0.0211443
\(841\) 10.8693 0.374804
\(842\) −14.5370 −0.500980
\(843\) −13.5672 −0.467280
\(844\) −12.4875 −0.429838
\(845\) 0.337566 0.0116126
\(846\) −10.3796 −0.356858
\(847\) −4.52675 −0.155541
\(848\) −23.5501 −0.808713
\(849\) 38.4604 1.31996
\(850\) −37.8957 −1.29981
\(851\) 34.7999 1.19292
\(852\) −32.8986 −1.12709
\(853\) 24.7408 0.847108 0.423554 0.905871i \(-0.360782\pi\)
0.423554 + 0.905871i \(0.360782\pi\)
\(854\) 14.0666 0.481349
\(855\) −2.47687 −0.0847070
\(856\) −9.80398 −0.335093
\(857\) 44.5905 1.52318 0.761592 0.648057i \(-0.224418\pi\)
0.761592 + 0.648057i \(0.224418\pi\)
\(858\) 15.2584 0.520915
\(859\) −30.3184 −1.03445 −0.517225 0.855849i \(-0.673035\pi\)
−0.517225 + 0.855849i \(0.673035\pi\)
\(860\) 0.126182 0.00430278
\(861\) −3.46608 −0.118124
\(862\) −37.4519 −1.27562
\(863\) 34.3245 1.16842 0.584210 0.811603i \(-0.301405\pi\)
0.584210 + 0.811603i \(0.301405\pi\)
\(864\) −25.7069 −0.874566
\(865\) −6.77595 −0.230389
\(866\) 24.2936 0.825530
\(867\) −0.906811 −0.0307969
\(868\) −0.307082 −0.0104230
\(869\) 1.82988 0.0620745
\(870\) 8.07717 0.273842
\(871\) −4.50319 −0.152585
\(872\) 3.44381 0.116622
\(873\) −9.21768 −0.311971
\(874\) 60.2804 2.03901
\(875\) 2.89798 0.0979696
\(876\) 38.8039 1.31106
\(877\) −22.4203 −0.757080 −0.378540 0.925585i \(-0.623574\pi\)
−0.378540 + 0.925585i \(0.623574\pi\)
\(878\) −14.2369 −0.480472
\(879\) −39.3875 −1.32851
\(880\) −6.52287 −0.219886
\(881\) −42.0849 −1.41788 −0.708939 0.705270i \(-0.750826\pi\)
−0.708939 + 0.705270i \(0.750826\pi\)
\(882\) −13.5053 −0.454747
\(883\) 6.14059 0.206647 0.103324 0.994648i \(-0.467052\pi\)
0.103324 + 0.994648i \(0.467052\pi\)
\(884\) −6.04916 −0.203455
\(885\) −0.461258 −0.0155050
\(886\) 27.2898 0.916819
\(887\) 10.8303 0.363646 0.181823 0.983331i \(-0.441800\pi\)
0.181823 + 0.983331i \(0.441800\pi\)
\(888\) −14.1223 −0.473914
\(889\) −11.1352 −0.373462
\(890\) 2.96342 0.0993340
\(891\) −44.8448 −1.50236
\(892\) −24.2436 −0.811734
\(893\) 30.2490 1.01225
\(894\) −30.9792 −1.03610
\(895\) −3.39500 −0.113482
\(896\) −6.84611 −0.228713
\(897\) −10.5126 −0.351005
\(898\) −57.7200 −1.92614
\(899\) −1.54167 −0.0514177
\(900\) −8.24005 −0.274668
\(901\) 20.4960 0.682821
\(902\) −14.6245 −0.486943
\(903\) 0.457365 0.0152201
\(904\) −8.39662 −0.279268
\(905\) 0.350836 0.0116622
\(906\) −2.33594 −0.0776065
\(907\) −14.3673 −0.477059 −0.238529 0.971135i \(-0.576665\pi\)
−0.238529 + 0.971135i \(0.576665\pi\)
\(908\) −12.4676 −0.413752
\(909\) 9.81198 0.325443
\(910\) 0.544349 0.0180450
\(911\) −35.6324 −1.18055 −0.590276 0.807201i \(-0.700981\pi\)
−0.590276 + 0.807201i \(0.700981\pi\)
\(912\) −61.7116 −2.04347
\(913\) 6.94613 0.229883
\(914\) −31.5329 −1.04301
\(915\) 6.00905 0.198653
\(916\) −28.6820 −0.947681
\(917\) 7.94117 0.262241
\(918\) 29.0526 0.958879
\(919\) −22.6521 −0.747224 −0.373612 0.927585i \(-0.621881\pi\)
−0.373612 + 0.927585i \(0.621881\pi\)
\(920\) 1.78146 0.0587329
\(921\) −40.8392 −1.34570
\(922\) −74.6142 −2.45729
\(923\) −11.1310 −0.366381
\(924\) 10.3344 0.339978
\(925\) 33.0068 1.08526
\(926\) 14.3322 0.470986
\(927\) −1.16441 −0.0382441
\(928\) 43.3327 1.42247
\(929\) −4.22358 −0.138571 −0.0692856 0.997597i \(-0.522072\pi\)
−0.0692856 + 0.997597i \(0.522072\pi\)
\(930\) −0.312329 −0.0102417
\(931\) 39.3581 1.28991
\(932\) 0.432806 0.0141770
\(933\) −21.0810 −0.690161
\(934\) 30.6412 1.00261
\(935\) 5.67696 0.185656
\(936\) 1.19286 0.0389898
\(937\) 55.9893 1.82909 0.914546 0.404483i \(-0.132548\pi\)
0.914546 + 0.404483i \(0.132548\pi\)
\(938\) −7.26171 −0.237103
\(939\) −0.876185 −0.0285932
\(940\) −2.34692 −0.0765481
\(941\) −33.3262 −1.08640 −0.543201 0.839603i \(-0.682788\pi\)
−0.543201 + 0.839603i \(0.682788\pi\)
\(942\) 42.0889 1.37133
\(943\) 10.0758 0.328114
\(944\) −3.21336 −0.104586
\(945\) −1.09806 −0.0357199
\(946\) 1.92977 0.0627423
\(947\) −9.46185 −0.307469 −0.153734 0.988112i \(-0.549130\pi\)
−0.153734 + 0.988112i \(0.549130\pi\)
\(948\) −1.34319 −0.0436248
\(949\) 13.1290 0.426185
\(950\) 57.1744 1.85498
\(951\) −34.3949 −1.11533
\(952\) 3.71557 0.120422
\(953\) −27.7988 −0.900492 −0.450246 0.892905i \(-0.648664\pi\)
−0.450246 + 0.892905i \(0.648664\pi\)
\(954\) 10.6109 0.343539
\(955\) 2.13072 0.0689484
\(956\) 39.7377 1.28521
\(957\) 51.8830 1.67714
\(958\) 54.8850 1.77325
\(959\) −12.5741 −0.406040
\(960\) 2.16707 0.0699419
\(961\) −30.9404 −0.998077
\(962\) 12.5444 0.404448
\(963\) 11.1435 0.359095
\(964\) 20.0819 0.646796
\(965\) 4.31020 0.138750
\(966\) −16.9523 −0.545430
\(967\) −54.5201 −1.75325 −0.876624 0.481175i \(-0.840210\pi\)
−0.876624 + 0.481175i \(0.840210\pi\)
\(968\) −5.34019 −0.171640
\(969\) 53.7085 1.72537
\(970\) −4.96227 −0.159329
\(971\) −16.2193 −0.520503 −0.260251 0.965541i \(-0.583805\pi\)
−0.260251 + 0.965541i \(0.583805\pi\)
\(972\) 16.6417 0.533783
\(973\) −6.37834 −0.204480
\(974\) −46.1195 −1.47777
\(975\) −9.97090 −0.319324
\(976\) 41.8621 1.33997
\(977\) −10.4155 −0.333221 −0.166610 0.986023i \(-0.553282\pi\)
−0.166610 + 0.986023i \(0.553282\pi\)
\(978\) −26.9060 −0.860359
\(979\) 19.0353 0.608370
\(980\) −3.05366 −0.0975457
\(981\) −3.91435 −0.124975
\(982\) −43.0057 −1.37237
\(983\) 60.0352 1.91483 0.957413 0.288722i \(-0.0932302\pi\)
0.957413 + 0.288722i \(0.0932302\pi\)
\(984\) −4.08892 −0.130350
\(985\) −2.99324 −0.0953726
\(986\) −48.9724 −1.55960
\(987\) −8.50673 −0.270772
\(988\) 9.12655 0.290354
\(989\) −1.32955 −0.0422772
\(990\) 2.93898 0.0934070
\(991\) 11.7059 0.371849 0.185925 0.982564i \(-0.440472\pi\)
0.185925 + 0.982564i \(0.440472\pi\)
\(992\) −1.67560 −0.0532002
\(993\) 31.1929 0.989878
\(994\) −17.9495 −0.569323
\(995\) −0.514436 −0.0163087
\(996\) −5.09868 −0.161558
\(997\) −16.3890 −0.519045 −0.259522 0.965737i \(-0.583565\pi\)
−0.259522 + 0.965737i \(0.583565\pi\)
\(998\) −42.4760 −1.34455
\(999\) −25.3045 −0.800600
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 1339.2.a.e.1.17 21
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1339.2.a.e.1.17 21 1.1 even 1 trivial