Properties

Label 2013.2.a.a.1.5
Level $2013$
Weight $2$
Character 2013.1
Self dual yes
Analytic conductor $16.074$
Analytic rank $1$
Dimension $11$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2013,2,Mod(1,2013)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2013, 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("2013.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2013 = 3 \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2013.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: \(16.0738859269\)
Analytic rank: \(1\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 4x^{10} - 6x^{9} + 37x^{8} - 2x^{7} - 109x^{6} + 55x^{5} + 115x^{4} - 76x^{3} - 29x^{2} + 14x + 3 \) 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 \(1.14470\) of defining polynomial
Character \(\chi\) \(=\) 2013.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.14470 q^{2} +1.00000 q^{3} -0.689651 q^{4} +1.33318 q^{5} -1.14470 q^{6} -3.52170 q^{7} +3.07886 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.14470 q^{2} +1.00000 q^{3} -0.689651 q^{4} +1.33318 q^{5} -1.14470 q^{6} -3.52170 q^{7} +3.07886 q^{8} +1.00000 q^{9} -1.52610 q^{10} -1.00000 q^{11} -0.689651 q^{12} -5.11237 q^{13} +4.03130 q^{14} +1.33318 q^{15} -2.14508 q^{16} +1.76849 q^{17} -1.14470 q^{18} +7.94593 q^{19} -0.919427 q^{20} -3.52170 q^{21} +1.14470 q^{22} +2.83929 q^{23} +3.07886 q^{24} -3.22263 q^{25} +5.85216 q^{26} +1.00000 q^{27} +2.42874 q^{28} +9.92659 q^{29} -1.52610 q^{30} -4.21272 q^{31} -3.70223 q^{32} -1.00000 q^{33} -2.02440 q^{34} -4.69505 q^{35} -0.689651 q^{36} -9.46041 q^{37} -9.09575 q^{38} -5.11237 q^{39} +4.10467 q^{40} -4.92390 q^{41} +4.03130 q^{42} -5.08527 q^{43} +0.689651 q^{44} +1.33318 q^{45} -3.25015 q^{46} -6.83907 q^{47} -2.14508 q^{48} +5.40234 q^{49} +3.68897 q^{50} +1.76849 q^{51} +3.52575 q^{52} +2.41931 q^{53} -1.14470 q^{54} -1.33318 q^{55} -10.8428 q^{56} +7.94593 q^{57} -11.3630 q^{58} -5.09379 q^{59} -0.919427 q^{60} +1.00000 q^{61} +4.82233 q^{62} -3.52170 q^{63} +8.52812 q^{64} -6.81570 q^{65} +1.14470 q^{66} -14.9335 q^{67} -1.21964 q^{68} +2.83929 q^{69} +5.37445 q^{70} -1.73224 q^{71} +3.07886 q^{72} -0.272088 q^{73} +10.8294 q^{74} -3.22263 q^{75} -5.47992 q^{76} +3.52170 q^{77} +5.85216 q^{78} +14.9608 q^{79} -2.85978 q^{80} +1.00000 q^{81} +5.63642 q^{82} -12.0244 q^{83} +2.42874 q^{84} +2.35772 q^{85} +5.82113 q^{86} +9.92659 q^{87} -3.07886 q^{88} -13.3848 q^{89} -1.52610 q^{90} +18.0042 q^{91} -1.95812 q^{92} -4.21272 q^{93} +7.82871 q^{94} +10.5933 q^{95} -3.70223 q^{96} -1.16994 q^{97} -6.18408 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q - 4 q^{2} + 11 q^{3} + 6 q^{4} - 13 q^{5} - 4 q^{6} - 5 q^{7} - 9 q^{8} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 11 q - 4 q^{2} + 11 q^{3} + 6 q^{4} - 13 q^{5} - 4 q^{6} - 5 q^{7} - 9 q^{8} + 11 q^{9} + 6 q^{10} - 11 q^{11} + 6 q^{12} - 3 q^{13} - 9 q^{14} - 13 q^{15} + 4 q^{16} - 7 q^{17} - 4 q^{18} - 8 q^{19} - 25 q^{20} - 5 q^{21} + 4 q^{22} - 15 q^{23} - 9 q^{24} + 4 q^{25} - 2 q^{26} + 11 q^{27} + 13 q^{28} - 8 q^{29} + 6 q^{30} - 17 q^{31} - 27 q^{32} - 11 q^{33} - 18 q^{34} - 2 q^{35} + 6 q^{36} - 10 q^{37} - 30 q^{38} - 3 q^{39} + 10 q^{40} - 25 q^{41} - 9 q^{42} - 7 q^{43} - 6 q^{44} - 13 q^{45} + 32 q^{46} - 30 q^{47} + 4 q^{48} - 2 q^{49} + 11 q^{50} - 7 q^{51} - 7 q^{52} - 18 q^{53} - 4 q^{54} + 13 q^{55} - 20 q^{56} - 8 q^{57} - 13 q^{58} - 43 q^{59} - 25 q^{60} + 11 q^{61} + 7 q^{62} - 5 q^{63} + 25 q^{64} - 27 q^{65} + 4 q^{66} - 30 q^{67} + 10 q^{68} - 15 q^{69} - 4 q^{70} - 7 q^{71} - 9 q^{72} + 6 q^{73} - 44 q^{74} + 4 q^{75} - 19 q^{76} + 5 q^{77} - 2 q^{78} + 17 q^{79} - 22 q^{80} + 11 q^{81} + 8 q^{82} - 34 q^{83} + 13 q^{84} + 10 q^{85} + 2 q^{86} - 8 q^{87} + 9 q^{88} - 41 q^{89} + 6 q^{90} - 39 q^{91} - 32 q^{92} - 17 q^{93} + 55 q^{94} - 9 q^{95} - 27 q^{96} - 41 q^{97} - 29 q^{98} - 11 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.14470 −0.809429 −0.404714 0.914443i \(-0.632629\pi\)
−0.404714 + 0.914443i \(0.632629\pi\)
\(3\) 1.00000 0.577350
\(4\) −0.689651 −0.344825
\(5\) 1.33318 0.596216 0.298108 0.954532i \(-0.403645\pi\)
0.298108 + 0.954532i \(0.403645\pi\)
\(6\) −1.14470 −0.467324
\(7\) −3.52170 −1.33108 −0.665538 0.746364i \(-0.731798\pi\)
−0.665538 + 0.746364i \(0.731798\pi\)
\(8\) 3.07886 1.08854
\(9\) 1.00000 0.333333
\(10\) −1.52610 −0.482594
\(11\) −1.00000 −0.301511
\(12\) −0.689651 −0.199085
\(13\) −5.11237 −1.41792 −0.708958 0.705250i \(-0.750835\pi\)
−0.708958 + 0.705250i \(0.750835\pi\)
\(14\) 4.03130 1.07741
\(15\) 1.33318 0.344225
\(16\) −2.14508 −0.536270
\(17\) 1.76849 0.428923 0.214461 0.976732i \(-0.431200\pi\)
0.214461 + 0.976732i \(0.431200\pi\)
\(18\) −1.14470 −0.269810
\(19\) 7.94593 1.82292 0.911461 0.411386i \(-0.134955\pi\)
0.911461 + 0.411386i \(0.134955\pi\)
\(20\) −0.919427 −0.205590
\(21\) −3.52170 −0.768497
\(22\) 1.14470 0.244052
\(23\) 2.83929 0.592033 0.296016 0.955183i \(-0.404342\pi\)
0.296016 + 0.955183i \(0.404342\pi\)
\(24\) 3.07886 0.628469
\(25\) −3.22263 −0.644527
\(26\) 5.85216 1.14770
\(27\) 1.00000 0.192450
\(28\) 2.42874 0.458989
\(29\) 9.92659 1.84332 0.921661 0.387996i \(-0.126833\pi\)
0.921661 + 0.387996i \(0.126833\pi\)
\(30\) −1.52610 −0.278626
\(31\) −4.21272 −0.756628 −0.378314 0.925677i \(-0.623496\pi\)
−0.378314 + 0.925677i \(0.623496\pi\)
\(32\) −3.70223 −0.654468
\(33\) −1.00000 −0.174078
\(34\) −2.02440 −0.347182
\(35\) −4.69505 −0.793608
\(36\) −0.689651 −0.114942
\(37\) −9.46041 −1.55528 −0.777641 0.628708i \(-0.783584\pi\)
−0.777641 + 0.628708i \(0.783584\pi\)
\(38\) −9.09575 −1.47553
\(39\) −5.11237 −0.818635
\(40\) 4.10467 0.649005
\(41\) −4.92390 −0.768984 −0.384492 0.923128i \(-0.625623\pi\)
−0.384492 + 0.923128i \(0.625623\pi\)
\(42\) 4.03130 0.622043
\(43\) −5.08527 −0.775496 −0.387748 0.921765i \(-0.626747\pi\)
−0.387748 + 0.921765i \(0.626747\pi\)
\(44\) 0.689651 0.103969
\(45\) 1.33318 0.198739
\(46\) −3.25015 −0.479208
\(47\) −6.83907 −0.997580 −0.498790 0.866723i \(-0.666222\pi\)
−0.498790 + 0.866723i \(0.666222\pi\)
\(48\) −2.14508 −0.309616
\(49\) 5.40234 0.771763
\(50\) 3.68897 0.521699
\(51\) 1.76849 0.247639
\(52\) 3.52575 0.488933
\(53\) 2.41931 0.332318 0.166159 0.986099i \(-0.446863\pi\)
0.166159 + 0.986099i \(0.446863\pi\)
\(54\) −1.14470 −0.155775
\(55\) −1.33318 −0.179766
\(56\) −10.8428 −1.44893
\(57\) 7.94593 1.05246
\(58\) −11.3630 −1.49204
\(59\) −5.09379 −0.663155 −0.331577 0.943428i \(-0.607581\pi\)
−0.331577 + 0.943428i \(0.607581\pi\)
\(60\) −0.919427 −0.118698
\(61\) 1.00000 0.128037
\(62\) 4.82233 0.612436
\(63\) −3.52170 −0.443692
\(64\) 8.52812 1.06602
\(65\) −6.81570 −0.845384
\(66\) 1.14470 0.140903
\(67\) −14.9335 −1.82441 −0.912206 0.409732i \(-0.865622\pi\)
−0.912206 + 0.409732i \(0.865622\pi\)
\(68\) −1.21964 −0.147903
\(69\) 2.83929 0.341810
\(70\) 5.37445 0.642369
\(71\) −1.73224 −0.205579 −0.102789 0.994703i \(-0.532777\pi\)
−0.102789 + 0.994703i \(0.532777\pi\)
\(72\) 3.07886 0.362847
\(73\) −0.272088 −0.0318455 −0.0159228 0.999873i \(-0.505069\pi\)
−0.0159228 + 0.999873i \(0.505069\pi\)
\(74\) 10.8294 1.25889
\(75\) −3.22263 −0.372118
\(76\) −5.47992 −0.628590
\(77\) 3.52170 0.401334
\(78\) 5.85216 0.662626
\(79\) 14.9608 1.68323 0.841613 0.540081i \(-0.181606\pi\)
0.841613 + 0.540081i \(0.181606\pi\)
\(80\) −2.85978 −0.319733
\(81\) 1.00000 0.111111
\(82\) 5.63642 0.622438
\(83\) −12.0244 −1.31985 −0.659924 0.751333i \(-0.729411\pi\)
−0.659924 + 0.751333i \(0.729411\pi\)
\(84\) 2.42874 0.264997
\(85\) 2.35772 0.255731
\(86\) 5.82113 0.627709
\(87\) 9.92659 1.06424
\(88\) −3.07886 −0.328207
\(89\) −13.3848 −1.41879 −0.709396 0.704811i \(-0.751032\pi\)
−0.709396 + 0.704811i \(0.751032\pi\)
\(90\) −1.52610 −0.160865
\(91\) 18.0042 1.88735
\(92\) −1.95812 −0.204148
\(93\) −4.21272 −0.436839
\(94\) 7.82871 0.807470
\(95\) 10.5933 1.08685
\(96\) −3.70223 −0.377857
\(97\) −1.16994 −0.118790 −0.0593949 0.998235i \(-0.518917\pi\)
−0.0593949 + 0.998235i \(0.518917\pi\)
\(98\) −6.18408 −0.624687
\(99\) −1.00000 −0.100504
\(100\) 2.22249 0.222249
\(101\) 1.20636 0.120037 0.0600186 0.998197i \(-0.480884\pi\)
0.0600186 + 0.998197i \(0.480884\pi\)
\(102\) −2.02440 −0.200446
\(103\) −7.97633 −0.785931 −0.392966 0.919553i \(-0.628551\pi\)
−0.392966 + 0.919553i \(0.628551\pi\)
\(104\) −15.7403 −1.54346
\(105\) −4.69505 −0.458190
\(106\) −2.76940 −0.268988
\(107\) 13.1670 1.27290 0.636451 0.771317i \(-0.280402\pi\)
0.636451 + 0.771317i \(0.280402\pi\)
\(108\) −0.689651 −0.0663617
\(109\) −7.42870 −0.711540 −0.355770 0.934574i \(-0.615781\pi\)
−0.355770 + 0.934574i \(0.615781\pi\)
\(110\) 1.52610 0.145508
\(111\) −9.46041 −0.897943
\(112\) 7.55432 0.713816
\(113\) −5.85940 −0.551206 −0.275603 0.961271i \(-0.588878\pi\)
−0.275603 + 0.961271i \(0.588878\pi\)
\(114\) −9.09575 −0.851895
\(115\) 3.78528 0.352979
\(116\) −6.84588 −0.635624
\(117\) −5.11237 −0.472639
\(118\) 5.83088 0.536776
\(119\) −6.22810 −0.570929
\(120\) 4.10467 0.374703
\(121\) 1.00000 0.0909091
\(122\) −1.14470 −0.103637
\(123\) −4.92390 −0.443973
\(124\) 2.90531 0.260904
\(125\) −10.9622 −0.980493
\(126\) 4.03130 0.359137
\(127\) −5.94779 −0.527781 −0.263890 0.964553i \(-0.585006\pi\)
−0.263890 + 0.964553i \(0.585006\pi\)
\(128\) −2.35773 −0.208396
\(129\) −5.08527 −0.447733
\(130\) 7.80197 0.684278
\(131\) −19.8009 −1.73002 −0.865008 0.501758i \(-0.832687\pi\)
−0.865008 + 0.501758i \(0.832687\pi\)
\(132\) 0.689651 0.0600264
\(133\) −27.9832 −2.42645
\(134\) 17.0944 1.47673
\(135\) 1.33318 0.114742
\(136\) 5.44494 0.466900
\(137\) −17.3461 −1.48198 −0.740988 0.671519i \(-0.765642\pi\)
−0.740988 + 0.671519i \(0.765642\pi\)
\(138\) −3.25015 −0.276671
\(139\) 10.2385 0.868417 0.434209 0.900812i \(-0.357028\pi\)
0.434209 + 0.900812i \(0.357028\pi\)
\(140\) 3.23794 0.273656
\(141\) −6.83907 −0.575953
\(142\) 1.98290 0.166401
\(143\) 5.11237 0.427518
\(144\) −2.14508 −0.178757
\(145\) 13.2339 1.09902
\(146\) 0.311461 0.0257767
\(147\) 5.40234 0.445577
\(148\) 6.52438 0.536301
\(149\) 11.0748 0.907280 0.453640 0.891185i \(-0.350125\pi\)
0.453640 + 0.891185i \(0.350125\pi\)
\(150\) 3.68897 0.301203
\(151\) −0.576802 −0.0469395 −0.0234698 0.999725i \(-0.507471\pi\)
−0.0234698 + 0.999725i \(0.507471\pi\)
\(152\) 24.4644 1.98432
\(153\) 1.76849 0.142974
\(154\) −4.03130 −0.324852
\(155\) −5.61631 −0.451113
\(156\) 3.52575 0.282286
\(157\) 0.409551 0.0326857 0.0163429 0.999866i \(-0.494798\pi\)
0.0163429 + 0.999866i \(0.494798\pi\)
\(158\) −17.1258 −1.36245
\(159\) 2.41931 0.191864
\(160\) −4.93573 −0.390204
\(161\) −9.99911 −0.788041
\(162\) −1.14470 −0.0899365
\(163\) 10.7263 0.840146 0.420073 0.907490i \(-0.362004\pi\)
0.420073 + 0.907490i \(0.362004\pi\)
\(164\) 3.39577 0.265165
\(165\) −1.33318 −0.103788
\(166\) 13.7644 1.06832
\(167\) −9.02703 −0.698533 −0.349266 0.937023i \(-0.613569\pi\)
−0.349266 + 0.937023i \(0.613569\pi\)
\(168\) −10.8428 −0.836540
\(169\) 13.1363 1.01049
\(170\) −2.69889 −0.206996
\(171\) 7.94593 0.607641
\(172\) 3.50706 0.267411
\(173\) 1.04546 0.0794852 0.0397426 0.999210i \(-0.487346\pi\)
0.0397426 + 0.999210i \(0.487346\pi\)
\(174\) −11.3630 −0.861428
\(175\) 11.3491 0.857914
\(176\) 2.14508 0.161692
\(177\) −5.09379 −0.382873
\(178\) 15.3217 1.14841
\(179\) −19.9008 −1.48745 −0.743727 0.668483i \(-0.766944\pi\)
−0.743727 + 0.668483i \(0.766944\pi\)
\(180\) −0.919427 −0.0685301
\(181\) 25.4805 1.89395 0.946975 0.321308i \(-0.104122\pi\)
0.946975 + 0.321308i \(0.104122\pi\)
\(182\) −20.6095 −1.52768
\(183\) 1.00000 0.0739221
\(184\) 8.74177 0.644452
\(185\) −12.6124 −0.927284
\(186\) 4.82233 0.353590
\(187\) −1.76849 −0.129325
\(188\) 4.71656 0.343991
\(189\) −3.52170 −0.256166
\(190\) −12.1263 −0.879731
\(191\) −3.08595 −0.223291 −0.111646 0.993748i \(-0.535612\pi\)
−0.111646 + 0.993748i \(0.535612\pi\)
\(192\) 8.52812 0.615464
\(193\) −20.2049 −1.45438 −0.727190 0.686437i \(-0.759174\pi\)
−0.727190 + 0.686437i \(0.759174\pi\)
\(194\) 1.33924 0.0961518
\(195\) −6.81570 −0.488083
\(196\) −3.72573 −0.266123
\(197\) 20.4684 1.45832 0.729158 0.684346i \(-0.239912\pi\)
0.729158 + 0.684346i \(0.239912\pi\)
\(198\) 1.14470 0.0813506
\(199\) −7.13828 −0.506019 −0.253009 0.967464i \(-0.581420\pi\)
−0.253009 + 0.967464i \(0.581420\pi\)
\(200\) −9.92203 −0.701593
\(201\) −14.9335 −1.05332
\(202\) −1.38093 −0.0971616
\(203\) −34.9584 −2.45360
\(204\) −1.21964 −0.0853921
\(205\) −6.56444 −0.458481
\(206\) 9.13055 0.636155
\(207\) 2.83929 0.197344
\(208\) 10.9665 0.760387
\(209\) −7.94593 −0.549632
\(210\) 5.37445 0.370872
\(211\) 2.60205 0.179133 0.0895664 0.995981i \(-0.471452\pi\)
0.0895664 + 0.995981i \(0.471452\pi\)
\(212\) −1.66848 −0.114592
\(213\) −1.73224 −0.118691
\(214\) −15.0723 −1.03032
\(215\) −6.77957 −0.462363
\(216\) 3.07886 0.209490
\(217\) 14.8359 1.00713
\(218\) 8.50367 0.575941
\(219\) −0.272088 −0.0183860
\(220\) 0.919427 0.0619878
\(221\) −9.04120 −0.608177
\(222\) 10.8294 0.726821
\(223\) 14.8878 0.996959 0.498480 0.866901i \(-0.333892\pi\)
0.498480 + 0.866901i \(0.333892\pi\)
\(224\) 13.0381 0.871146
\(225\) −3.22263 −0.214842
\(226\) 6.70729 0.446162
\(227\) −24.1004 −1.59960 −0.799800 0.600267i \(-0.795061\pi\)
−0.799800 + 0.600267i \(0.795061\pi\)
\(228\) −5.47992 −0.362916
\(229\) −13.3300 −0.880872 −0.440436 0.897784i \(-0.645176\pi\)
−0.440436 + 0.897784i \(0.645176\pi\)
\(230\) −4.33303 −0.285712
\(231\) 3.52170 0.231711
\(232\) 30.5626 2.00653
\(233\) 2.31713 0.151801 0.0759003 0.997115i \(-0.475817\pi\)
0.0759003 + 0.997115i \(0.475817\pi\)
\(234\) 5.85216 0.382567
\(235\) −9.11770 −0.594773
\(236\) 3.51293 0.228672
\(237\) 14.9608 0.971811
\(238\) 7.12934 0.462126
\(239\) −13.3420 −0.863021 −0.431510 0.902108i \(-0.642019\pi\)
−0.431510 + 0.902108i \(0.642019\pi\)
\(240\) −2.85978 −0.184598
\(241\) −14.0952 −0.907951 −0.453975 0.891014i \(-0.649995\pi\)
−0.453975 + 0.891014i \(0.649995\pi\)
\(242\) −1.14470 −0.0735844
\(243\) 1.00000 0.0641500
\(244\) −0.689651 −0.0441503
\(245\) 7.20228 0.460137
\(246\) 5.63642 0.359365
\(247\) −40.6226 −2.58475
\(248\) −12.9704 −0.823620
\(249\) −12.0244 −0.762014
\(250\) 12.5485 0.793639
\(251\) −10.1102 −0.638150 −0.319075 0.947730i \(-0.603372\pi\)
−0.319075 + 0.947730i \(0.603372\pi\)
\(252\) 2.42874 0.152996
\(253\) −2.83929 −0.178505
\(254\) 6.80846 0.427201
\(255\) 2.35772 0.147646
\(256\) −14.3573 −0.897334
\(257\) 5.00734 0.312349 0.156175 0.987729i \(-0.450084\pi\)
0.156175 + 0.987729i \(0.450084\pi\)
\(258\) 5.82113 0.362408
\(259\) 33.3167 2.07020
\(260\) 4.70045 0.291510
\(261\) 9.92659 0.614441
\(262\) 22.6662 1.40032
\(263\) −11.7522 −0.724670 −0.362335 0.932048i \(-0.618020\pi\)
−0.362335 + 0.932048i \(0.618020\pi\)
\(264\) −3.07886 −0.189491
\(265\) 3.22538 0.198133
\(266\) 32.0325 1.96404
\(267\) −13.3848 −0.819139
\(268\) 10.2989 0.629103
\(269\) 25.2506 1.53956 0.769779 0.638311i \(-0.220367\pi\)
0.769779 + 0.638311i \(0.220367\pi\)
\(270\) −1.52610 −0.0928753
\(271\) −0.247119 −0.0150114 −0.00750571 0.999972i \(-0.502389\pi\)
−0.00750571 + 0.999972i \(0.502389\pi\)
\(272\) −3.79356 −0.230019
\(273\) 18.0042 1.08966
\(274\) 19.8561 1.19955
\(275\) 3.22263 0.194332
\(276\) −1.95812 −0.117865
\(277\) −8.44029 −0.507128 −0.253564 0.967319i \(-0.581603\pi\)
−0.253564 + 0.967319i \(0.581603\pi\)
\(278\) −11.7201 −0.702922
\(279\) −4.21272 −0.252209
\(280\) −14.4554 −0.863874
\(281\) 17.3091 1.03257 0.516286 0.856416i \(-0.327314\pi\)
0.516286 + 0.856416i \(0.327314\pi\)
\(282\) 7.82871 0.466193
\(283\) −17.2268 −1.02403 −0.512013 0.858978i \(-0.671100\pi\)
−0.512013 + 0.858978i \(0.671100\pi\)
\(284\) 1.19464 0.0708888
\(285\) 10.5933 0.627496
\(286\) −5.85216 −0.346045
\(287\) 17.3405 1.02358
\(288\) −3.70223 −0.218156
\(289\) −13.8724 −0.816025
\(290\) −15.1489 −0.889576
\(291\) −1.16994 −0.0685833
\(292\) 0.187646 0.0109811
\(293\) 20.3320 1.18781 0.593905 0.804535i \(-0.297586\pi\)
0.593905 + 0.804535i \(0.297586\pi\)
\(294\) −6.18408 −0.360663
\(295\) −6.79093 −0.395383
\(296\) −29.1273 −1.69299
\(297\) −1.00000 −0.0580259
\(298\) −12.6773 −0.734378
\(299\) −14.5155 −0.839453
\(300\) 2.22249 0.128316
\(301\) 17.9088 1.03224
\(302\) 0.660268 0.0379942
\(303\) 1.20636 0.0693035
\(304\) −17.0447 −0.977579
\(305\) 1.33318 0.0763376
\(306\) −2.02440 −0.115727
\(307\) −15.3499 −0.876063 −0.438031 0.898960i \(-0.644324\pi\)
−0.438031 + 0.898960i \(0.644324\pi\)
\(308\) −2.42874 −0.138390
\(309\) −7.97633 −0.453758
\(310\) 6.42902 0.365144
\(311\) 9.55462 0.541793 0.270896 0.962608i \(-0.412680\pi\)
0.270896 + 0.962608i \(0.412680\pi\)
\(312\) −15.7403 −0.891117
\(313\) 15.5623 0.879635 0.439817 0.898087i \(-0.355043\pi\)
0.439817 + 0.898087i \(0.355043\pi\)
\(314\) −0.468815 −0.0264568
\(315\) −4.69505 −0.264536
\(316\) −10.3178 −0.580419
\(317\) 15.1622 0.851596 0.425798 0.904818i \(-0.359993\pi\)
0.425798 + 0.904818i \(0.359993\pi\)
\(318\) −2.76940 −0.155300
\(319\) −9.92659 −0.555783
\(320\) 11.3695 0.635575
\(321\) 13.1670 0.734910
\(322\) 11.4460 0.637863
\(323\) 14.0523 0.781893
\(324\) −0.689651 −0.0383139
\(325\) 16.4753 0.913886
\(326\) −12.2784 −0.680038
\(327\) −7.42870 −0.410808
\(328\) −15.1600 −0.837070
\(329\) 24.0851 1.32785
\(330\) 1.52610 0.0840088
\(331\) −31.4774 −1.73015 −0.865077 0.501639i \(-0.832731\pi\)
−0.865077 + 0.501639i \(0.832731\pi\)
\(332\) 8.29262 0.455117
\(333\) −9.46041 −0.518428
\(334\) 10.3333 0.565412
\(335\) −19.9090 −1.08774
\(336\) 7.55432 0.412122
\(337\) 5.47362 0.298167 0.149083 0.988825i \(-0.452368\pi\)
0.149083 + 0.988825i \(0.452368\pi\)
\(338\) −15.0372 −0.817918
\(339\) −5.85940 −0.318239
\(340\) −1.62600 −0.0881823
\(341\) 4.21272 0.228132
\(342\) −9.09575 −0.491842
\(343\) 5.62647 0.303801
\(344\) −15.6568 −0.844159
\(345\) 3.78528 0.203793
\(346\) −1.19675 −0.0643376
\(347\) 20.0402 1.07581 0.537906 0.843005i \(-0.319215\pi\)
0.537906 + 0.843005i \(0.319215\pi\)
\(348\) −6.84588 −0.366978
\(349\) −4.99181 −0.267205 −0.133603 0.991035i \(-0.542655\pi\)
−0.133603 + 0.991035i \(0.542655\pi\)
\(350\) −12.9914 −0.694420
\(351\) −5.11237 −0.272878
\(352\) 3.70223 0.197329
\(353\) 4.32905 0.230412 0.115206 0.993342i \(-0.463247\pi\)
0.115206 + 0.993342i \(0.463247\pi\)
\(354\) 5.83088 0.309908
\(355\) −2.30938 −0.122569
\(356\) 9.23087 0.489235
\(357\) −6.22810 −0.329626
\(358\) 22.7805 1.20399
\(359\) −13.5801 −0.716731 −0.358366 0.933581i \(-0.616666\pi\)
−0.358366 + 0.933581i \(0.616666\pi\)
\(360\) 4.10467 0.216335
\(361\) 44.1379 2.32304
\(362\) −29.1676 −1.53302
\(363\) 1.00000 0.0524864
\(364\) −12.4166 −0.650808
\(365\) −0.362742 −0.0189868
\(366\) −1.14470 −0.0598347
\(367\) 15.9859 0.834459 0.417230 0.908801i \(-0.363001\pi\)
0.417230 + 0.908801i \(0.363001\pi\)
\(368\) −6.09051 −0.317490
\(369\) −4.92390 −0.256328
\(370\) 14.4375 0.750570
\(371\) −8.52008 −0.442341
\(372\) 2.90531 0.150633
\(373\) −4.07689 −0.211093 −0.105547 0.994414i \(-0.533659\pi\)
−0.105547 + 0.994414i \(0.533659\pi\)
\(374\) 2.02440 0.104679
\(375\) −10.9622 −0.566088
\(376\) −21.0565 −1.08591
\(377\) −50.7484 −2.61368
\(378\) 4.03130 0.207348
\(379\) −8.06236 −0.414136 −0.207068 0.978327i \(-0.566392\pi\)
−0.207068 + 0.978327i \(0.566392\pi\)
\(380\) −7.30571 −0.374775
\(381\) −5.94779 −0.304714
\(382\) 3.53250 0.180738
\(383\) −4.49317 −0.229591 −0.114795 0.993389i \(-0.536621\pi\)
−0.114795 + 0.993389i \(0.536621\pi\)
\(384\) −2.35773 −0.120317
\(385\) 4.69505 0.239282
\(386\) 23.1286 1.17722
\(387\) −5.08527 −0.258499
\(388\) 0.806852 0.0409617
\(389\) 21.7065 1.10056 0.550282 0.834979i \(-0.314520\pi\)
0.550282 + 0.834979i \(0.314520\pi\)
\(390\) 7.80197 0.395068
\(391\) 5.02127 0.253936
\(392\) 16.6330 0.840095
\(393\) −19.8009 −0.998825
\(394\) −23.4303 −1.18040
\(395\) 19.9455 1.00357
\(396\) 0.689651 0.0346562
\(397\) −0.434918 −0.0218279 −0.0109140 0.999940i \(-0.503474\pi\)
−0.0109140 + 0.999940i \(0.503474\pi\)
\(398\) 8.17122 0.409586
\(399\) −27.9832 −1.40091
\(400\) 6.91281 0.345641
\(401\) 23.6376 1.18041 0.590203 0.807255i \(-0.299048\pi\)
0.590203 + 0.807255i \(0.299048\pi\)
\(402\) 17.0944 0.852591
\(403\) 21.5370 1.07284
\(404\) −0.831966 −0.0413919
\(405\) 1.33318 0.0662462
\(406\) 40.0171 1.98602
\(407\) 9.46041 0.468935
\(408\) 5.44494 0.269565
\(409\) −32.9182 −1.62770 −0.813849 0.581076i \(-0.802632\pi\)
−0.813849 + 0.581076i \(0.802632\pi\)
\(410\) 7.51435 0.371107
\(411\) −17.3461 −0.855619
\(412\) 5.50088 0.271009
\(413\) 17.9388 0.882709
\(414\) −3.25015 −0.159736
\(415\) −16.0307 −0.786914
\(416\) 18.9272 0.927981
\(417\) 10.2385 0.501381
\(418\) 9.09575 0.444888
\(419\) 28.6995 1.40206 0.701030 0.713132i \(-0.252724\pi\)
0.701030 + 0.713132i \(0.252724\pi\)
\(420\) 3.23794 0.157995
\(421\) 36.0867 1.75876 0.879380 0.476120i \(-0.157957\pi\)
0.879380 + 0.476120i \(0.157957\pi\)
\(422\) −2.97858 −0.144995
\(423\) −6.83907 −0.332527
\(424\) 7.44871 0.361742
\(425\) −5.69921 −0.276452
\(426\) 1.98290 0.0960719
\(427\) −3.52170 −0.170427
\(428\) −9.08063 −0.438929
\(429\) 5.11237 0.246828
\(430\) 7.76061 0.374250
\(431\) 32.5593 1.56833 0.784163 0.620555i \(-0.213093\pi\)
0.784163 + 0.620555i \(0.213093\pi\)
\(432\) −2.14508 −0.103205
\(433\) −29.5779 −1.42142 −0.710711 0.703484i \(-0.751627\pi\)
−0.710711 + 0.703484i \(0.751627\pi\)
\(434\) −16.9828 −0.815199
\(435\) 13.2339 0.634518
\(436\) 5.12320 0.245357
\(437\) 22.5608 1.07923
\(438\) 0.311461 0.0148822
\(439\) 20.9170 0.998316 0.499158 0.866511i \(-0.333643\pi\)
0.499158 + 0.866511i \(0.333643\pi\)
\(440\) −4.10467 −0.195682
\(441\) 5.40234 0.257254
\(442\) 10.3495 0.492276
\(443\) 13.9300 0.661833 0.330917 0.943660i \(-0.392642\pi\)
0.330917 + 0.943660i \(0.392642\pi\)
\(444\) 6.52438 0.309633
\(445\) −17.8444 −0.845905
\(446\) −17.0421 −0.806967
\(447\) 11.0748 0.523818
\(448\) −30.0334 −1.41895
\(449\) −15.8421 −0.747633 −0.373817 0.927503i \(-0.621951\pi\)
−0.373817 + 0.927503i \(0.621951\pi\)
\(450\) 3.68897 0.173900
\(451\) 4.92390 0.231858
\(452\) 4.04094 0.190070
\(453\) −0.576802 −0.0271005
\(454\) 27.5879 1.29476
\(455\) 24.0028 1.12527
\(456\) 24.4644 1.14565
\(457\) 31.1875 1.45889 0.729444 0.684040i \(-0.239779\pi\)
0.729444 + 0.684040i \(0.239779\pi\)
\(458\) 15.2589 0.713003
\(459\) 1.76849 0.0825462
\(460\) −2.61052 −0.121716
\(461\) −38.3496 −1.78612 −0.893059 0.449940i \(-0.851445\pi\)
−0.893059 + 0.449940i \(0.851445\pi\)
\(462\) −4.03130 −0.187553
\(463\) 3.32069 0.154325 0.0771627 0.997019i \(-0.475414\pi\)
0.0771627 + 0.997019i \(0.475414\pi\)
\(464\) −21.2933 −0.988519
\(465\) −5.61631 −0.260450
\(466\) −2.65244 −0.122872
\(467\) −16.9438 −0.784068 −0.392034 0.919951i \(-0.628228\pi\)
−0.392034 + 0.919951i \(0.628228\pi\)
\(468\) 3.52575 0.162978
\(469\) 52.5911 2.42843
\(470\) 10.4371 0.481426
\(471\) 0.409551 0.0188711
\(472\) −15.6830 −0.721870
\(473\) 5.08527 0.233821
\(474\) −17.1258 −0.786612
\(475\) −25.6068 −1.17492
\(476\) 4.29521 0.196871
\(477\) 2.41931 0.110773
\(478\) 15.2726 0.698554
\(479\) −5.47364 −0.250097 −0.125049 0.992151i \(-0.539909\pi\)
−0.125049 + 0.992151i \(0.539909\pi\)
\(480\) −4.93573 −0.225284
\(481\) 48.3652 2.20526
\(482\) 16.1348 0.734921
\(483\) −9.99911 −0.454975
\(484\) −0.689651 −0.0313478
\(485\) −1.55974 −0.0708243
\(486\) −1.14470 −0.0519249
\(487\) 37.1479 1.68333 0.841666 0.539998i \(-0.181575\pi\)
0.841666 + 0.539998i \(0.181575\pi\)
\(488\) 3.07886 0.139373
\(489\) 10.7263 0.485059
\(490\) −8.24449 −0.372448
\(491\) −36.9552 −1.66777 −0.833883 0.551941i \(-0.813887\pi\)
−0.833883 + 0.551941i \(0.813887\pi\)
\(492\) 3.39577 0.153093
\(493\) 17.5551 0.790643
\(494\) 46.5009 2.09217
\(495\) −1.33318 −0.0599219
\(496\) 9.03664 0.405757
\(497\) 6.10041 0.273641
\(498\) 13.7644 0.616796
\(499\) −20.2310 −0.905662 −0.452831 0.891596i \(-0.649586\pi\)
−0.452831 + 0.891596i \(0.649586\pi\)
\(500\) 7.56012 0.338099
\(501\) −9.02703 −0.403298
\(502\) 11.5732 0.516537
\(503\) −19.6455 −0.875952 −0.437976 0.898987i \(-0.644304\pi\)
−0.437976 + 0.898987i \(0.644304\pi\)
\(504\) −10.8428 −0.482976
\(505\) 1.60829 0.0715681
\(506\) 3.25015 0.144487
\(507\) 13.1363 0.583405
\(508\) 4.10189 0.181992
\(509\) 35.8608 1.58950 0.794752 0.606935i \(-0.207601\pi\)
0.794752 + 0.606935i \(0.207601\pi\)
\(510\) −2.69889 −0.119509
\(511\) 0.958213 0.0423888
\(512\) 21.1504 0.934723
\(513\) 7.94593 0.350822
\(514\) −5.73193 −0.252825
\(515\) −10.6339 −0.468585
\(516\) 3.50706 0.154390
\(517\) 6.83907 0.300782
\(518\) −38.1378 −1.67568
\(519\) 1.04546 0.0458908
\(520\) −20.9846 −0.920235
\(521\) −22.2297 −0.973900 −0.486950 0.873430i \(-0.661891\pi\)
−0.486950 + 0.873430i \(0.661891\pi\)
\(522\) −11.3630 −0.497346
\(523\) 8.21955 0.359416 0.179708 0.983720i \(-0.442485\pi\)
0.179708 + 0.983720i \(0.442485\pi\)
\(524\) 13.6557 0.596553
\(525\) 11.3491 0.495317
\(526\) 13.4528 0.586569
\(527\) −7.45018 −0.324535
\(528\) 2.14508 0.0933527
\(529\) −14.9384 −0.649497
\(530\) −3.69210 −0.160375
\(531\) −5.09379 −0.221052
\(532\) 19.2986 0.836700
\(533\) 25.1728 1.09036
\(534\) 15.3217 0.663035
\(535\) 17.5540 0.758924
\(536\) −45.9780 −1.98595
\(537\) −19.9008 −0.858782
\(538\) −28.9045 −1.24616
\(539\) −5.40234 −0.232695
\(540\) −0.919427 −0.0395659
\(541\) 9.01502 0.387586 0.193793 0.981042i \(-0.437921\pi\)
0.193793 + 0.981042i \(0.437921\pi\)
\(542\) 0.282879 0.0121507
\(543\) 25.4805 1.09347
\(544\) −6.54737 −0.280716
\(545\) −9.90378 −0.424231
\(546\) −20.6095 −0.882006
\(547\) 21.1680 0.905079 0.452540 0.891744i \(-0.350518\pi\)
0.452540 + 0.891744i \(0.350518\pi\)
\(548\) 11.9627 0.511022
\(549\) 1.00000 0.0426790
\(550\) −3.68897 −0.157298
\(551\) 78.8761 3.36023
\(552\) 8.74177 0.372074
\(553\) −52.6875 −2.24050
\(554\) 9.66164 0.410484
\(555\) −12.6124 −0.535368
\(556\) −7.06098 −0.299452
\(557\) 13.3758 0.566749 0.283375 0.959009i \(-0.408546\pi\)
0.283375 + 0.959009i \(0.408546\pi\)
\(558\) 4.82233 0.204145
\(559\) 25.9978 1.09959
\(560\) 10.0713 0.425588
\(561\) −1.76849 −0.0746659
\(562\) −19.8138 −0.835793
\(563\) 1.28336 0.0540874 0.0270437 0.999634i \(-0.491391\pi\)
0.0270437 + 0.999634i \(0.491391\pi\)
\(564\) 4.71656 0.198603
\(565\) −7.81163 −0.328638
\(566\) 19.7196 0.828876
\(567\) −3.52170 −0.147897
\(568\) −5.33331 −0.223781
\(569\) −13.6521 −0.572324 −0.286162 0.958181i \(-0.592380\pi\)
−0.286162 + 0.958181i \(0.592380\pi\)
\(570\) −12.1263 −0.507913
\(571\) 11.7612 0.492191 0.246095 0.969246i \(-0.420852\pi\)
0.246095 + 0.969246i \(0.420852\pi\)
\(572\) −3.52575 −0.147419
\(573\) −3.08595 −0.128917
\(574\) −19.8497 −0.828512
\(575\) −9.14999 −0.381581
\(576\) 8.52812 0.355338
\(577\) −4.18009 −0.174019 −0.0870096 0.996207i \(-0.527731\pi\)
−0.0870096 + 0.996207i \(0.527731\pi\)
\(578\) 15.8798 0.660514
\(579\) −20.2049 −0.839686
\(580\) −9.12678 −0.378969
\(581\) 42.3462 1.75682
\(582\) 1.33924 0.0555133
\(583\) −2.41931 −0.100198
\(584\) −0.837721 −0.0346651
\(585\) −6.81570 −0.281795
\(586\) −23.2742 −0.961447
\(587\) 29.1341 1.20249 0.601246 0.799064i \(-0.294671\pi\)
0.601246 + 0.799064i \(0.294671\pi\)
\(588\) −3.72573 −0.153646
\(589\) −33.4740 −1.37927
\(590\) 7.77361 0.320034
\(591\) 20.4684 0.841959
\(592\) 20.2934 0.834052
\(593\) 12.7057 0.521761 0.260880 0.965371i \(-0.415987\pi\)
0.260880 + 0.965371i \(0.415987\pi\)
\(594\) 1.14470 0.0469678
\(595\) −8.30317 −0.340397
\(596\) −7.63772 −0.312853
\(597\) −7.13828 −0.292150
\(598\) 16.6160 0.679478
\(599\) −16.5948 −0.678046 −0.339023 0.940778i \(-0.610096\pi\)
−0.339023 + 0.940778i \(0.610096\pi\)
\(600\) −9.92203 −0.405065
\(601\) 30.7501 1.25432 0.627161 0.778889i \(-0.284217\pi\)
0.627161 + 0.778889i \(0.284217\pi\)
\(602\) −20.5003 −0.835528
\(603\) −14.9335 −0.608137
\(604\) 0.397792 0.0161859
\(605\) 1.33318 0.0542014
\(606\) −1.38093 −0.0560962
\(607\) −21.0480 −0.854312 −0.427156 0.904178i \(-0.640484\pi\)
−0.427156 + 0.904178i \(0.640484\pi\)
\(608\) −29.4177 −1.19304
\(609\) −34.9584 −1.41659
\(610\) −1.52610 −0.0617898
\(611\) 34.9638 1.41449
\(612\) −1.21964 −0.0493011
\(613\) 27.1514 1.09664 0.548318 0.836270i \(-0.315268\pi\)
0.548318 + 0.836270i \(0.315268\pi\)
\(614\) 17.5711 0.709110
\(615\) −6.56444 −0.264704
\(616\) 10.8428 0.436869
\(617\) 26.8583 1.08127 0.540636 0.841256i \(-0.318183\pi\)
0.540636 + 0.841256i \(0.318183\pi\)
\(618\) 9.13055 0.367285
\(619\) −14.0051 −0.562914 −0.281457 0.959574i \(-0.590818\pi\)
−0.281457 + 0.959574i \(0.590818\pi\)
\(620\) 3.87329 0.155555
\(621\) 2.83929 0.113937
\(622\) −10.9372 −0.438543
\(623\) 47.1374 1.88852
\(624\) 10.9665 0.439009
\(625\) 1.49855 0.0599419
\(626\) −17.8143 −0.712002
\(627\) −7.94593 −0.317330
\(628\) −0.282447 −0.0112709
\(629\) −16.7307 −0.667096
\(630\) 5.37445 0.214123
\(631\) 24.5099 0.975723 0.487862 0.872921i \(-0.337777\pi\)
0.487862 + 0.872921i \(0.337777\pi\)
\(632\) 46.0623 1.83226
\(633\) 2.60205 0.103422
\(634\) −17.3563 −0.689307
\(635\) −7.92946 −0.314671
\(636\) −1.66848 −0.0661595
\(637\) −27.6188 −1.09430
\(638\) 11.3630 0.449866
\(639\) −1.73224 −0.0685263
\(640\) −3.14327 −0.124249
\(641\) 1.69206 0.0668324 0.0334162 0.999442i \(-0.489361\pi\)
0.0334162 + 0.999442i \(0.489361\pi\)
\(642\) −15.0723 −0.594857
\(643\) −23.6988 −0.934591 −0.467295 0.884101i \(-0.654772\pi\)
−0.467295 + 0.884101i \(0.654772\pi\)
\(644\) 6.89589 0.271736
\(645\) −6.77957 −0.266945
\(646\) −16.0858 −0.632887
\(647\) −3.21842 −0.126529 −0.0632646 0.997997i \(-0.520151\pi\)
−0.0632646 + 0.997997i \(0.520151\pi\)
\(648\) 3.07886 0.120949
\(649\) 5.09379 0.199949
\(650\) −18.8594 −0.739725
\(651\) 14.8359 0.581466
\(652\) −7.39738 −0.289704
\(653\) 1.62340 0.0635286 0.0317643 0.999495i \(-0.489887\pi\)
0.0317643 + 0.999495i \(0.489887\pi\)
\(654\) 8.50367 0.332520
\(655\) −26.3982 −1.03146
\(656\) 10.5622 0.412384
\(657\) −0.272088 −0.0106152
\(658\) −27.5703 −1.07480
\(659\) −21.7938 −0.848965 −0.424483 0.905436i \(-0.639544\pi\)
−0.424483 + 0.905436i \(0.639544\pi\)
\(660\) 0.919427 0.0357887
\(661\) 33.5998 1.30688 0.653440 0.756979i \(-0.273325\pi\)
0.653440 + 0.756979i \(0.273325\pi\)
\(662\) 36.0323 1.40044
\(663\) −9.04120 −0.351131
\(664\) −37.0214 −1.43671
\(665\) −37.3065 −1.44669
\(666\) 10.8294 0.419630
\(667\) 28.1845 1.09131
\(668\) 6.22550 0.240872
\(669\) 14.8878 0.575595
\(670\) 22.7899 0.880450
\(671\) −1.00000 −0.0386046
\(672\) 13.0381 0.502956
\(673\) 23.6911 0.913225 0.456613 0.889666i \(-0.349063\pi\)
0.456613 + 0.889666i \(0.349063\pi\)
\(674\) −6.26568 −0.241345
\(675\) −3.22263 −0.124039
\(676\) −9.05949 −0.348442
\(677\) 20.4021 0.784115 0.392058 0.919941i \(-0.371763\pi\)
0.392058 + 0.919941i \(0.371763\pi\)
\(678\) 6.70729 0.257592
\(679\) 4.12018 0.158118
\(680\) 7.25908 0.278373
\(681\) −24.1004 −0.923529
\(682\) −4.82233 −0.184656
\(683\) 14.2024 0.543441 0.271720 0.962376i \(-0.412407\pi\)
0.271720 + 0.962376i \(0.412407\pi\)
\(684\) −5.47992 −0.209530
\(685\) −23.1254 −0.883577
\(686\) −6.44065 −0.245905
\(687\) −13.3300 −0.508572
\(688\) 10.9083 0.415876
\(689\) −12.3684 −0.471199
\(690\) −4.33303 −0.164956
\(691\) −35.6324 −1.35552 −0.677759 0.735284i \(-0.737049\pi\)
−0.677759 + 0.735284i \(0.737049\pi\)
\(692\) −0.721005 −0.0274085
\(693\) 3.52170 0.133778
\(694\) −22.9401 −0.870793
\(695\) 13.6497 0.517764
\(696\) 30.5626 1.15847
\(697\) −8.70789 −0.329835
\(698\) 5.71414 0.216284
\(699\) 2.31713 0.0876421
\(700\) −7.82694 −0.295830
\(701\) −8.93051 −0.337301 −0.168650 0.985676i \(-0.553941\pi\)
−0.168650 + 0.985676i \(0.553941\pi\)
\(702\) 5.85216 0.220875
\(703\) −75.1718 −2.83516
\(704\) −8.52812 −0.321416
\(705\) −9.11770 −0.343392
\(706\) −4.95548 −0.186502
\(707\) −4.24843 −0.159779
\(708\) 3.51293 0.132024
\(709\) −37.0126 −1.39004 −0.695019 0.718992i \(-0.744604\pi\)
−0.695019 + 0.718992i \(0.744604\pi\)
\(710\) 2.64356 0.0992111
\(711\) 14.9608 0.561075
\(712\) −41.2100 −1.54441
\(713\) −11.9611 −0.447948
\(714\) 7.12934 0.266809
\(715\) 6.81570 0.254893
\(716\) 13.7246 0.512912
\(717\) −13.3420 −0.498265
\(718\) 15.5452 0.580143
\(719\) −24.6974 −0.921058 −0.460529 0.887645i \(-0.652340\pi\)
−0.460529 + 0.887645i \(0.652340\pi\)
\(720\) −2.85978 −0.106578
\(721\) 28.0902 1.04613
\(722\) −50.5248 −1.88034
\(723\) −14.0952 −0.524206
\(724\) −17.5726 −0.653082
\(725\) −31.9898 −1.18807
\(726\) −1.14470 −0.0424840
\(727\) −22.5756 −0.837282 −0.418641 0.908152i \(-0.637493\pi\)
−0.418641 + 0.908152i \(0.637493\pi\)
\(728\) 55.4324 2.05446
\(729\) 1.00000 0.0370370
\(730\) 0.415233 0.0153685
\(731\) −8.99327 −0.332628
\(732\) −0.689651 −0.0254902
\(733\) 36.8572 1.36135 0.680676 0.732585i \(-0.261686\pi\)
0.680676 + 0.732585i \(0.261686\pi\)
\(734\) −18.2992 −0.675435
\(735\) 7.20228 0.265660
\(736\) −10.5117 −0.387466
\(737\) 14.9335 0.550081
\(738\) 5.63642 0.207479
\(739\) 9.87637 0.363308 0.181654 0.983362i \(-0.441855\pi\)
0.181654 + 0.983362i \(0.441855\pi\)
\(740\) 8.69816 0.319751
\(741\) −40.6226 −1.49231
\(742\) 9.75298 0.358043
\(743\) −25.0671 −0.919622 −0.459811 0.888017i \(-0.652083\pi\)
−0.459811 + 0.888017i \(0.652083\pi\)
\(744\) −12.9704 −0.475517
\(745\) 14.7646 0.540934
\(746\) 4.66684 0.170865
\(747\) −12.0244 −0.439949
\(748\) 1.21964 0.0445946
\(749\) −46.3702 −1.69433
\(750\) 12.5485 0.458208
\(751\) −9.07144 −0.331021 −0.165511 0.986208i \(-0.552927\pi\)
−0.165511 + 0.986208i \(0.552927\pi\)
\(752\) 14.6704 0.534973
\(753\) −10.1102 −0.368436
\(754\) 58.0920 2.11559
\(755\) −0.768980 −0.0279861
\(756\) 2.42874 0.0883324
\(757\) 29.3753 1.06766 0.533832 0.845591i \(-0.320752\pi\)
0.533832 + 0.845591i \(0.320752\pi\)
\(758\) 9.22902 0.335213
\(759\) −2.83929 −0.103060
\(760\) 32.6154 1.18308
\(761\) 35.8397 1.29919 0.649594 0.760281i \(-0.274939\pi\)
0.649594 + 0.760281i \(0.274939\pi\)
\(762\) 6.80846 0.246645
\(763\) 26.1616 0.947114
\(764\) 2.12823 0.0769965
\(765\) 2.35772 0.0852435
\(766\) 5.14336 0.185837
\(767\) 26.0413 0.940298
\(768\) −14.3573 −0.518076
\(769\) −20.3758 −0.734772 −0.367386 0.930069i \(-0.619747\pi\)
−0.367386 + 0.930069i \(0.619747\pi\)
\(770\) −5.37445 −0.193682
\(771\) 5.00734 0.180335
\(772\) 13.9343 0.501507
\(773\) −43.4010 −1.56103 −0.780513 0.625140i \(-0.785042\pi\)
−0.780513 + 0.625140i \(0.785042\pi\)
\(774\) 5.82113 0.209236
\(775\) 13.5761 0.487667
\(776\) −3.60209 −0.129307
\(777\) 33.3167 1.19523
\(778\) −24.8476 −0.890829
\(779\) −39.1250 −1.40180
\(780\) 4.70045 0.168303
\(781\) 1.73224 0.0619843
\(782\) −5.74787 −0.205543
\(783\) 9.92659 0.354748
\(784\) −11.5885 −0.413873
\(785\) 0.546004 0.0194877
\(786\) 22.6662 0.808478
\(787\) 14.3584 0.511823 0.255912 0.966700i \(-0.417624\pi\)
0.255912 + 0.966700i \(0.417624\pi\)
\(788\) −14.1161 −0.502864
\(789\) −11.7522 −0.418388
\(790\) −22.8317 −0.812315
\(791\) 20.6350 0.733697
\(792\) −3.07886 −0.109402
\(793\) −5.11237 −0.181546
\(794\) 0.497853 0.0176682
\(795\) 3.22538 0.114392
\(796\) 4.92292 0.174488
\(797\) −34.5788 −1.22484 −0.612421 0.790532i \(-0.709804\pi\)
−0.612421 + 0.790532i \(0.709804\pi\)
\(798\) 32.0325 1.13394
\(799\) −12.0948 −0.427885
\(800\) 11.9309 0.421822
\(801\) −13.3848 −0.472930
\(802\) −27.0581 −0.955455
\(803\) 0.272088 0.00960179
\(804\) 10.2989 0.363213
\(805\) −13.3306 −0.469842
\(806\) −24.6535 −0.868383
\(807\) 25.2506 0.888864
\(808\) 3.71421 0.130665
\(809\) 7.62680 0.268144 0.134072 0.990972i \(-0.457195\pi\)
0.134072 + 0.990972i \(0.457195\pi\)
\(810\) −1.52610 −0.0536216
\(811\) 19.4077 0.681496 0.340748 0.940155i \(-0.389320\pi\)
0.340748 + 0.940155i \(0.389320\pi\)
\(812\) 24.1091 0.846064
\(813\) −0.247119 −0.00866685
\(814\) −10.8294 −0.379570
\(815\) 14.3000 0.500908
\(816\) −3.79356 −0.132801
\(817\) −40.4072 −1.41367
\(818\) 37.6816 1.31751
\(819\) 18.0042 0.629118
\(820\) 4.52717 0.158096
\(821\) 4.76831 0.166415 0.0832076 0.996532i \(-0.473484\pi\)
0.0832076 + 0.996532i \(0.473484\pi\)
\(822\) 19.8561 0.692562
\(823\) −39.4737 −1.37597 −0.687984 0.725726i \(-0.741504\pi\)
−0.687984 + 0.725726i \(0.741504\pi\)
\(824\) −24.5580 −0.855518
\(825\) 3.22263 0.112198
\(826\) −20.5346 −0.714490
\(827\) 2.07065 0.0720036 0.0360018 0.999352i \(-0.488538\pi\)
0.0360018 + 0.999352i \(0.488538\pi\)
\(828\) −1.95812 −0.0680493
\(829\) 2.50352 0.0869507 0.0434753 0.999055i \(-0.486157\pi\)
0.0434753 + 0.999055i \(0.486157\pi\)
\(830\) 18.3504 0.636951
\(831\) −8.44029 −0.292790
\(832\) −43.5989 −1.51152
\(833\) 9.55401 0.331027
\(834\) −11.7201 −0.405832
\(835\) −12.0346 −0.416476
\(836\) 5.47992 0.189527
\(837\) −4.21272 −0.145613
\(838\) −32.8524 −1.13487
\(839\) 12.2823 0.424032 0.212016 0.977266i \(-0.431997\pi\)
0.212016 + 0.977266i \(0.431997\pi\)
\(840\) −14.4554 −0.498758
\(841\) 69.5373 2.39784
\(842\) −41.3087 −1.42359
\(843\) 17.3091 0.596155
\(844\) −1.79451 −0.0617695
\(845\) 17.5131 0.602469
\(846\) 7.82871 0.269157
\(847\) −3.52170 −0.121007
\(848\) −5.18962 −0.178212
\(849\) −17.2268 −0.591222
\(850\) 6.52392 0.223768
\(851\) −26.8609 −0.920778
\(852\) 1.19464 0.0409276
\(853\) 55.1509 1.88833 0.944166 0.329470i \(-0.106870\pi\)
0.944166 + 0.329470i \(0.106870\pi\)
\(854\) 4.03130 0.137948
\(855\) 10.5933 0.362285
\(856\) 40.5393 1.38560
\(857\) −45.0798 −1.53990 −0.769948 0.638106i \(-0.779718\pi\)
−0.769948 + 0.638106i \(0.779718\pi\)
\(858\) −5.85216 −0.199789
\(859\) 10.3830 0.354264 0.177132 0.984187i \(-0.443318\pi\)
0.177132 + 0.984187i \(0.443318\pi\)
\(860\) 4.67553 0.159434
\(861\) 17.3405 0.590962
\(862\) −37.2708 −1.26945
\(863\) −56.8511 −1.93523 −0.967617 0.252421i \(-0.918773\pi\)
−0.967617 + 0.252421i \(0.918773\pi\)
\(864\) −3.70223 −0.125952
\(865\) 1.39379 0.0473903
\(866\) 33.8579 1.15054
\(867\) −13.8724 −0.471132
\(868\) −10.2316 −0.347283
\(869\) −14.9608 −0.507512
\(870\) −15.1489 −0.513597
\(871\) 76.3454 2.58686
\(872\) −22.8719 −0.774540
\(873\) −1.16994 −0.0395966
\(874\) −25.8255 −0.873560
\(875\) 38.6057 1.30511
\(876\) 0.187646 0.00633997
\(877\) 11.5941 0.391506 0.195753 0.980653i \(-0.437285\pi\)
0.195753 + 0.980653i \(0.437285\pi\)
\(878\) −23.9438 −0.808066
\(879\) 20.3320 0.685782
\(880\) 2.85978 0.0964030
\(881\) −4.12729 −0.139052 −0.0695259 0.997580i \(-0.522149\pi\)
−0.0695259 + 0.997580i \(0.522149\pi\)
\(882\) −6.18408 −0.208229
\(883\) −16.0992 −0.541782 −0.270891 0.962610i \(-0.587318\pi\)
−0.270891 + 0.962610i \(0.587318\pi\)
\(884\) 6.23527 0.209715
\(885\) −6.79093 −0.228275
\(886\) −15.9457 −0.535707
\(887\) 16.9188 0.568077 0.284038 0.958813i \(-0.408326\pi\)
0.284038 + 0.958813i \(0.408326\pi\)
\(888\) −29.1273 −0.977447
\(889\) 20.9463 0.702516
\(890\) 20.4266 0.684700
\(891\) −1.00000 −0.0335013
\(892\) −10.2674 −0.343777
\(893\) −54.3428 −1.81851
\(894\) −12.6773 −0.423994
\(895\) −26.5313 −0.886844
\(896\) 8.30320 0.277390
\(897\) −14.5155 −0.484659
\(898\) 18.1345 0.605156
\(899\) −41.8180 −1.39471
\(900\) 2.22249 0.0740831
\(901\) 4.27854 0.142539
\(902\) −5.63642 −0.187672
\(903\) 17.9088 0.595966
\(904\) −18.0403 −0.600010
\(905\) 33.9700 1.12920
\(906\) 0.660268 0.0219359
\(907\) 24.6453 0.818334 0.409167 0.912459i \(-0.365819\pi\)
0.409167 + 0.912459i \(0.365819\pi\)
\(908\) 16.6209 0.551582
\(909\) 1.20636 0.0400124
\(910\) −27.4762 −0.910826
\(911\) −40.8445 −1.35324 −0.676620 0.736333i \(-0.736556\pi\)
−0.676620 + 0.736333i \(0.736556\pi\)
\(912\) −17.0447 −0.564405
\(913\) 12.0244 0.397949
\(914\) −35.7005 −1.18087
\(915\) 1.33318 0.0440735
\(916\) 9.19305 0.303747
\(917\) 69.7329 2.30278
\(918\) −2.02440 −0.0668153
\(919\) 18.4138 0.607416 0.303708 0.952765i \(-0.401775\pi\)
0.303708 + 0.952765i \(0.401775\pi\)
\(920\) 11.6543 0.384232
\(921\) −15.3499 −0.505795
\(922\) 43.8989 1.44573
\(923\) 8.85584 0.291494
\(924\) −2.42874 −0.0798997
\(925\) 30.4875 1.00242
\(926\) −3.80121 −0.124915
\(927\) −7.97633 −0.261977
\(928\) −36.7505 −1.20639
\(929\) −6.06386 −0.198949 −0.0994744 0.995040i \(-0.531716\pi\)
−0.0994744 + 0.995040i \(0.531716\pi\)
\(930\) 6.42902 0.210816
\(931\) 42.9266 1.40686
\(932\) −1.59801 −0.0523447
\(933\) 9.55462 0.312804
\(934\) 19.3957 0.634647
\(935\) −2.35772 −0.0771056
\(936\) −15.7403 −0.514486
\(937\) 32.3089 1.05549 0.527743 0.849404i \(-0.323039\pi\)
0.527743 + 0.849404i \(0.323039\pi\)
\(938\) −60.2013 −1.96564
\(939\) 15.5623 0.507857
\(940\) 6.28802 0.205093
\(941\) 32.3579 1.05484 0.527419 0.849605i \(-0.323160\pi\)
0.527419 + 0.849605i \(0.323160\pi\)
\(942\) −0.468815 −0.0152748
\(943\) −13.9804 −0.455264
\(944\) 10.9266 0.355630
\(945\) −4.69505 −0.152730
\(946\) −5.82113 −0.189261
\(947\) 13.7872 0.448025 0.224013 0.974586i \(-0.428084\pi\)
0.224013 + 0.974586i \(0.428084\pi\)
\(948\) −10.3178 −0.335105
\(949\) 1.39102 0.0451543
\(950\) 29.3123 0.951016
\(951\) 15.1622 0.491669
\(952\) −19.1754 −0.621479
\(953\) −51.7560 −1.67654 −0.838270 0.545256i \(-0.816433\pi\)
−0.838270 + 0.545256i \(0.816433\pi\)
\(954\) −2.76940 −0.0896626
\(955\) −4.11412 −0.133130
\(956\) 9.20130 0.297591
\(957\) −9.92659 −0.320881
\(958\) 6.26571 0.202436
\(959\) 61.0876 1.97262
\(960\) 11.3695 0.366949
\(961\) −13.2529 −0.427514
\(962\) −55.3638 −1.78500
\(963\) 13.1670 0.424301
\(964\) 9.72075 0.313084
\(965\) −26.9367 −0.867124
\(966\) 11.4460 0.368270
\(967\) −44.6069 −1.43446 −0.717231 0.696836i \(-0.754591\pi\)
−0.717231 + 0.696836i \(0.754591\pi\)
\(968\) 3.07886 0.0989582
\(969\) 14.0523 0.451426
\(970\) 1.78545 0.0573272
\(971\) 16.2610 0.521839 0.260920 0.965361i \(-0.415974\pi\)
0.260920 + 0.965361i \(0.415974\pi\)
\(972\) −0.689651 −0.0221206
\(973\) −36.0568 −1.15593
\(974\) −42.5234 −1.36254
\(975\) 16.4753 0.527632
\(976\) −2.14508 −0.0686624
\(977\) −37.5432 −1.20111 −0.600557 0.799582i \(-0.705054\pi\)
−0.600557 + 0.799582i \(0.705054\pi\)
\(978\) −12.2784 −0.392620
\(979\) 13.3848 0.427782
\(980\) −4.96706 −0.158667
\(981\) −7.42870 −0.237180
\(982\) 42.3028 1.34994
\(983\) 52.4689 1.67350 0.836750 0.547585i \(-0.184453\pi\)
0.836750 + 0.547585i \(0.184453\pi\)
\(984\) −15.1600 −0.483283
\(985\) 27.2881 0.869470
\(986\) −20.0954 −0.639969
\(987\) 24.0851 0.766637
\(988\) 28.0154 0.891288
\(989\) −14.4386 −0.459119
\(990\) 1.52610 0.0485025
\(991\) −45.0110 −1.42982 −0.714910 0.699216i \(-0.753532\pi\)
−0.714910 + 0.699216i \(0.753532\pi\)
\(992\) 15.5965 0.495188
\(993\) −31.4774 −0.998905
\(994\) −6.98317 −0.221493
\(995\) −9.51660 −0.301696
\(996\) 8.29262 0.262762
\(997\) −11.8012 −0.373747 −0.186873 0.982384i \(-0.559835\pi\)
−0.186873 + 0.982384i \(0.559835\pi\)
\(998\) 23.1585 0.733069
\(999\) −9.46041 −0.299314
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 2013.2.a.a.1.5 11
3.2 odd 2 6039.2.a.d.1.7 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2013.2.a.a.1.5 11 1.1 even 1 trivial
6039.2.a.d.1.7 11 3.2 odd 2