Properties

Label 3549.2.a.z.1.5
Level $3549$
Weight $2$
Character 3549.1
Self dual yes
Analytic conductor $28.339$
Analytic rank $1$
Dimension $6$
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] = [3549,2,Mod(1,3549)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3549, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3549.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3549 = 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3549.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: \(28.3389076774\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.121819537.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} - 25x^{4} + 55x^{3} + 224x^{2} - 252x - 728 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-2.07801\) of defining polynomial
Character \(\chi\) \(=\) 3549.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.80194 q^{2} -1.00000 q^{3} +1.24698 q^{4} -3.07801 q^{5} -1.80194 q^{6} +1.00000 q^{7} -1.35690 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.80194 q^{2} -1.00000 q^{3} +1.24698 q^{4} -3.07801 q^{5} -1.80194 q^{6} +1.00000 q^{7} -1.35690 q^{8} +1.00000 q^{9} -5.54638 q^{10} +6.47118 q^{11} -1.24698 q^{12} +1.80194 q^{14} +3.07801 q^{15} -4.93900 q^{16} -2.47976 q^{17} +1.80194 q^{18} -5.83821 q^{19} -3.83821 q^{20} -1.00000 q^{21} +11.6607 q^{22} +3.64015 q^{23} +1.35690 q^{24} +4.47413 q^{25} -1.00000 q^{27} +1.24698 q^{28} +4.48732 q^{29} +5.54638 q^{30} +2.78617 q^{31} -6.18598 q^{32} -6.47118 q^{33} -4.46837 q^{34} -3.07801 q^{35} +1.24698 q^{36} -1.16039 q^{37} -10.5201 q^{38} +4.17654 q^{40} -1.53218 q^{41} -1.80194 q^{42} -6.89765 q^{43} +8.06943 q^{44} -3.07801 q^{45} +6.55933 q^{46} -5.15603 q^{47} +4.93900 q^{48} +1.00000 q^{49} +8.06211 q^{50} +2.47976 q^{51} -9.65435 q^{53} -1.80194 q^{54} -19.9183 q^{55} -1.35690 q^{56} +5.83821 q^{57} +8.08588 q^{58} +11.5467 q^{59} +3.83821 q^{60} -7.28676 q^{61} +5.02051 q^{62} +1.00000 q^{63} -1.26875 q^{64} -11.6607 q^{66} +2.56252 q^{67} -3.09221 q^{68} -3.64015 q^{69} -5.54638 q^{70} -11.8167 q^{71} -1.35690 q^{72} -11.4677 q^{73} -2.09095 q^{74} -4.47413 q^{75} -7.28013 q^{76} +6.47118 q^{77} -11.8875 q^{79} +15.2023 q^{80} +1.00000 q^{81} -2.76089 q^{82} +7.58647 q^{83} -1.24698 q^{84} +7.63272 q^{85} -12.4291 q^{86} -4.48732 q^{87} -8.78072 q^{88} -18.6906 q^{89} -5.54638 q^{90} +4.53919 q^{92} -2.78617 q^{93} -9.29084 q^{94} +17.9701 q^{95} +6.18598 q^{96} -1.08433 q^{97} +1.80194 q^{98} +6.47118 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{2} - 6 q^{3} - 2 q^{4} - 3 q^{5} - 2 q^{6} + 6 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{2} - 6 q^{3} - 2 q^{4} - 3 q^{5} - 2 q^{6} + 6 q^{7} + 6 q^{9} - q^{10} + 2 q^{12} + 2 q^{14} + 3 q^{15} - 10 q^{16} - 9 q^{17} + 2 q^{18} - 11 q^{19} + q^{20} - 6 q^{21} + 7 q^{22} - 11 q^{23} + 29 q^{25} - 6 q^{27} - 2 q^{28} - 10 q^{29} + q^{30} - 7 q^{31} - 8 q^{32} - 10 q^{34} - 3 q^{35} - 2 q^{36} + 20 q^{37} - 6 q^{38} + 10 q^{41} - 2 q^{42} - 9 q^{43} - 3 q^{45} + 8 q^{46} - 36 q^{47} + 10 q^{48} + 6 q^{49} - 9 q^{50} + 9 q^{51} - 12 q^{53} - 2 q^{54} - 29 q^{55} + 11 q^{57} + 6 q^{58} + 41 q^{59} - q^{60} - 24 q^{61} + 6 q^{63} + 8 q^{64} - 7 q^{66} - 15 q^{67} + 10 q^{68} + 11 q^{69} - q^{70} - 13 q^{71} - 25 q^{73} + 2 q^{74} - 29 q^{75} - q^{76} - 16 q^{79} + 5 q^{80} + 6 q^{81} + q^{82} + 2 q^{83} + 2 q^{84} + 33 q^{85} + 4 q^{86} + 10 q^{87} - 14 q^{88} - 15 q^{89} - q^{90} + 13 q^{92} + 7 q^{93} - 33 q^{94} - 23 q^{95} + 8 q^{96} - 10 q^{97} + 2 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.80194 1.27416 0.637081 0.770797i \(-0.280142\pi\)
0.637081 + 0.770797i \(0.280142\pi\)
\(3\) −1.00000 −0.577350
\(4\) 1.24698 0.623490
\(5\) −3.07801 −1.37653 −0.688263 0.725461i \(-0.741627\pi\)
−0.688263 + 0.725461i \(0.741627\pi\)
\(6\) −1.80194 −0.735638
\(7\) 1.00000 0.377964
\(8\) −1.35690 −0.479735
\(9\) 1.00000 0.333333
\(10\) −5.54638 −1.75392
\(11\) 6.47118 1.95113 0.975567 0.219702i \(-0.0705086\pi\)
0.975567 + 0.219702i \(0.0705086\pi\)
\(12\) −1.24698 −0.359972
\(13\) 0 0
\(14\) 1.80194 0.481588
\(15\) 3.07801 0.794738
\(16\) −4.93900 −1.23475
\(17\) −2.47976 −0.601430 −0.300715 0.953714i \(-0.597225\pi\)
−0.300715 + 0.953714i \(0.597225\pi\)
\(18\) 1.80194 0.424721
\(19\) −5.83821 −1.33938 −0.669689 0.742642i \(-0.733573\pi\)
−0.669689 + 0.742642i \(0.733573\pi\)
\(20\) −3.83821 −0.858251
\(21\) −1.00000 −0.218218
\(22\) 11.6607 2.48606
\(23\) 3.64015 0.759024 0.379512 0.925187i \(-0.376092\pi\)
0.379512 + 0.925187i \(0.376092\pi\)
\(24\) 1.35690 0.276975
\(25\) 4.47413 0.894826
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 1.24698 0.235657
\(29\) 4.48732 0.833275 0.416637 0.909073i \(-0.363208\pi\)
0.416637 + 0.909073i \(0.363208\pi\)
\(30\) 5.54638 1.01263
\(31\) 2.78617 0.500412 0.250206 0.968193i \(-0.419502\pi\)
0.250206 + 0.968193i \(0.419502\pi\)
\(32\) −6.18598 −1.09354
\(33\) −6.47118 −1.12649
\(34\) −4.46837 −0.766319
\(35\) −3.07801 −0.520278
\(36\) 1.24698 0.207830
\(37\) −1.16039 −0.190767 −0.0953836 0.995441i \(-0.530408\pi\)
−0.0953836 + 0.995441i \(0.530408\pi\)
\(38\) −10.5201 −1.70658
\(39\) 0 0
\(40\) 4.17654 0.660368
\(41\) −1.53218 −0.239286 −0.119643 0.992817i \(-0.538175\pi\)
−0.119643 + 0.992817i \(0.538175\pi\)
\(42\) −1.80194 −0.278045
\(43\) −6.89765 −1.05188 −0.525941 0.850521i \(-0.676287\pi\)
−0.525941 + 0.850521i \(0.676287\pi\)
\(44\) 8.06943 1.21651
\(45\) −3.07801 −0.458842
\(46\) 6.55933 0.967120
\(47\) −5.15603 −0.752084 −0.376042 0.926603i \(-0.622715\pi\)
−0.376042 + 0.926603i \(0.622715\pi\)
\(48\) 4.93900 0.712883
\(49\) 1.00000 0.142857
\(50\) 8.06211 1.14015
\(51\) 2.47976 0.347236
\(52\) 0 0
\(53\) −9.65435 −1.32613 −0.663064 0.748563i \(-0.730744\pi\)
−0.663064 + 0.748563i \(0.730744\pi\)
\(54\) −1.80194 −0.245213
\(55\) −19.9183 −2.68579
\(56\) −1.35690 −0.181323
\(57\) 5.83821 0.773290
\(58\) 8.08588 1.06173
\(59\) 11.5467 1.50325 0.751625 0.659591i \(-0.229270\pi\)
0.751625 + 0.659591i \(0.229270\pi\)
\(60\) 3.83821 0.495511
\(61\) −7.28676 −0.932974 −0.466487 0.884528i \(-0.654480\pi\)
−0.466487 + 0.884528i \(0.654480\pi\)
\(62\) 5.02051 0.637606
\(63\) 1.00000 0.125988
\(64\) −1.26875 −0.158594
\(65\) 0 0
\(66\) −11.6607 −1.43533
\(67\) 2.56252 0.313062 0.156531 0.987673i \(-0.449969\pi\)
0.156531 + 0.987673i \(0.449969\pi\)
\(68\) −3.09221 −0.374985
\(69\) −3.64015 −0.438223
\(70\) −5.54638 −0.662919
\(71\) −11.8167 −1.40238 −0.701191 0.712973i \(-0.747348\pi\)
−0.701191 + 0.712973i \(0.747348\pi\)
\(72\) −1.35690 −0.159912
\(73\) −11.4677 −1.34219 −0.671095 0.741371i \(-0.734176\pi\)
−0.671095 + 0.741371i \(0.734176\pi\)
\(74\) −2.09095 −0.243069
\(75\) −4.47413 −0.516628
\(76\) −7.28013 −0.835088
\(77\) 6.47118 0.737459
\(78\) 0 0
\(79\) −11.8875 −1.33745 −0.668725 0.743510i \(-0.733160\pi\)
−0.668725 + 0.743510i \(0.733160\pi\)
\(80\) 15.2023 1.69967
\(81\) 1.00000 0.111111
\(82\) −2.76089 −0.304889
\(83\) 7.58647 0.832723 0.416362 0.909199i \(-0.363305\pi\)
0.416362 + 0.909199i \(0.363305\pi\)
\(84\) −1.24698 −0.136057
\(85\) 7.63272 0.827884
\(86\) −12.4291 −1.34027
\(87\) −4.48732 −0.481092
\(88\) −8.78072 −0.936027
\(89\) −18.6906 −1.98120 −0.990601 0.136785i \(-0.956323\pi\)
−0.990601 + 0.136785i \(0.956323\pi\)
\(90\) −5.54638 −0.584640
\(91\) 0 0
\(92\) 4.53919 0.473244
\(93\) −2.78617 −0.288913
\(94\) −9.29084 −0.958277
\(95\) 17.9701 1.84369
\(96\) 6.18598 0.631354
\(97\) −1.08433 −0.110097 −0.0550484 0.998484i \(-0.517531\pi\)
−0.0550484 + 0.998484i \(0.517531\pi\)
\(98\) 1.80194 0.182023
\(99\) 6.47118 0.650378
\(100\) 5.57915 0.557915
\(101\) −18.9676 −1.88735 −0.943675 0.330873i \(-0.892657\pi\)
−0.943675 + 0.330873i \(0.892657\pi\)
\(102\) 4.46837 0.442435
\(103\) 0.207896 0.0204846 0.0102423 0.999948i \(-0.496740\pi\)
0.0102423 + 0.999948i \(0.496740\pi\)
\(104\) 0 0
\(105\) 3.07801 0.300383
\(106\) −17.3965 −1.68970
\(107\) −6.92160 −0.669136 −0.334568 0.942372i \(-0.608590\pi\)
−0.334568 + 0.942372i \(0.608590\pi\)
\(108\) −1.24698 −0.119991
\(109\) −9.79742 −0.938423 −0.469211 0.883086i \(-0.655462\pi\)
−0.469211 + 0.883086i \(0.655462\pi\)
\(110\) −35.8916 −3.42213
\(111\) 1.16039 0.110140
\(112\) −4.93900 −0.466692
\(113\) −11.9913 −1.12805 −0.564023 0.825759i \(-0.690747\pi\)
−0.564023 + 0.825759i \(0.690747\pi\)
\(114\) 10.5201 0.985297
\(115\) −11.2044 −1.04482
\(116\) 5.59560 0.519538
\(117\) 0 0
\(118\) 20.8064 1.91538
\(119\) −2.47976 −0.227319
\(120\) −4.17654 −0.381264
\(121\) 30.8762 2.80692
\(122\) −13.1303 −1.18876
\(123\) 1.53218 0.138152
\(124\) 3.47430 0.312001
\(125\) 1.61862 0.144774
\(126\) 1.80194 0.160529
\(127\) 19.4114 1.72249 0.861243 0.508194i \(-0.169687\pi\)
0.861243 + 0.508194i \(0.169687\pi\)
\(128\) 10.0858 0.891463
\(129\) 6.89765 0.607304
\(130\) 0 0
\(131\) −18.4298 −1.61022 −0.805111 0.593124i \(-0.797894\pi\)
−0.805111 + 0.593124i \(0.797894\pi\)
\(132\) −8.06943 −0.702354
\(133\) −5.83821 −0.506237
\(134\) 4.61751 0.398892
\(135\) 3.07801 0.264913
\(136\) 3.36477 0.288527
\(137\) 8.25681 0.705427 0.352714 0.935731i \(-0.385259\pi\)
0.352714 + 0.935731i \(0.385259\pi\)
\(138\) −6.55933 −0.558367
\(139\) 12.8927 1.09355 0.546774 0.837280i \(-0.315856\pi\)
0.546774 + 0.837280i \(0.315856\pi\)
\(140\) −3.83821 −0.324388
\(141\) 5.15603 0.434216
\(142\) −21.2929 −1.78686
\(143\) 0 0
\(144\) −4.93900 −0.411583
\(145\) −13.8120 −1.14703
\(146\) −20.6640 −1.71017
\(147\) −1.00000 −0.0824786
\(148\) −1.44699 −0.118941
\(149\) −0.731647 −0.0599389 −0.0299694 0.999551i \(-0.509541\pi\)
−0.0299694 + 0.999551i \(0.509541\pi\)
\(150\) −8.06211 −0.658268
\(151\) 1.03330 0.0840886 0.0420443 0.999116i \(-0.486613\pi\)
0.0420443 + 0.999116i \(0.486613\pi\)
\(152\) 7.92185 0.642547
\(153\) −2.47976 −0.200477
\(154\) 11.6607 0.939643
\(155\) −8.57586 −0.688830
\(156\) 0 0
\(157\) 6.14425 0.490364 0.245182 0.969477i \(-0.421152\pi\)
0.245182 + 0.969477i \(0.421152\pi\)
\(158\) −21.4206 −1.70413
\(159\) 9.65435 0.765640
\(160\) 19.0405 1.50528
\(161\) 3.64015 0.286884
\(162\) 1.80194 0.141574
\(163\) 7.70203 0.603269 0.301635 0.953424i \(-0.402468\pi\)
0.301635 + 0.953424i \(0.402468\pi\)
\(164\) −1.91059 −0.149192
\(165\) 19.9183 1.55064
\(166\) 13.6704 1.06102
\(167\) 12.9563 1.00259 0.501293 0.865278i \(-0.332858\pi\)
0.501293 + 0.865278i \(0.332858\pi\)
\(168\) 1.35690 0.104687
\(169\) 0 0
\(170\) 13.7537 1.05486
\(171\) −5.83821 −0.446459
\(172\) −8.60123 −0.655837
\(173\) −10.0988 −0.767799 −0.383900 0.923375i \(-0.625419\pi\)
−0.383900 + 0.923375i \(0.625419\pi\)
\(174\) −8.08588 −0.612989
\(175\) 4.47413 0.338213
\(176\) −31.9612 −2.40916
\(177\) −11.5467 −0.867902
\(178\) −33.6793 −2.52437
\(179\) −24.2768 −1.81453 −0.907267 0.420556i \(-0.861835\pi\)
−0.907267 + 0.420556i \(0.861835\pi\)
\(180\) −3.83821 −0.286084
\(181\) 7.33317 0.545070 0.272535 0.962146i \(-0.412138\pi\)
0.272535 + 0.962146i \(0.412138\pi\)
\(182\) 0 0
\(183\) 7.28676 0.538653
\(184\) −4.93931 −0.364130
\(185\) 3.57170 0.262596
\(186\) −5.02051 −0.368122
\(187\) −16.0470 −1.17347
\(188\) −6.42946 −0.468916
\(189\) −1.00000 −0.0727393
\(190\) 32.3809 2.34916
\(191\) −0.103045 −0.00745611 −0.00372805 0.999993i \(-0.501187\pi\)
−0.00372805 + 0.999993i \(0.501187\pi\)
\(192\) 1.26875 0.0915641
\(193\) −11.8062 −0.849828 −0.424914 0.905234i \(-0.639696\pi\)
−0.424914 + 0.905234i \(0.639696\pi\)
\(194\) −1.95389 −0.140281
\(195\) 0 0
\(196\) 1.24698 0.0890700
\(197\) −22.8672 −1.62922 −0.814609 0.580011i \(-0.803048\pi\)
−0.814609 + 0.580011i \(0.803048\pi\)
\(198\) 11.6607 0.828687
\(199\) 0.821199 0.0582132 0.0291066 0.999576i \(-0.490734\pi\)
0.0291066 + 0.999576i \(0.490734\pi\)
\(200\) −6.07093 −0.429280
\(201\) −2.56252 −0.180746
\(202\) −34.1785 −2.40479
\(203\) 4.48732 0.314948
\(204\) 3.09221 0.216498
\(205\) 4.71606 0.329384
\(206\) 0.374616 0.0261007
\(207\) 3.64015 0.253008
\(208\) 0 0
\(209\) −37.7801 −2.61331
\(210\) 5.54638 0.382736
\(211\) −24.8233 −1.70891 −0.854453 0.519528i \(-0.826108\pi\)
−0.854453 + 0.519528i \(0.826108\pi\)
\(212\) −12.0388 −0.826827
\(213\) 11.8167 0.809666
\(214\) −12.4723 −0.852588
\(215\) 21.2310 1.44794
\(216\) 1.35690 0.0923251
\(217\) 2.78617 0.189138
\(218\) −17.6543 −1.19570
\(219\) 11.4677 0.774914
\(220\) −24.8378 −1.67456
\(221\) 0 0
\(222\) 2.09095 0.140336
\(223\) 21.2058 1.42004 0.710022 0.704179i \(-0.248685\pi\)
0.710022 + 0.704179i \(0.248685\pi\)
\(224\) −6.18598 −0.413318
\(225\) 4.47413 0.298275
\(226\) −21.6075 −1.43731
\(227\) 19.0786 1.26629 0.633144 0.774034i \(-0.281764\pi\)
0.633144 + 0.774034i \(0.281764\pi\)
\(228\) 7.28013 0.482139
\(229\) 13.5257 0.893805 0.446902 0.894583i \(-0.352527\pi\)
0.446902 + 0.894583i \(0.352527\pi\)
\(230\) −20.1897 −1.33127
\(231\) −6.47118 −0.425772
\(232\) −6.08883 −0.399751
\(233\) −17.1313 −1.12231 −0.561154 0.827711i \(-0.689643\pi\)
−0.561154 + 0.827711i \(0.689643\pi\)
\(234\) 0 0
\(235\) 15.8703 1.03526
\(236\) 14.3985 0.937261
\(237\) 11.8875 0.772177
\(238\) −4.46837 −0.289641
\(239\) −17.8428 −1.15416 −0.577078 0.816689i \(-0.695807\pi\)
−0.577078 + 0.816689i \(0.695807\pi\)
\(240\) −15.2023 −0.981303
\(241\) 17.0689 1.09950 0.549752 0.835328i \(-0.314722\pi\)
0.549752 + 0.835328i \(0.314722\pi\)
\(242\) 55.6369 3.57648
\(243\) −1.00000 −0.0641500
\(244\) −9.08644 −0.581700
\(245\) −3.07801 −0.196647
\(246\) 2.76089 0.176028
\(247\) 0 0
\(248\) −3.78055 −0.240065
\(249\) −7.58647 −0.480773
\(250\) 2.91666 0.184466
\(251\) −22.0735 −1.39327 −0.696633 0.717428i \(-0.745319\pi\)
−0.696633 + 0.717428i \(0.745319\pi\)
\(252\) 1.24698 0.0785523
\(253\) 23.5561 1.48096
\(254\) 34.9782 2.19473
\(255\) −7.63272 −0.477979
\(256\) 20.7114 1.29446
\(257\) −10.5585 −0.658619 −0.329309 0.944222i \(-0.606816\pi\)
−0.329309 + 0.944222i \(0.606816\pi\)
\(258\) 12.4291 0.773804
\(259\) −1.16039 −0.0721033
\(260\) 0 0
\(261\) 4.48732 0.277758
\(262\) −33.2094 −2.05168
\(263\) 24.6204 1.51816 0.759081 0.650997i \(-0.225649\pi\)
0.759081 + 0.650997i \(0.225649\pi\)
\(264\) 8.78072 0.540416
\(265\) 29.7162 1.82545
\(266\) −10.5201 −0.645028
\(267\) 18.6906 1.14385
\(268\) 3.19541 0.195191
\(269\) −12.5227 −0.763525 −0.381762 0.924260i \(-0.624683\pi\)
−0.381762 + 0.924260i \(0.624683\pi\)
\(270\) 5.54638 0.337542
\(271\) 12.3506 0.750243 0.375122 0.926976i \(-0.377601\pi\)
0.375122 + 0.926976i \(0.377601\pi\)
\(272\) 12.2475 0.742616
\(273\) 0 0
\(274\) 14.8783 0.898829
\(275\) 28.9529 1.74593
\(276\) −4.53919 −0.273227
\(277\) 16.3139 0.980204 0.490102 0.871665i \(-0.336959\pi\)
0.490102 + 0.871665i \(0.336959\pi\)
\(278\) 23.2319 1.39336
\(279\) 2.78617 0.166804
\(280\) 4.17654 0.249596
\(281\) −10.5157 −0.627315 −0.313657 0.949536i \(-0.601554\pi\)
−0.313657 + 0.949536i \(0.601554\pi\)
\(282\) 9.29084 0.553261
\(283\) 25.4773 1.51447 0.757233 0.653145i \(-0.226551\pi\)
0.757233 + 0.653145i \(0.226551\pi\)
\(284\) −14.7352 −0.874371
\(285\) −17.9701 −1.06445
\(286\) 0 0
\(287\) −1.53218 −0.0904416
\(288\) −6.18598 −0.364512
\(289\) −10.8508 −0.638282
\(290\) −24.8884 −1.46150
\(291\) 1.08433 0.0635644
\(292\) −14.3000 −0.836842
\(293\) −4.73420 −0.276575 −0.138288 0.990392i \(-0.544160\pi\)
−0.138288 + 0.990392i \(0.544160\pi\)
\(294\) −1.80194 −0.105091
\(295\) −35.5408 −2.06926
\(296\) 1.57453 0.0915178
\(297\) −6.47118 −0.375496
\(298\) −1.31838 −0.0763718
\(299\) 0 0
\(300\) −5.57915 −0.322112
\(301\) −6.89765 −0.397574
\(302\) 1.86194 0.107143
\(303\) 18.9676 1.08966
\(304\) 28.8349 1.65380
\(305\) 22.4287 1.28426
\(306\) −4.46837 −0.255440
\(307\) −5.13876 −0.293284 −0.146642 0.989190i \(-0.546847\pi\)
−0.146642 + 0.989190i \(0.546847\pi\)
\(308\) 8.06943 0.459798
\(309\) −0.207896 −0.0118268
\(310\) −15.4532 −0.877681
\(311\) −14.4284 −0.818161 −0.409081 0.912498i \(-0.634150\pi\)
−0.409081 + 0.912498i \(0.634150\pi\)
\(312\) 0 0
\(313\) −3.77751 −0.213518 −0.106759 0.994285i \(-0.534047\pi\)
−0.106759 + 0.994285i \(0.534047\pi\)
\(314\) 11.0716 0.624804
\(315\) −3.07801 −0.173426
\(316\) −14.8235 −0.833886
\(317\) 17.6020 0.988627 0.494313 0.869284i \(-0.335420\pi\)
0.494313 + 0.869284i \(0.335420\pi\)
\(318\) 17.3965 0.975550
\(319\) 29.0383 1.62583
\(320\) 3.90522 0.218309
\(321\) 6.92160 0.386326
\(322\) 6.55933 0.365537
\(323\) 14.4774 0.805542
\(324\) 1.24698 0.0692766
\(325\) 0 0
\(326\) 13.8786 0.768663
\(327\) 9.79742 0.541799
\(328\) 2.07901 0.114794
\(329\) −5.15603 −0.284261
\(330\) 35.8916 1.97577
\(331\) 8.93862 0.491311 0.245656 0.969357i \(-0.420997\pi\)
0.245656 + 0.969357i \(0.420997\pi\)
\(332\) 9.46018 0.519195
\(333\) −1.16039 −0.0635891
\(334\) 23.3464 1.27746
\(335\) −7.88746 −0.430938
\(336\) 4.93900 0.269445
\(337\) 4.36539 0.237798 0.118899 0.992906i \(-0.462064\pi\)
0.118899 + 0.992906i \(0.462064\pi\)
\(338\) 0 0
\(339\) 11.9913 0.651277
\(340\) 9.51784 0.516177
\(341\) 18.0298 0.976370
\(342\) −10.5201 −0.568862
\(343\) 1.00000 0.0539949
\(344\) 9.35939 0.504625
\(345\) 11.2044 0.603225
\(346\) −18.1975 −0.978301
\(347\) 31.8740 1.71109 0.855544 0.517730i \(-0.173223\pi\)
0.855544 + 0.517730i \(0.173223\pi\)
\(348\) −5.59560 −0.299956
\(349\) −6.86699 −0.367582 −0.183791 0.982965i \(-0.558837\pi\)
−0.183791 + 0.982965i \(0.558837\pi\)
\(350\) 8.06211 0.430938
\(351\) 0 0
\(352\) −40.0306 −2.13364
\(353\) 4.18244 0.222609 0.111304 0.993786i \(-0.464497\pi\)
0.111304 + 0.993786i \(0.464497\pi\)
\(354\) −20.8064 −1.10585
\(355\) 36.3719 1.93042
\(356\) −23.3068 −1.23526
\(357\) 2.47976 0.131243
\(358\) −43.7453 −2.31201
\(359\) 36.3813 1.92013 0.960067 0.279771i \(-0.0902586\pi\)
0.960067 + 0.279771i \(0.0902586\pi\)
\(360\) 4.17654 0.220123
\(361\) 15.0847 0.793933
\(362\) 13.2139 0.694508
\(363\) −30.8762 −1.62058
\(364\) 0 0
\(365\) 35.2976 1.84756
\(366\) 13.1303 0.686331
\(367\) 9.37858 0.489558 0.244779 0.969579i \(-0.421285\pi\)
0.244779 + 0.969579i \(0.421285\pi\)
\(368\) −17.9787 −0.937205
\(369\) −1.53218 −0.0797620
\(370\) 6.43597 0.334590
\(371\) −9.65435 −0.501229
\(372\) −3.47430 −0.180134
\(373\) −25.2895 −1.30944 −0.654721 0.755871i \(-0.727214\pi\)
−0.654721 + 0.755871i \(0.727214\pi\)
\(374\) −28.9156 −1.49519
\(375\) −1.61862 −0.0835854
\(376\) 6.99619 0.360801
\(377\) 0 0
\(378\) −1.80194 −0.0926817
\(379\) −12.7741 −0.656163 −0.328081 0.944649i \(-0.606402\pi\)
−0.328081 + 0.944649i \(0.606402\pi\)
\(380\) 22.4083 1.14952
\(381\) −19.4114 −0.994477
\(382\) −0.185682 −0.00950029
\(383\) 5.12177 0.261710 0.130855 0.991402i \(-0.458228\pi\)
0.130855 + 0.991402i \(0.458228\pi\)
\(384\) −10.0858 −0.514686
\(385\) −19.9183 −1.01513
\(386\) −21.2740 −1.08282
\(387\) −6.89765 −0.350627
\(388\) −1.35213 −0.0686442
\(389\) −9.81899 −0.497843 −0.248921 0.968524i \(-0.580076\pi\)
−0.248921 + 0.968524i \(0.580076\pi\)
\(390\) 0 0
\(391\) −9.02670 −0.456500
\(392\) −1.35690 −0.0685336
\(393\) 18.4298 0.929662
\(394\) −41.2052 −2.07589
\(395\) 36.5898 1.84104
\(396\) 8.06943 0.405504
\(397\) −3.04602 −0.152876 −0.0764378 0.997074i \(-0.524355\pi\)
−0.0764378 + 0.997074i \(0.524355\pi\)
\(398\) 1.47975 0.0741731
\(399\) 5.83821 0.292276
\(400\) −22.0977 −1.10489
\(401\) 0.864643 0.0431782 0.0215891 0.999767i \(-0.493127\pi\)
0.0215891 + 0.999767i \(0.493127\pi\)
\(402\) −4.61751 −0.230300
\(403\) 0 0
\(404\) −23.6523 −1.17674
\(405\) −3.07801 −0.152947
\(406\) 8.08588 0.401295
\(407\) −7.50911 −0.372213
\(408\) −3.36477 −0.166581
\(409\) −3.61305 −0.178654 −0.0893269 0.996002i \(-0.528472\pi\)
−0.0893269 + 0.996002i \(0.528472\pi\)
\(410\) 8.49804 0.419688
\(411\) −8.25681 −0.407279
\(412\) 0.259242 0.0127719
\(413\) 11.5467 0.568175
\(414\) 6.55933 0.322373
\(415\) −23.3512 −1.14627
\(416\) 0 0
\(417\) −12.8927 −0.631360
\(418\) −68.0774 −3.32978
\(419\) −24.8403 −1.21353 −0.606763 0.794883i \(-0.707532\pi\)
−0.606763 + 0.794883i \(0.707532\pi\)
\(420\) 3.83821 0.187286
\(421\) 21.4647 1.04612 0.523062 0.852295i \(-0.324790\pi\)
0.523062 + 0.852295i \(0.324790\pi\)
\(422\) −44.7300 −2.17742
\(423\) −5.15603 −0.250695
\(424\) 13.0999 0.636190
\(425\) −11.0948 −0.538175
\(426\) 21.2929 1.03165
\(427\) −7.28676 −0.352631
\(428\) −8.63109 −0.417200
\(429\) 0 0
\(430\) 38.2570 1.84491
\(431\) 15.8820 0.765008 0.382504 0.923954i \(-0.375062\pi\)
0.382504 + 0.923954i \(0.375062\pi\)
\(432\) 4.93900 0.237628
\(433\) −3.11055 −0.149483 −0.0747417 0.997203i \(-0.523813\pi\)
−0.0747417 + 0.997203i \(0.523813\pi\)
\(434\) 5.02051 0.240992
\(435\) 13.8120 0.662235
\(436\) −12.2172 −0.585097
\(437\) −21.2520 −1.01662
\(438\) 20.6640 0.987366
\(439\) −6.37921 −0.304463 −0.152232 0.988345i \(-0.548646\pi\)
−0.152232 + 0.988345i \(0.548646\pi\)
\(440\) 27.0271 1.28847
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 16.9667 0.806112 0.403056 0.915175i \(-0.367948\pi\)
0.403056 + 0.915175i \(0.367948\pi\)
\(444\) 1.44699 0.0686709
\(445\) 57.5299 2.72718
\(446\) 38.2115 1.80937
\(447\) 0.731647 0.0346057
\(448\) −1.26875 −0.0599428
\(449\) −4.45098 −0.210055 −0.105027 0.994469i \(-0.533493\pi\)
−0.105027 + 0.994469i \(0.533493\pi\)
\(450\) 8.06211 0.380051
\(451\) −9.91500 −0.466879
\(452\) −14.9529 −0.703325
\(453\) −1.03330 −0.0485486
\(454\) 34.3784 1.61346
\(455\) 0 0
\(456\) −7.92185 −0.370974
\(457\) 5.95579 0.278600 0.139300 0.990250i \(-0.455515\pi\)
0.139300 + 0.990250i \(0.455515\pi\)
\(458\) 24.3725 1.13885
\(459\) 2.47976 0.115745
\(460\) −13.9717 −0.651433
\(461\) −2.58879 −0.120572 −0.0602860 0.998181i \(-0.519201\pi\)
−0.0602860 + 0.998181i \(0.519201\pi\)
\(462\) −11.6607 −0.542503
\(463\) 36.0018 1.67315 0.836573 0.547855i \(-0.184555\pi\)
0.836573 + 0.547855i \(0.184555\pi\)
\(464\) −22.1629 −1.02889
\(465\) 8.57586 0.397696
\(466\) −30.8695 −1.43000
\(467\) 35.6189 1.64825 0.824123 0.566411i \(-0.191669\pi\)
0.824123 + 0.566411i \(0.191669\pi\)
\(468\) 0 0
\(469\) 2.56252 0.118326
\(470\) 28.5973 1.31909
\(471\) −6.14425 −0.283112
\(472\) −15.6676 −0.721162
\(473\) −44.6359 −2.05236
\(474\) 21.4206 0.983879
\(475\) −26.1209 −1.19851
\(476\) −3.09221 −0.141731
\(477\) −9.65435 −0.442042
\(478\) −32.1516 −1.47058
\(479\) 35.6402 1.62844 0.814221 0.580556i \(-0.197165\pi\)
0.814221 + 0.580556i \(0.197165\pi\)
\(480\) −19.0405 −0.869076
\(481\) 0 0
\(482\) 30.7571 1.40095
\(483\) −3.64015 −0.165633
\(484\) 38.5019 1.75009
\(485\) 3.33757 0.151551
\(486\) −1.80194 −0.0817376
\(487\) 10.6754 0.483750 0.241875 0.970307i \(-0.422238\pi\)
0.241875 + 0.970307i \(0.422238\pi\)
\(488\) 9.88737 0.447580
\(489\) −7.70203 −0.348298
\(490\) −5.54638 −0.250560
\(491\) −36.8647 −1.66368 −0.831840 0.555016i \(-0.812712\pi\)
−0.831840 + 0.555016i \(0.812712\pi\)
\(492\) 1.91059 0.0861363
\(493\) −11.1275 −0.501156
\(494\) 0 0
\(495\) −19.9183 −0.895263
\(496\) −13.7609 −0.617883
\(497\) −11.8167 −0.530051
\(498\) −13.6704 −0.612583
\(499\) −3.60337 −0.161309 −0.0806545 0.996742i \(-0.525701\pi\)
−0.0806545 + 0.996742i \(0.525701\pi\)
\(500\) 2.01839 0.0902652
\(501\) −12.9563 −0.578843
\(502\) −39.7750 −1.77525
\(503\) −34.9369 −1.55776 −0.778879 0.627174i \(-0.784212\pi\)
−0.778879 + 0.627174i \(0.784212\pi\)
\(504\) −1.35690 −0.0604409
\(505\) 58.3826 2.59799
\(506\) 42.4466 1.88698
\(507\) 0 0
\(508\) 24.2057 1.07395
\(509\) 12.8076 0.567687 0.283844 0.958871i \(-0.408390\pi\)
0.283844 + 0.958871i \(0.408390\pi\)
\(510\) −13.7537 −0.609023
\(511\) −11.4677 −0.507300
\(512\) 17.1491 0.757892
\(513\) 5.83821 0.257763
\(514\) −19.0257 −0.839187
\(515\) −0.639905 −0.0281976
\(516\) 8.60123 0.378648
\(517\) −33.3656 −1.46742
\(518\) −2.09095 −0.0918713
\(519\) 10.0988 0.443289
\(520\) 0 0
\(521\) 8.90891 0.390307 0.195153 0.980773i \(-0.437480\pi\)
0.195153 + 0.980773i \(0.437480\pi\)
\(522\) 8.08588 0.353909
\(523\) −4.62938 −0.202429 −0.101214 0.994865i \(-0.532273\pi\)
−0.101214 + 0.994865i \(0.532273\pi\)
\(524\) −22.9816 −1.00396
\(525\) −4.47413 −0.195267
\(526\) 44.3645 1.93438
\(527\) −6.90904 −0.300962
\(528\) 31.9612 1.39093
\(529\) −9.74930 −0.423883
\(530\) 53.5467 2.32592
\(531\) 11.5467 0.501083
\(532\) −7.28013 −0.315634
\(533\) 0 0
\(534\) 33.6793 1.45745
\(535\) 21.3047 0.921084
\(536\) −3.47708 −0.150187
\(537\) 24.2768 1.04762
\(538\) −22.5652 −0.972855
\(539\) 6.47118 0.278733
\(540\) 3.83821 0.165170
\(541\) 6.69318 0.287762 0.143881 0.989595i \(-0.454042\pi\)
0.143881 + 0.989595i \(0.454042\pi\)
\(542\) 22.2549 0.955932
\(543\) −7.33317 −0.314696
\(544\) 15.3397 0.657686
\(545\) 30.1565 1.29176
\(546\) 0 0
\(547\) −11.3091 −0.483541 −0.241771 0.970333i \(-0.577728\pi\)
−0.241771 + 0.970333i \(0.577728\pi\)
\(548\) 10.2961 0.439827
\(549\) −7.28676 −0.310991
\(550\) 52.1713 2.22459
\(551\) −26.1979 −1.11607
\(552\) 4.93931 0.210231
\(553\) −11.8875 −0.505508
\(554\) 29.3965 1.24894
\(555\) −3.57170 −0.151610
\(556\) 16.0770 0.681816
\(557\) 6.71339 0.284456 0.142228 0.989834i \(-0.454573\pi\)
0.142228 + 0.989834i \(0.454573\pi\)
\(558\) 5.02051 0.212535
\(559\) 0 0
\(560\) 15.2023 0.642414
\(561\) 16.0470 0.677503
\(562\) −18.9487 −0.799301
\(563\) −12.5571 −0.529218 −0.264609 0.964356i \(-0.585243\pi\)
−0.264609 + 0.964356i \(0.585243\pi\)
\(564\) 6.42946 0.270729
\(565\) 36.9093 1.55278
\(566\) 45.9084 1.92968
\(567\) 1.00000 0.0419961
\(568\) 16.0340 0.672772
\(569\) 22.9022 0.960109 0.480055 0.877239i \(-0.340617\pi\)
0.480055 + 0.877239i \(0.340617\pi\)
\(570\) −32.3809 −1.35629
\(571\) −22.8646 −0.956854 −0.478427 0.878127i \(-0.658793\pi\)
−0.478427 + 0.878127i \(0.658793\pi\)
\(572\) 0 0
\(573\) 0.103045 0.00430479
\(574\) −2.76089 −0.115237
\(575\) 16.2865 0.679195
\(576\) −1.26875 −0.0528646
\(577\) 14.3282 0.596492 0.298246 0.954489i \(-0.403598\pi\)
0.298246 + 0.954489i \(0.403598\pi\)
\(578\) −19.5525 −0.813275
\(579\) 11.8062 0.490648
\(580\) −17.2233 −0.715159
\(581\) 7.58647 0.314740
\(582\) 1.95389 0.0809914
\(583\) −62.4750 −2.58745
\(584\) 15.5604 0.643896
\(585\) 0 0
\(586\) −8.53074 −0.352402
\(587\) −19.9113 −0.821826 −0.410913 0.911674i \(-0.634790\pi\)
−0.410913 + 0.911674i \(0.634790\pi\)
\(588\) −1.24698 −0.0514246
\(589\) −16.2663 −0.670240
\(590\) −64.0423 −2.63658
\(591\) 22.8672 0.940629
\(592\) 5.73118 0.235550
\(593\) −8.92435 −0.366479 −0.183239 0.983068i \(-0.558658\pi\)
−0.183239 + 0.983068i \(0.558658\pi\)
\(594\) −11.6607 −0.478443
\(595\) 7.63272 0.312911
\(596\) −0.912349 −0.0373713
\(597\) −0.821199 −0.0336094
\(598\) 0 0
\(599\) 9.43773 0.385615 0.192808 0.981237i \(-0.438241\pi\)
0.192808 + 0.981237i \(0.438241\pi\)
\(600\) 6.07093 0.247845
\(601\) −9.25969 −0.377711 −0.188855 0.982005i \(-0.560478\pi\)
−0.188855 + 0.982005i \(0.560478\pi\)
\(602\) −12.4291 −0.506574
\(603\) 2.56252 0.104354
\(604\) 1.28850 0.0524284
\(605\) −95.0371 −3.86381
\(606\) 34.1785 1.38841
\(607\) 1.08268 0.0439447 0.0219724 0.999759i \(-0.493005\pi\)
0.0219724 + 0.999759i \(0.493005\pi\)
\(608\) 36.1151 1.46466
\(609\) −4.48732 −0.181836
\(610\) 40.4151 1.63636
\(611\) 0 0
\(612\) −3.09221 −0.124995
\(613\) −35.5525 −1.43595 −0.717976 0.696067i \(-0.754931\pi\)
−0.717976 + 0.696067i \(0.754931\pi\)
\(614\) −9.25972 −0.373692
\(615\) −4.71606 −0.190170
\(616\) −8.78072 −0.353785
\(617\) −9.15107 −0.368408 −0.184204 0.982888i \(-0.558971\pi\)
−0.184204 + 0.982888i \(0.558971\pi\)
\(618\) −0.374616 −0.0150692
\(619\) −19.8832 −0.799175 −0.399587 0.916695i \(-0.630847\pi\)
−0.399587 + 0.916695i \(0.630847\pi\)
\(620\) −10.6939 −0.429478
\(621\) −3.64015 −0.146074
\(622\) −25.9991 −1.04247
\(623\) −18.6906 −0.748824
\(624\) 0 0
\(625\) −27.3528 −1.09411
\(626\) −6.80684 −0.272056
\(627\) 37.7801 1.50879
\(628\) 7.66175 0.305737
\(629\) 2.87749 0.114733
\(630\) −5.54638 −0.220973
\(631\) −5.93855 −0.236410 −0.118205 0.992989i \(-0.537714\pi\)
−0.118205 + 0.992989i \(0.537714\pi\)
\(632\) 16.1301 0.641621
\(633\) 24.8233 0.986638
\(634\) 31.7177 1.25967
\(635\) −59.7485 −2.37105
\(636\) 12.0388 0.477369
\(637\) 0 0
\(638\) 52.3252 2.07157
\(639\) −11.8167 −0.467461
\(640\) −31.0440 −1.22712
\(641\) 40.5393 1.60121 0.800604 0.599194i \(-0.204512\pi\)
0.800604 + 0.599194i \(0.204512\pi\)
\(642\) 12.4723 0.492242
\(643\) −48.4977 −1.91256 −0.956282 0.292446i \(-0.905531\pi\)
−0.956282 + 0.292446i \(0.905531\pi\)
\(644\) 4.53919 0.178869
\(645\) −21.2310 −0.835970
\(646\) 26.0873 1.02639
\(647\) −22.0190 −0.865656 −0.432828 0.901476i \(-0.642484\pi\)
−0.432828 + 0.901476i \(0.642484\pi\)
\(648\) −1.35690 −0.0533039
\(649\) 74.7207 2.93304
\(650\) 0 0
\(651\) −2.78617 −0.109199
\(652\) 9.60427 0.376132
\(653\) −3.53681 −0.138406 −0.0692031 0.997603i \(-0.522046\pi\)
−0.0692031 + 0.997603i \(0.522046\pi\)
\(654\) 17.6543 0.690339
\(655\) 56.7272 2.21651
\(656\) 7.56743 0.295458
\(657\) −11.4677 −0.447397
\(658\) −9.29084 −0.362195
\(659\) −31.8032 −1.23888 −0.619438 0.785045i \(-0.712640\pi\)
−0.619438 + 0.785045i \(0.712640\pi\)
\(660\) 24.8378 0.966809
\(661\) −28.1365 −1.09438 −0.547191 0.837008i \(-0.684303\pi\)
−0.547191 + 0.837008i \(0.684303\pi\)
\(662\) 16.1068 0.626010
\(663\) 0 0
\(664\) −10.2941 −0.399487
\(665\) 17.9701 0.696849
\(666\) −2.09095 −0.0810228
\(667\) 16.3345 0.632476
\(668\) 16.1562 0.625102
\(669\) −21.2058 −0.819863
\(670\) −14.2127 −0.549085
\(671\) −47.1539 −1.82036
\(672\) 6.18598 0.238629
\(673\) 15.9373 0.614338 0.307169 0.951655i \(-0.400618\pi\)
0.307169 + 0.951655i \(0.400618\pi\)
\(674\) 7.86615 0.302993
\(675\) −4.47413 −0.172209
\(676\) 0 0
\(677\) −38.8890 −1.49463 −0.747314 0.664471i \(-0.768657\pi\)
−0.747314 + 0.664471i \(0.768657\pi\)
\(678\) 21.6075 0.829833
\(679\) −1.08433 −0.0416127
\(680\) −10.3568 −0.397165
\(681\) −19.0786 −0.731092
\(682\) 32.4886 1.24405
\(683\) 37.7568 1.44472 0.722362 0.691515i \(-0.243056\pi\)
0.722362 + 0.691515i \(0.243056\pi\)
\(684\) −7.28013 −0.278363
\(685\) −25.4145 −0.971040
\(686\) 1.80194 0.0687983
\(687\) −13.5257 −0.516039
\(688\) 34.0675 1.29881
\(689\) 0 0
\(690\) 20.1897 0.768607
\(691\) 23.0633 0.877370 0.438685 0.898641i \(-0.355444\pi\)
0.438685 + 0.898641i \(0.355444\pi\)
\(692\) −12.5930 −0.478715
\(693\) 6.47118 0.245820
\(694\) 57.4350 2.18020
\(695\) −39.6840 −1.50530
\(696\) 6.08883 0.230797
\(697\) 3.79943 0.143914
\(698\) −12.3739 −0.468359
\(699\) 17.1313 0.647965
\(700\) 5.57915 0.210872
\(701\) −42.7123 −1.61322 −0.806610 0.591084i \(-0.798700\pi\)
−0.806610 + 0.591084i \(0.798700\pi\)
\(702\) 0 0
\(703\) 6.77462 0.255509
\(704\) −8.21031 −0.309438
\(705\) −15.8703 −0.597710
\(706\) 7.53650 0.283640
\(707\) −18.9676 −0.713352
\(708\) −14.3985 −0.541128
\(709\) 5.88615 0.221059 0.110529 0.993873i \(-0.464745\pi\)
0.110529 + 0.993873i \(0.464745\pi\)
\(710\) 65.5398 2.45967
\(711\) −11.8875 −0.445816
\(712\) 25.3612 0.950452
\(713\) 10.1421 0.379824
\(714\) 4.46837 0.167225
\(715\) 0 0
\(716\) −30.2727 −1.13134
\(717\) 17.8428 0.666352
\(718\) 65.5569 2.44656
\(719\) −22.3734 −0.834387 −0.417194 0.908818i \(-0.636986\pi\)
−0.417194 + 0.908818i \(0.636986\pi\)
\(720\) 15.2023 0.566556
\(721\) 0.207896 0.00774245
\(722\) 27.1817 1.01160
\(723\) −17.0689 −0.634799
\(724\) 9.14431 0.339846
\(725\) 20.0769 0.745637
\(726\) −55.6369 −2.06488
\(727\) −27.1276 −1.00611 −0.503053 0.864256i \(-0.667790\pi\)
−0.503053 + 0.864256i \(0.667790\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 63.6041 2.35409
\(731\) 17.1045 0.632633
\(732\) 9.08644 0.335844
\(733\) −35.5147 −1.31177 −0.655884 0.754862i \(-0.727704\pi\)
−0.655884 + 0.754862i \(0.727704\pi\)
\(734\) 16.8996 0.623776
\(735\) 3.07801 0.113534
\(736\) −22.5179 −0.830021
\(737\) 16.5825 0.610826
\(738\) −2.76089 −0.101630
\(739\) −35.1297 −1.29227 −0.646134 0.763224i \(-0.723615\pi\)
−0.646134 + 0.763224i \(0.723615\pi\)
\(740\) 4.45383 0.163726
\(741\) 0 0
\(742\) −17.3965 −0.638647
\(743\) 9.33606 0.342507 0.171253 0.985227i \(-0.445218\pi\)
0.171253 + 0.985227i \(0.445218\pi\)
\(744\) 3.78055 0.138602
\(745\) 2.25202 0.0825074
\(746\) −45.5701 −1.66844
\(747\) 7.58647 0.277574
\(748\) −20.0102 −0.731647
\(749\) −6.92160 −0.252910
\(750\) −2.91666 −0.106501
\(751\) −14.9393 −0.545144 −0.272572 0.962135i \(-0.587874\pi\)
−0.272572 + 0.962135i \(0.587874\pi\)
\(752\) 25.4656 0.928635
\(753\) 22.0735 0.804402
\(754\) 0 0
\(755\) −3.18050 −0.115750
\(756\) −1.24698 −0.0453522
\(757\) 30.8483 1.12120 0.560601 0.828086i \(-0.310570\pi\)
0.560601 + 0.828086i \(0.310570\pi\)
\(758\) −23.0182 −0.836058
\(759\) −23.5561 −0.855031
\(760\) −24.3835 −0.884483
\(761\) 0.975789 0.0353723 0.0176862 0.999844i \(-0.494370\pi\)
0.0176862 + 0.999844i \(0.494370\pi\)
\(762\) −34.9782 −1.26713
\(763\) −9.79742 −0.354690
\(764\) −0.128496 −0.00464881
\(765\) 7.63272 0.275961
\(766\) 9.22911 0.333461
\(767\) 0 0
\(768\) −20.7114 −0.747358
\(769\) 51.9223 1.87237 0.936184 0.351511i \(-0.114332\pi\)
0.936184 + 0.351511i \(0.114332\pi\)
\(770\) −35.8916 −1.29344
\(771\) 10.5585 0.380254
\(772\) −14.7221 −0.529859
\(773\) −9.10468 −0.327473 −0.163736 0.986504i \(-0.552355\pi\)
−0.163736 + 0.986504i \(0.552355\pi\)
\(774\) −12.4291 −0.446756
\(775\) 12.4657 0.447781
\(776\) 1.47132 0.0528173
\(777\) 1.16039 0.0416288
\(778\) −17.6932 −0.634332
\(779\) 8.94518 0.320494
\(780\) 0 0
\(781\) −76.4679 −2.73624
\(782\) −16.2655 −0.581655
\(783\) −4.48732 −0.160364
\(784\) −4.93900 −0.176393
\(785\) −18.9120 −0.675000
\(786\) 33.2094 1.18454
\(787\) 41.4199 1.47646 0.738230 0.674550i \(-0.235662\pi\)
0.738230 + 0.674550i \(0.235662\pi\)
\(788\) −28.5149 −1.01580
\(789\) −24.6204 −0.876511
\(790\) 65.9326 2.34578
\(791\) −11.9913 −0.426361
\(792\) −8.78072 −0.312009
\(793\) 0 0
\(794\) −5.48875 −0.194788
\(795\) −29.7162 −1.05392
\(796\) 1.02402 0.0362954
\(797\) 16.8411 0.596542 0.298271 0.954481i \(-0.403590\pi\)
0.298271 + 0.954481i \(0.403590\pi\)
\(798\) 10.5201 0.372407
\(799\) 12.7857 0.452325
\(800\) −27.6769 −0.978526
\(801\) −18.6906 −0.660400
\(802\) 1.55803 0.0550160
\(803\) −74.2094 −2.61879
\(804\) −3.19541 −0.112694
\(805\) −11.2044 −0.394904
\(806\) 0 0
\(807\) 12.5227 0.440821
\(808\) 25.7371 0.905429
\(809\) −21.4075 −0.752646 −0.376323 0.926488i \(-0.622812\pi\)
−0.376323 + 0.926488i \(0.622812\pi\)
\(810\) −5.54638 −0.194880
\(811\) −27.5993 −0.969141 −0.484571 0.874752i \(-0.661024\pi\)
−0.484571 + 0.874752i \(0.661024\pi\)
\(812\) 5.59560 0.196367
\(813\) −12.3506 −0.433153
\(814\) −13.5309 −0.474259
\(815\) −23.7069 −0.830416
\(816\) −12.2475 −0.428749
\(817\) 40.2699 1.40887
\(818\) −6.51049 −0.227634
\(819\) 0 0
\(820\) 5.88083 0.205367
\(821\) −13.3942 −0.467461 −0.233731 0.972301i \(-0.575093\pi\)
−0.233731 + 0.972301i \(0.575093\pi\)
\(822\) −14.8783 −0.518939
\(823\) −21.4686 −0.748349 −0.374175 0.927358i \(-0.622074\pi\)
−0.374175 + 0.927358i \(0.622074\pi\)
\(824\) −0.282093 −0.00982718
\(825\) −28.9529 −1.00801
\(826\) 20.8064 0.723947
\(827\) 21.3923 0.743882 0.371941 0.928256i \(-0.378692\pi\)
0.371941 + 0.928256i \(0.378692\pi\)
\(828\) 4.53919 0.157748
\(829\) 16.9639 0.589181 0.294591 0.955624i \(-0.404817\pi\)
0.294591 + 0.955624i \(0.404817\pi\)
\(830\) −42.0774 −1.46053
\(831\) −16.3139 −0.565921
\(832\) 0 0
\(833\) −2.47976 −0.0859185
\(834\) −23.2319 −0.804456
\(835\) −39.8795 −1.38009
\(836\) −47.1110 −1.62937
\(837\) −2.78617 −0.0963042
\(838\) −44.7606 −1.54623
\(839\) 28.1419 0.971565 0.485782 0.874080i \(-0.338535\pi\)
0.485782 + 0.874080i \(0.338535\pi\)
\(840\) −4.17654 −0.144104
\(841\) −8.86393 −0.305653
\(842\) 38.6780 1.33293
\(843\) 10.5157 0.362180
\(844\) −30.9542 −1.06549
\(845\) 0 0
\(846\) −9.29084 −0.319426
\(847\) 30.8762 1.06092
\(848\) 47.6829 1.63744
\(849\) −25.4773 −0.874377
\(850\) −19.9921 −0.685723
\(851\) −4.22400 −0.144797
\(852\) 14.7352 0.504819
\(853\) −14.3944 −0.492856 −0.246428 0.969161i \(-0.579257\pi\)
−0.246428 + 0.969161i \(0.579257\pi\)
\(854\) −13.1303 −0.449309
\(855\) 17.9701 0.614563
\(856\) 9.39189 0.321008
\(857\) 31.7777 1.08550 0.542752 0.839893i \(-0.317382\pi\)
0.542752 + 0.839893i \(0.317382\pi\)
\(858\) 0 0
\(859\) 14.0645 0.479876 0.239938 0.970788i \(-0.422873\pi\)
0.239938 + 0.970788i \(0.422873\pi\)
\(860\) 26.4746 0.902778
\(861\) 1.53218 0.0522165
\(862\) 28.6183 0.974745
\(863\) −41.6977 −1.41941 −0.709704 0.704500i \(-0.751171\pi\)
−0.709704 + 0.704500i \(0.751171\pi\)
\(864\) 6.18598 0.210451
\(865\) 31.0843 1.05690
\(866\) −5.60502 −0.190466
\(867\) 10.8508 0.368512
\(868\) 3.47430 0.117925
\(869\) −76.9262 −2.60954
\(870\) 24.8884 0.843796
\(871\) 0 0
\(872\) 13.2941 0.450194
\(873\) −1.08433 −0.0366989
\(874\) −38.2947 −1.29534
\(875\) 1.61862 0.0547195
\(876\) 14.3000 0.483151
\(877\) 33.9320 1.14580 0.572901 0.819625i \(-0.305818\pi\)
0.572901 + 0.819625i \(0.305818\pi\)
\(878\) −11.4949 −0.387935
\(879\) 4.73420 0.159681
\(880\) 98.3767 3.31628
\(881\) −33.4358 −1.12648 −0.563240 0.826294i \(-0.690445\pi\)
−0.563240 + 0.826294i \(0.690445\pi\)
\(882\) 1.80194 0.0606744
\(883\) −14.8739 −0.500545 −0.250273 0.968175i \(-0.580520\pi\)
−0.250273 + 0.968175i \(0.580520\pi\)
\(884\) 0 0
\(885\) 35.5408 1.19469
\(886\) 30.5729 1.02712
\(887\) −1.91185 −0.0641937 −0.0320968 0.999485i \(-0.510218\pi\)
−0.0320968 + 0.999485i \(0.510218\pi\)
\(888\) −1.57453 −0.0528378
\(889\) 19.4114 0.651038
\(890\) 103.665 3.47487
\(891\) 6.47118 0.216793
\(892\) 26.4432 0.885383
\(893\) 30.1020 1.00732
\(894\) 1.31838 0.0440933
\(895\) 74.7242 2.49775
\(896\) 10.0858 0.336941
\(897\) 0 0
\(898\) −8.02040 −0.267644
\(899\) 12.5025 0.416980
\(900\) 5.57915 0.185972
\(901\) 23.9405 0.797572
\(902\) −17.8662 −0.594880
\(903\) 6.89765 0.229539
\(904\) 16.2709 0.541163
\(905\) −22.5716 −0.750304
\(906\) −1.86194 −0.0618588
\(907\) 26.8212 0.890583 0.445292 0.895386i \(-0.353100\pi\)
0.445292 + 0.895386i \(0.353100\pi\)
\(908\) 23.7906 0.789518
\(909\) −18.9676 −0.629117
\(910\) 0 0
\(911\) −5.59749 −0.185453 −0.0927265 0.995692i \(-0.529558\pi\)
−0.0927265 + 0.995692i \(0.529558\pi\)
\(912\) −28.8349 −0.954820
\(913\) 49.0934 1.62476
\(914\) 10.7320 0.354982
\(915\) −22.4287 −0.741470
\(916\) 16.8663 0.557278
\(917\) −18.4298 −0.608607
\(918\) 4.46837 0.147478
\(919\) 45.7248 1.50832 0.754160 0.656691i \(-0.228044\pi\)
0.754160 + 0.656691i \(0.228044\pi\)
\(920\) 15.2032 0.501235
\(921\) 5.13876 0.169328
\(922\) −4.66484 −0.153628
\(923\) 0 0
\(924\) −8.06943 −0.265465
\(925\) −5.19175 −0.170704
\(926\) 64.8730 2.13186
\(927\) 0.207896 0.00682820
\(928\) −27.7585 −0.911217
\(929\) −43.0428 −1.41219 −0.706094 0.708118i \(-0.749544\pi\)
−0.706094 + 0.708118i \(0.749544\pi\)
\(930\) 15.4532 0.506729
\(931\) −5.83821 −0.191340
\(932\) −21.3624 −0.699748
\(933\) 14.4284 0.472365
\(934\) 64.1830 2.10013
\(935\) 49.3927 1.61531
\(936\) 0 0
\(937\) 34.5984 1.13028 0.565141 0.824994i \(-0.308822\pi\)
0.565141 + 0.824994i \(0.308822\pi\)
\(938\) 4.61751 0.150767
\(939\) 3.77751 0.123274
\(940\) 19.7899 0.645476
\(941\) −0.138429 −0.00451266 −0.00225633 0.999997i \(-0.500718\pi\)
−0.00225633 + 0.999997i \(0.500718\pi\)
\(942\) −11.0716 −0.360731
\(943\) −5.57736 −0.181624
\(944\) −57.0291 −1.85614
\(945\) 3.07801 0.100128
\(946\) −80.4311 −2.61504
\(947\) 2.66158 0.0864896 0.0432448 0.999065i \(-0.486230\pi\)
0.0432448 + 0.999065i \(0.486230\pi\)
\(948\) 14.8235 0.481444
\(949\) 0 0
\(950\) −47.0683 −1.52710
\(951\) −17.6020 −0.570784
\(952\) 3.36477 0.109053
\(953\) 16.1600 0.523474 0.261737 0.965139i \(-0.415705\pi\)
0.261737 + 0.965139i \(0.415705\pi\)
\(954\) −17.3965 −0.563234
\(955\) 0.317175 0.0102635
\(956\) −22.2496 −0.719604
\(957\) −29.0383 −0.938674
\(958\) 64.2214 2.07490
\(959\) 8.25681 0.266626
\(960\) −3.90522 −0.126041
\(961\) −23.2372 −0.749588
\(962\) 0 0
\(963\) −6.92160 −0.223045
\(964\) 21.2846 0.685530
\(965\) 36.3395 1.16981
\(966\) −6.55933 −0.211043
\(967\) −25.1632 −0.809194 −0.404597 0.914495i \(-0.632588\pi\)
−0.404597 + 0.914495i \(0.632588\pi\)
\(968\) −41.8957 −1.34658
\(969\) −14.4774 −0.465080
\(970\) 6.01409 0.193101
\(971\) −3.58992 −0.115206 −0.0576029 0.998340i \(-0.518346\pi\)
−0.0576029 + 0.998340i \(0.518346\pi\)
\(972\) −1.24698 −0.0399969
\(973\) 12.8927 0.413322
\(974\) 19.2365 0.616376
\(975\) 0 0
\(976\) 35.9893 1.15199
\(977\) 26.3689 0.843615 0.421807 0.906685i \(-0.361396\pi\)
0.421807 + 0.906685i \(0.361396\pi\)
\(978\) −13.8786 −0.443788
\(979\) −120.950 −3.86559
\(980\) −3.83821 −0.122607
\(981\) −9.79742 −0.312808
\(982\) −66.4279 −2.11980
\(983\) −17.9645 −0.572978 −0.286489 0.958084i \(-0.592488\pi\)
−0.286489 + 0.958084i \(0.592488\pi\)
\(984\) −2.07901 −0.0662763
\(985\) 70.3853 2.24266
\(986\) −20.0510 −0.638555
\(987\) 5.15603 0.164118
\(988\) 0 0
\(989\) −25.1085 −0.798403
\(990\) −35.8916 −1.14071
\(991\) 10.3327 0.328230 0.164115 0.986441i \(-0.447523\pi\)
0.164115 + 0.986441i \(0.447523\pi\)
\(992\) −17.2352 −0.547219
\(993\) −8.93862 −0.283659
\(994\) −21.2929 −0.675371
\(995\) −2.52766 −0.0801321
\(996\) −9.46018 −0.299757
\(997\) 12.8061 0.405574 0.202787 0.979223i \(-0.435000\pi\)
0.202787 + 0.979223i \(0.435000\pi\)
\(998\) −6.49305 −0.205534
\(999\) 1.16039 0.0367132
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 3549.2.a.z.1.5 yes 6
13.12 even 2 3549.2.a.y.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3549.2.a.y.1.2 6 13.12 even 2
3549.2.a.z.1.5 yes 6 1.1 even 1 trivial