Properties

Label 4235.2.a.bl.1.7
Level $4235$
Weight $2$
Character 4235.1
Self dual yes
Analytic conductor $33.817$
Analytic rank $0$
Dimension $10$
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] = [4235,2,Mod(1,4235)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4235, 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("4235.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4235 = 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4235.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: \(33.8166452560\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} - 14x^{8} + 26x^{7} + 63x^{6} - 106x^{5} - 96x^{4} + 140x^{3} + 38x^{2} - 38x - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(1.04772\) of defining polynomial
Character \(\chi\) \(=\) 4235.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.04772 q^{2} -2.73241 q^{3} -0.902293 q^{4} -1.00000 q^{5} -2.86278 q^{6} +1.00000 q^{7} -3.04078 q^{8} +4.46605 q^{9} +O(q^{10})\) \(q+1.04772 q^{2} -2.73241 q^{3} -0.902293 q^{4} -1.00000 q^{5} -2.86278 q^{6} +1.00000 q^{7} -3.04078 q^{8} +4.46605 q^{9} -1.04772 q^{10} +2.46543 q^{12} +4.59617 q^{13} +1.04772 q^{14} +2.73241 q^{15} -1.38128 q^{16} -1.64156 q^{17} +4.67915 q^{18} -2.01283 q^{19} +0.902293 q^{20} -2.73241 q^{21} -5.80988 q^{23} +8.30864 q^{24} +1.00000 q^{25} +4.81547 q^{26} -4.00585 q^{27} -0.902293 q^{28} +4.38164 q^{29} +2.86278 q^{30} -4.78061 q^{31} +4.63436 q^{32} -1.71988 q^{34} -1.00000 q^{35} -4.02969 q^{36} -9.12654 q^{37} -2.10887 q^{38} -12.5586 q^{39} +3.04078 q^{40} -6.29590 q^{41} -2.86278 q^{42} -7.18190 q^{43} -4.46605 q^{45} -6.08710 q^{46} +5.94587 q^{47} +3.77422 q^{48} +1.00000 q^{49} +1.04772 q^{50} +4.48540 q^{51} -4.14709 q^{52} -1.11529 q^{53} -4.19699 q^{54} -3.04078 q^{56} +5.49988 q^{57} +4.59071 q^{58} -0.538897 q^{59} -2.46543 q^{60} -12.6439 q^{61} -5.00871 q^{62} +4.46605 q^{63} +7.61805 q^{64} -4.59617 q^{65} +7.28624 q^{67} +1.48117 q^{68} +15.8750 q^{69} -1.04772 q^{70} +0.579271 q^{71} -13.5803 q^{72} +2.20321 q^{73} -9.56201 q^{74} -2.73241 q^{75} +1.81616 q^{76} -13.1578 q^{78} -14.5839 q^{79} +1.38128 q^{80} -2.45255 q^{81} -6.59631 q^{82} +12.9399 q^{83} +2.46543 q^{84} +1.64156 q^{85} -7.52459 q^{86} -11.9724 q^{87} +18.6247 q^{89} -4.67915 q^{90} +4.59617 q^{91} +5.24222 q^{92} +13.0626 q^{93} +6.22957 q^{94} +2.01283 q^{95} -12.6630 q^{96} +19.1597 q^{97} +1.04772 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 2 q^{2} + 4 q^{3} + 12 q^{4} - 10 q^{5} + 10 q^{7} + 6 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 2 q^{2} + 4 q^{3} + 12 q^{4} - 10 q^{5} + 10 q^{7} + 6 q^{8} + 6 q^{9} - 2 q^{10} + 16 q^{12} + 26 q^{13} + 2 q^{14} - 4 q^{15} + 20 q^{16} + 4 q^{17} + 8 q^{18} + 6 q^{19} - 12 q^{20} + 4 q^{21} - 8 q^{23} + 22 q^{24} + 10 q^{25} - 6 q^{26} + 10 q^{27} + 12 q^{28} - 4 q^{29} + 18 q^{31} + 24 q^{32} + 8 q^{34} - 10 q^{35} - 10 q^{36} - 16 q^{37} - 2 q^{38} + 16 q^{39} - 6 q^{40} - 30 q^{41} + 22 q^{43} - 6 q^{45} + 28 q^{46} + 14 q^{47} - 4 q^{48} + 10 q^{49} + 2 q^{50} + 36 q^{51} + 34 q^{52} - 10 q^{53} + 6 q^{54} + 6 q^{56} - 2 q^{57} - 38 q^{58} + 22 q^{59} - 16 q^{60} + 12 q^{61} - 6 q^{62} + 6 q^{63} + 8 q^{64} - 26 q^{65} - 14 q^{67} + 70 q^{68} + 8 q^{69} - 2 q^{70} - 8 q^{71} + 26 q^{72} + 30 q^{73} - 20 q^{74} + 4 q^{75} + 18 q^{76} - 32 q^{78} + 8 q^{79} - 20 q^{80} + 10 q^{81} - 28 q^{82} - 14 q^{83} + 16 q^{84} - 4 q^{85} - 14 q^{86} - 24 q^{87} - 6 q^{89} - 8 q^{90} + 26 q^{91} - 20 q^{92} + 14 q^{93} + 16 q^{94} - 6 q^{95} + 24 q^{96} + 20 q^{97} + 2 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.04772 0.740846 0.370423 0.928863i \(-0.379213\pi\)
0.370423 + 0.928863i \(0.379213\pi\)
\(3\) −2.73241 −1.57756 −0.788778 0.614678i \(-0.789286\pi\)
−0.788778 + 0.614678i \(0.789286\pi\)
\(4\) −0.902293 −0.451147
\(5\) −1.00000 −0.447214
\(6\) −2.86278 −1.16873
\(7\) 1.00000 0.377964
\(8\) −3.04078 −1.07508
\(9\) 4.46605 1.48868
\(10\) −1.04772 −0.331317
\(11\) 0 0
\(12\) 2.46543 0.711709
\(13\) 4.59617 1.27475 0.637374 0.770555i \(-0.280021\pi\)
0.637374 + 0.770555i \(0.280021\pi\)
\(14\) 1.04772 0.280014
\(15\) 2.73241 0.705505
\(16\) −1.38128 −0.345320
\(17\) −1.64156 −0.398136 −0.199068 0.979986i \(-0.563791\pi\)
−0.199068 + 0.979986i \(0.563791\pi\)
\(18\) 4.67915 1.10289
\(19\) −2.01283 −0.461775 −0.230888 0.972980i \(-0.574163\pi\)
−0.230888 + 0.972980i \(0.574163\pi\)
\(20\) 0.902293 0.201759
\(21\) −2.73241 −0.596260
\(22\) 0 0
\(23\) −5.80988 −1.21144 −0.605722 0.795676i \(-0.707116\pi\)
−0.605722 + 0.795676i \(0.707116\pi\)
\(24\) 8.30864 1.69599
\(25\) 1.00000 0.200000
\(26\) 4.81547 0.944392
\(27\) −4.00585 −0.770925
\(28\) −0.902293 −0.170517
\(29\) 4.38164 0.813650 0.406825 0.913506i \(-0.366636\pi\)
0.406825 + 0.913506i \(0.366636\pi\)
\(30\) 2.86278 0.522671
\(31\) −4.78061 −0.858622 −0.429311 0.903157i \(-0.641244\pi\)
−0.429311 + 0.903157i \(0.641244\pi\)
\(32\) 4.63436 0.819248
\(33\) 0 0
\(34\) −1.71988 −0.294958
\(35\) −1.00000 −0.169031
\(36\) −4.02969 −0.671614
\(37\) −9.12654 −1.50039 −0.750197 0.661214i \(-0.770041\pi\)
−0.750197 + 0.661214i \(0.770041\pi\)
\(38\) −2.10887 −0.342105
\(39\) −12.5586 −2.01099
\(40\) 3.04078 0.480789
\(41\) −6.29590 −0.983255 −0.491627 0.870806i \(-0.663598\pi\)
−0.491627 + 0.870806i \(0.663598\pi\)
\(42\) −2.86278 −0.441737
\(43\) −7.18190 −1.09523 −0.547615 0.836730i \(-0.684464\pi\)
−0.547615 + 0.836730i \(0.684464\pi\)
\(44\) 0 0
\(45\) −4.46605 −0.665759
\(46\) −6.08710 −0.897494
\(47\) 5.94587 0.867294 0.433647 0.901083i \(-0.357227\pi\)
0.433647 + 0.901083i \(0.357227\pi\)
\(48\) 3.77422 0.544762
\(49\) 1.00000 0.142857
\(50\) 1.04772 0.148169
\(51\) 4.48540 0.628082
\(52\) −4.14709 −0.575098
\(53\) −1.11529 −0.153197 −0.0765987 0.997062i \(-0.524406\pi\)
−0.0765987 + 0.997062i \(0.524406\pi\)
\(54\) −4.19699 −0.571137
\(55\) 0 0
\(56\) −3.04078 −0.406341
\(57\) 5.49988 0.728476
\(58\) 4.59071 0.602789
\(59\) −0.538897 −0.0701584 −0.0350792 0.999385i \(-0.511168\pi\)
−0.0350792 + 0.999385i \(0.511168\pi\)
\(60\) −2.46543 −0.318286
\(61\) −12.6439 −1.61888 −0.809441 0.587201i \(-0.800230\pi\)
−0.809441 + 0.587201i \(0.800230\pi\)
\(62\) −5.00871 −0.636107
\(63\) 4.46605 0.562669
\(64\) 7.61805 0.952257
\(65\) −4.59617 −0.570084
\(66\) 0 0
\(67\) 7.28624 0.890155 0.445078 0.895492i \(-0.353176\pi\)
0.445078 + 0.895492i \(0.353176\pi\)
\(68\) 1.48117 0.179618
\(69\) 15.8750 1.91112
\(70\) −1.04772 −0.125226
\(71\) 0.579271 0.0687468 0.0343734 0.999409i \(-0.489056\pi\)
0.0343734 + 0.999409i \(0.489056\pi\)
\(72\) −13.5803 −1.60045
\(73\) 2.20321 0.257866 0.128933 0.991653i \(-0.458845\pi\)
0.128933 + 0.991653i \(0.458845\pi\)
\(74\) −9.56201 −1.11156
\(75\) −2.73241 −0.315511
\(76\) 1.81616 0.208328
\(77\) 0 0
\(78\) −13.1578 −1.48983
\(79\) −14.5839 −1.64081 −0.820406 0.571781i \(-0.806253\pi\)
−0.820406 + 0.571781i \(0.806253\pi\)
\(80\) 1.38128 0.154432
\(81\) −2.45255 −0.272505
\(82\) −6.59631 −0.728441
\(83\) 12.9399 1.42034 0.710171 0.704029i \(-0.248618\pi\)
0.710171 + 0.704029i \(0.248618\pi\)
\(84\) 2.46543 0.269001
\(85\) 1.64156 0.178052
\(86\) −7.52459 −0.811397
\(87\) −11.9724 −1.28358
\(88\) 0 0
\(89\) 18.6247 1.97421 0.987106 0.160069i \(-0.0511718\pi\)
0.987106 + 0.160069i \(0.0511718\pi\)
\(90\) −4.67915 −0.493226
\(91\) 4.59617 0.481809
\(92\) 5.24222 0.546539
\(93\) 13.0626 1.35452
\(94\) 6.22957 0.642531
\(95\) 2.01283 0.206512
\(96\) −12.6630 −1.29241
\(97\) 19.1597 1.94537 0.972685 0.232129i \(-0.0745693\pi\)
0.972685 + 0.232129i \(0.0745693\pi\)
\(98\) 1.04772 0.105835
\(99\) 0 0
\(100\) −0.902293 −0.0902293
\(101\) 14.8769 1.48031 0.740154 0.672438i \(-0.234753\pi\)
0.740154 + 0.672438i \(0.234753\pi\)
\(102\) 4.69942 0.465312
\(103\) −4.95404 −0.488136 −0.244068 0.969758i \(-0.578482\pi\)
−0.244068 + 0.969758i \(0.578482\pi\)
\(104\) −13.9759 −1.37045
\(105\) 2.73241 0.266656
\(106\) −1.16851 −0.113496
\(107\) −4.85171 −0.469032 −0.234516 0.972112i \(-0.575351\pi\)
−0.234516 + 0.972112i \(0.575351\pi\)
\(108\) 3.61445 0.347800
\(109\) 10.0461 0.962245 0.481123 0.876653i \(-0.340229\pi\)
0.481123 + 0.876653i \(0.340229\pi\)
\(110\) 0 0
\(111\) 24.9374 2.36696
\(112\) −1.38128 −0.130519
\(113\) 9.97995 0.938835 0.469417 0.882976i \(-0.344464\pi\)
0.469417 + 0.882976i \(0.344464\pi\)
\(114\) 5.76230 0.539689
\(115\) 5.80988 0.541774
\(116\) −3.95352 −0.367075
\(117\) 20.5267 1.89769
\(118\) −0.564610 −0.0519766
\(119\) −1.64156 −0.150481
\(120\) −8.30864 −0.758472
\(121\) 0 0
\(122\) −13.2472 −1.19934
\(123\) 17.2030 1.55114
\(124\) 4.31351 0.387364
\(125\) −1.00000 −0.0894427
\(126\) 4.67915 0.416852
\(127\) 11.7630 1.04380 0.521899 0.853008i \(-0.325224\pi\)
0.521899 + 0.853008i \(0.325224\pi\)
\(128\) −1.28718 −0.113772
\(129\) 19.6239 1.72779
\(130\) −4.81547 −0.422345
\(131\) −12.9246 −1.12923 −0.564615 0.825355i \(-0.690975\pi\)
−0.564615 + 0.825355i \(0.690975\pi\)
\(132\) 0 0
\(133\) −2.01283 −0.174535
\(134\) 7.63390 0.659468
\(135\) 4.00585 0.344768
\(136\) 4.99161 0.428027
\(137\) −19.5900 −1.67369 −0.836845 0.547440i \(-0.815603\pi\)
−0.836845 + 0.547440i \(0.815603\pi\)
\(138\) 16.6324 1.41585
\(139\) −2.63936 −0.223868 −0.111934 0.993716i \(-0.535704\pi\)
−0.111934 + 0.993716i \(0.535704\pi\)
\(140\) 0.902293 0.0762577
\(141\) −16.2465 −1.36820
\(142\) 0.606911 0.0509308
\(143\) 0 0
\(144\) −6.16887 −0.514072
\(145\) −4.38164 −0.363875
\(146\) 2.30833 0.191039
\(147\) −2.73241 −0.225365
\(148\) 8.23481 0.676898
\(149\) −8.51343 −0.697447 −0.348724 0.937226i \(-0.613385\pi\)
−0.348724 + 0.937226i \(0.613385\pi\)
\(150\) −2.86278 −0.233745
\(151\) −4.15842 −0.338408 −0.169204 0.985581i \(-0.554120\pi\)
−0.169204 + 0.985581i \(0.554120\pi\)
\(152\) 6.12057 0.496444
\(153\) −7.33127 −0.592698
\(154\) 0 0
\(155\) 4.78061 0.383988
\(156\) 11.3315 0.907249
\(157\) 14.7170 1.17454 0.587271 0.809391i \(-0.300202\pi\)
0.587271 + 0.809391i \(0.300202\pi\)
\(158\) −15.2797 −1.21559
\(159\) 3.04744 0.241677
\(160\) −4.63436 −0.366379
\(161\) −5.80988 −0.457883
\(162\) −2.56957 −0.201884
\(163\) −10.3649 −0.811838 −0.405919 0.913909i \(-0.633048\pi\)
−0.405919 + 0.913909i \(0.633048\pi\)
\(164\) 5.68075 0.443592
\(165\) 0 0
\(166\) 13.5574 1.05226
\(167\) −15.6253 −1.20912 −0.604561 0.796558i \(-0.706652\pi\)
−0.604561 + 0.796558i \(0.706652\pi\)
\(168\) 8.30864 0.641025
\(169\) 8.12474 0.624980
\(170\) 1.71988 0.131909
\(171\) −8.98941 −0.687437
\(172\) 6.48018 0.494109
\(173\) 6.72314 0.511151 0.255575 0.966789i \(-0.417735\pi\)
0.255575 + 0.966789i \(0.417735\pi\)
\(174\) −12.5437 −0.950934
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 1.47249 0.110679
\(178\) 19.5134 1.46259
\(179\) 2.10575 0.157391 0.0786955 0.996899i \(-0.474925\pi\)
0.0786955 + 0.996899i \(0.474925\pi\)
\(180\) 4.02969 0.300355
\(181\) −3.31433 −0.246352 −0.123176 0.992385i \(-0.539308\pi\)
−0.123176 + 0.992385i \(0.539308\pi\)
\(182\) 4.81547 0.356947
\(183\) 34.5482 2.55388
\(184\) 17.6665 1.30240
\(185\) 9.12654 0.670997
\(186\) 13.6858 1.00349
\(187\) 0 0
\(188\) −5.36491 −0.391277
\(189\) −4.00585 −0.291382
\(190\) 2.10887 0.152994
\(191\) 22.6133 1.63624 0.818122 0.575045i \(-0.195016\pi\)
0.818122 + 0.575045i \(0.195016\pi\)
\(192\) −20.8156 −1.50224
\(193\) 8.80536 0.633824 0.316912 0.948455i \(-0.397354\pi\)
0.316912 + 0.948455i \(0.397354\pi\)
\(194\) 20.0739 1.44122
\(195\) 12.5586 0.899340
\(196\) −0.902293 −0.0644495
\(197\) 4.47451 0.318796 0.159398 0.987214i \(-0.449045\pi\)
0.159398 + 0.987214i \(0.449045\pi\)
\(198\) 0 0
\(199\) 16.1857 1.14737 0.573686 0.819075i \(-0.305513\pi\)
0.573686 + 0.819075i \(0.305513\pi\)
\(200\) −3.04078 −0.215015
\(201\) −19.9090 −1.40427
\(202\) 15.5868 1.09668
\(203\) 4.38164 0.307531
\(204\) −4.04715 −0.283357
\(205\) 6.29590 0.439725
\(206\) −5.19042 −0.361634
\(207\) −25.9472 −1.80346
\(208\) −6.34860 −0.440196
\(209\) 0 0
\(210\) 2.86278 0.197551
\(211\) 9.05146 0.623128 0.311564 0.950225i \(-0.399147\pi\)
0.311564 + 0.950225i \(0.399147\pi\)
\(212\) 1.00632 0.0691145
\(213\) −1.58280 −0.108452
\(214\) −5.08321 −0.347481
\(215\) 7.18190 0.489802
\(216\) 12.1809 0.828804
\(217\) −4.78061 −0.324529
\(218\) 10.5255 0.712876
\(219\) −6.02006 −0.406798
\(220\) 0 0
\(221\) −7.54487 −0.507523
\(222\) 26.1273 1.75355
\(223\) 20.9541 1.40319 0.701596 0.712575i \(-0.252471\pi\)
0.701596 + 0.712575i \(0.252471\pi\)
\(224\) 4.63436 0.309646
\(225\) 4.46605 0.297737
\(226\) 10.4561 0.695532
\(227\) −13.9832 −0.928100 −0.464050 0.885809i \(-0.653604\pi\)
−0.464050 + 0.885809i \(0.653604\pi\)
\(228\) −4.96250 −0.328650
\(229\) 18.7995 1.24230 0.621152 0.783690i \(-0.286665\pi\)
0.621152 + 0.783690i \(0.286665\pi\)
\(230\) 6.08710 0.401372
\(231\) 0 0
\(232\) −13.3236 −0.874736
\(233\) −4.22660 −0.276894 −0.138447 0.990370i \(-0.544211\pi\)
−0.138447 + 0.990370i \(0.544211\pi\)
\(234\) 21.5061 1.40590
\(235\) −5.94587 −0.387866
\(236\) 0.486243 0.0316517
\(237\) 39.8490 2.58847
\(238\) −1.71988 −0.111483
\(239\) 15.9435 1.03130 0.515648 0.856800i \(-0.327551\pi\)
0.515648 + 0.856800i \(0.327551\pi\)
\(240\) −3.77422 −0.243625
\(241\) 18.1054 1.16627 0.583137 0.812374i \(-0.301825\pi\)
0.583137 + 0.812374i \(0.301825\pi\)
\(242\) 0 0
\(243\) 18.7189 1.20082
\(244\) 11.4085 0.730353
\(245\) −1.00000 −0.0638877
\(246\) 18.0238 1.14916
\(247\) −9.25131 −0.588647
\(248\) 14.5368 0.923085
\(249\) −35.3572 −2.24067
\(250\) −1.04772 −0.0662633
\(251\) −1.36675 −0.0862683 −0.0431342 0.999069i \(-0.513734\pi\)
−0.0431342 + 0.999069i \(0.513734\pi\)
\(252\) −4.02969 −0.253846
\(253\) 0 0
\(254\) 12.3243 0.773293
\(255\) −4.48540 −0.280887
\(256\) −16.5847 −1.03654
\(257\) −7.73846 −0.482712 −0.241356 0.970437i \(-0.577592\pi\)
−0.241356 + 0.970437i \(0.577592\pi\)
\(258\) 20.5602 1.28002
\(259\) −9.12654 −0.567096
\(260\) 4.14709 0.257192
\(261\) 19.5686 1.21127
\(262\) −13.5413 −0.836585
\(263\) 17.2350 1.06275 0.531377 0.847135i \(-0.321675\pi\)
0.531377 + 0.847135i \(0.321675\pi\)
\(264\) 0 0
\(265\) 1.11529 0.0685119
\(266\) −2.10887 −0.129303
\(267\) −50.8902 −3.11443
\(268\) −6.57432 −0.401591
\(269\) 7.53103 0.459175 0.229588 0.973288i \(-0.426262\pi\)
0.229588 + 0.973288i \(0.426262\pi\)
\(270\) 4.19699 0.255420
\(271\) 8.01922 0.487133 0.243567 0.969884i \(-0.421683\pi\)
0.243567 + 0.969884i \(0.421683\pi\)
\(272\) 2.26745 0.137484
\(273\) −12.5586 −0.760081
\(274\) −20.5248 −1.23995
\(275\) 0 0
\(276\) −14.3239 −0.862196
\(277\) 16.0763 0.965930 0.482965 0.875640i \(-0.339560\pi\)
0.482965 + 0.875640i \(0.339560\pi\)
\(278\) −2.76530 −0.165852
\(279\) −21.3504 −1.27822
\(280\) 3.04078 0.181721
\(281\) −8.74780 −0.521850 −0.260925 0.965359i \(-0.584028\pi\)
−0.260925 + 0.965359i \(0.584028\pi\)
\(282\) −17.0217 −1.01363
\(283\) 25.2988 1.50386 0.751930 0.659243i \(-0.229123\pi\)
0.751930 + 0.659243i \(0.229123\pi\)
\(284\) −0.522672 −0.0310149
\(285\) −5.49988 −0.325785
\(286\) 0 0
\(287\) −6.29590 −0.371635
\(288\) 20.6973 1.21960
\(289\) −14.3053 −0.841488
\(290\) −4.59071 −0.269576
\(291\) −52.3520 −3.06893
\(292\) −1.98794 −0.116335
\(293\) 22.2043 1.29719 0.648593 0.761135i \(-0.275358\pi\)
0.648593 + 0.761135i \(0.275358\pi\)
\(294\) −2.86278 −0.166961
\(295\) 0.538897 0.0313758
\(296\) 27.7518 1.61304
\(297\) 0 0
\(298\) −8.91965 −0.516701
\(299\) −26.7032 −1.54428
\(300\) 2.46543 0.142342
\(301\) −7.18190 −0.413958
\(302\) −4.35684 −0.250708
\(303\) −40.6498 −2.33527
\(304\) 2.78029 0.159460
\(305\) 12.6439 0.723986
\(306\) −7.68108 −0.439098
\(307\) 18.1007 1.03306 0.516530 0.856269i \(-0.327224\pi\)
0.516530 + 0.856269i \(0.327224\pi\)
\(308\) 0 0
\(309\) 13.5365 0.770062
\(310\) 5.00871 0.284476
\(311\) −29.0659 −1.64818 −0.824088 0.566462i \(-0.808312\pi\)
−0.824088 + 0.566462i \(0.808312\pi\)
\(312\) 38.1879 2.16196
\(313\) 26.6162 1.50444 0.752219 0.658913i \(-0.228983\pi\)
0.752219 + 0.658913i \(0.228983\pi\)
\(314\) 15.4192 0.870155
\(315\) −4.46605 −0.251633
\(316\) 13.1589 0.740247
\(317\) −30.2170 −1.69716 −0.848579 0.529069i \(-0.822541\pi\)
−0.848579 + 0.529069i \(0.822541\pi\)
\(318\) 3.19284 0.179046
\(319\) 0 0
\(320\) −7.61805 −0.425862
\(321\) 13.2568 0.739925
\(322\) −6.08710 −0.339221
\(323\) 3.30418 0.183849
\(324\) 2.21292 0.122940
\(325\) 4.59617 0.254949
\(326\) −10.8594 −0.601447
\(327\) −27.4501 −1.51800
\(328\) 19.1444 1.05707
\(329\) 5.94587 0.327806
\(330\) 0 0
\(331\) 17.1535 0.942843 0.471422 0.881908i \(-0.343741\pi\)
0.471422 + 0.881908i \(0.343741\pi\)
\(332\) −11.6756 −0.640783
\(333\) −40.7596 −2.23361
\(334\) −16.3709 −0.895774
\(335\) −7.28624 −0.398090
\(336\) 3.77422 0.205901
\(337\) 25.5559 1.39212 0.696058 0.717985i \(-0.254935\pi\)
0.696058 + 0.717985i \(0.254935\pi\)
\(338\) 8.51242 0.463014
\(339\) −27.2693 −1.48106
\(340\) −1.48117 −0.0803275
\(341\) 0 0
\(342\) −9.41834 −0.509285
\(343\) 1.00000 0.0539949
\(344\) 21.8386 1.17746
\(345\) −15.8750 −0.854679
\(346\) 7.04393 0.378684
\(347\) −34.7316 −1.86449 −0.932246 0.361825i \(-0.882154\pi\)
−0.932246 + 0.361825i \(0.882154\pi\)
\(348\) 10.8026 0.579082
\(349\) −0.287691 −0.0153997 −0.00769987 0.999970i \(-0.502451\pi\)
−0.00769987 + 0.999970i \(0.502451\pi\)
\(350\) 1.04772 0.0560027
\(351\) −18.4115 −0.982735
\(352\) 0 0
\(353\) 3.86597 0.205765 0.102882 0.994694i \(-0.467193\pi\)
0.102882 + 0.994694i \(0.467193\pi\)
\(354\) 1.54275 0.0819960
\(355\) −0.579271 −0.0307445
\(356\) −16.8049 −0.890659
\(357\) 4.48540 0.237393
\(358\) 2.20622 0.116603
\(359\) −15.6846 −0.827802 −0.413901 0.910322i \(-0.635834\pi\)
−0.413901 + 0.910322i \(0.635834\pi\)
\(360\) 13.5803 0.715743
\(361\) −14.9485 −0.786764
\(362\) −3.47247 −0.182509
\(363\) 0 0
\(364\) −4.14709 −0.217367
\(365\) −2.20321 −0.115321
\(366\) 36.1967 1.89203
\(367\) −8.65212 −0.451637 −0.225819 0.974169i \(-0.572506\pi\)
−0.225819 + 0.974169i \(0.572506\pi\)
\(368\) 8.02508 0.418336
\(369\) −28.1178 −1.46376
\(370\) 9.56201 0.497105
\(371\) −1.11529 −0.0579032
\(372\) −11.7863 −0.611089
\(373\) 31.5858 1.63545 0.817726 0.575607i \(-0.195234\pi\)
0.817726 + 0.575607i \(0.195234\pi\)
\(374\) 0 0
\(375\) 2.73241 0.141101
\(376\) −18.0800 −0.932407
\(377\) 20.1387 1.03720
\(378\) −4.19699 −0.215870
\(379\) −21.0511 −1.08132 −0.540661 0.841240i \(-0.681826\pi\)
−0.540661 + 0.841240i \(0.681826\pi\)
\(380\) −1.81616 −0.0931673
\(381\) −32.1413 −1.64665
\(382\) 23.6923 1.21221
\(383\) −8.01880 −0.409741 −0.204871 0.978789i \(-0.565677\pi\)
−0.204871 + 0.978789i \(0.565677\pi\)
\(384\) 3.51709 0.179481
\(385\) 0 0
\(386\) 9.22551 0.469566
\(387\) −32.0747 −1.63045
\(388\) −17.2876 −0.877647
\(389\) −7.90323 −0.400710 −0.200355 0.979723i \(-0.564209\pi\)
−0.200355 + 0.979723i \(0.564209\pi\)
\(390\) 13.1578 0.666273
\(391\) 9.53725 0.482319
\(392\) −3.04078 −0.153582
\(393\) 35.3153 1.78142
\(394\) 4.68801 0.236179
\(395\) 14.5839 0.733794
\(396\) 0 0
\(397\) 32.8305 1.64772 0.823858 0.566797i \(-0.191818\pi\)
0.823858 + 0.566797i \(0.191818\pi\)
\(398\) 16.9580 0.850027
\(399\) 5.49988 0.275338
\(400\) −1.38128 −0.0690640
\(401\) −36.1779 −1.80664 −0.903319 0.428969i \(-0.858877\pi\)
−0.903319 + 0.428969i \(0.858877\pi\)
\(402\) −20.8589 −1.04035
\(403\) −21.9725 −1.09453
\(404\) −13.4233 −0.667836
\(405\) 2.45255 0.121868
\(406\) 4.59071 0.227833
\(407\) 0 0
\(408\) −13.6391 −0.675236
\(409\) −11.1641 −0.552027 −0.276013 0.961154i \(-0.589013\pi\)
−0.276013 + 0.961154i \(0.589013\pi\)
\(410\) 6.59631 0.325769
\(411\) 53.5280 2.64034
\(412\) 4.47000 0.220221
\(413\) −0.538897 −0.0265174
\(414\) −27.1853 −1.33608
\(415\) −12.9399 −0.635196
\(416\) 21.3003 1.04433
\(417\) 7.21181 0.353164
\(418\) 0 0
\(419\) 28.1200 1.37375 0.686877 0.726774i \(-0.258981\pi\)
0.686877 + 0.726774i \(0.258981\pi\)
\(420\) −2.46543 −0.120301
\(421\) −0.681076 −0.0331936 −0.0165968 0.999862i \(-0.505283\pi\)
−0.0165968 + 0.999862i \(0.505283\pi\)
\(422\) 9.48335 0.461642
\(423\) 26.5545 1.29113
\(424\) 3.39136 0.164699
\(425\) −1.64156 −0.0796272
\(426\) −1.65833 −0.0803463
\(427\) −12.6439 −0.611880
\(428\) 4.37766 0.211602
\(429\) 0 0
\(430\) 7.52459 0.362868
\(431\) −29.2893 −1.41081 −0.705407 0.708803i \(-0.749236\pi\)
−0.705407 + 0.708803i \(0.749236\pi\)
\(432\) 5.53320 0.266216
\(433\) −30.6718 −1.47399 −0.736996 0.675897i \(-0.763756\pi\)
−0.736996 + 0.675897i \(0.763756\pi\)
\(434\) −5.00871 −0.240426
\(435\) 11.9724 0.574034
\(436\) −9.06456 −0.434114
\(437\) 11.6943 0.559415
\(438\) −6.30731 −0.301375
\(439\) −13.4053 −0.639801 −0.319901 0.947451i \(-0.603650\pi\)
−0.319901 + 0.947451i \(0.603650\pi\)
\(440\) 0 0
\(441\) 4.46605 0.212669
\(442\) −7.90487 −0.375996
\(443\) 28.2929 1.34424 0.672118 0.740444i \(-0.265385\pi\)
0.672118 + 0.740444i \(0.265385\pi\)
\(444\) −22.5009 −1.06784
\(445\) −18.6247 −0.882894
\(446\) 21.9540 1.03955
\(447\) 23.2621 1.10026
\(448\) 7.61805 0.359919
\(449\) 21.3998 1.00992 0.504960 0.863143i \(-0.331507\pi\)
0.504960 + 0.863143i \(0.331507\pi\)
\(450\) 4.67915 0.220577
\(451\) 0 0
\(452\) −9.00484 −0.423552
\(453\) 11.3625 0.533857
\(454\) −14.6504 −0.687580
\(455\) −4.59617 −0.215472
\(456\) −16.7239 −0.783168
\(457\) 32.5047 1.52051 0.760254 0.649626i \(-0.225075\pi\)
0.760254 + 0.649626i \(0.225075\pi\)
\(458\) 19.6965 0.920357
\(459\) 6.57582 0.306933
\(460\) −5.24222 −0.244420
\(461\) 17.1768 0.800002 0.400001 0.916515i \(-0.369010\pi\)
0.400001 + 0.916515i \(0.369010\pi\)
\(462\) 0 0
\(463\) 7.30879 0.339668 0.169834 0.985473i \(-0.445677\pi\)
0.169834 + 0.985473i \(0.445677\pi\)
\(464\) −6.05227 −0.280970
\(465\) −13.0626 −0.605762
\(466\) −4.42828 −0.205136
\(467\) −22.7764 −1.05397 −0.526984 0.849875i \(-0.676677\pi\)
−0.526984 + 0.849875i \(0.676677\pi\)
\(468\) −18.5211 −0.856139
\(469\) 7.28624 0.336447
\(470\) −6.22957 −0.287349
\(471\) −40.2127 −1.85290
\(472\) 1.63866 0.0754257
\(473\) 0 0
\(474\) 41.7504 1.91766
\(475\) −2.01283 −0.0923551
\(476\) 1.48117 0.0678891
\(477\) −4.98096 −0.228062
\(478\) 16.7042 0.764033
\(479\) 2.28452 0.104382 0.0521911 0.998637i \(-0.483380\pi\)
0.0521911 + 0.998637i \(0.483380\pi\)
\(480\) 12.6630 0.577983
\(481\) −41.9471 −1.91262
\(482\) 18.9693 0.864030
\(483\) 15.8750 0.722336
\(484\) 0 0
\(485\) −19.1597 −0.869996
\(486\) 19.6121 0.889621
\(487\) 40.9710 1.85657 0.928286 0.371867i \(-0.121282\pi\)
0.928286 + 0.371867i \(0.121282\pi\)
\(488\) 38.4472 1.74042
\(489\) 28.3210 1.28072
\(490\) −1.04772 −0.0473309
\(491\) −5.25167 −0.237004 −0.118502 0.992954i \(-0.537809\pi\)
−0.118502 + 0.992954i \(0.537809\pi\)
\(492\) −15.5221 −0.699791
\(493\) −7.19270 −0.323943
\(494\) −9.69274 −0.436097
\(495\) 0 0
\(496\) 6.60336 0.296500
\(497\) 0.579271 0.0259839
\(498\) −37.0442 −1.65999
\(499\) −17.3766 −0.777885 −0.388943 0.921262i \(-0.627160\pi\)
−0.388943 + 0.921262i \(0.627160\pi\)
\(500\) 0.902293 0.0403518
\(501\) 42.6947 1.90746
\(502\) −1.43196 −0.0639116
\(503\) 2.47225 0.110232 0.0551162 0.998480i \(-0.482447\pi\)
0.0551162 + 0.998480i \(0.482447\pi\)
\(504\) −13.5803 −0.604913
\(505\) −14.8769 −0.662014
\(506\) 0 0
\(507\) −22.2001 −0.985941
\(508\) −10.6137 −0.470906
\(509\) 1.26310 0.0559860 0.0279930 0.999608i \(-0.491088\pi\)
0.0279930 + 0.999608i \(0.491088\pi\)
\(510\) −4.69942 −0.208094
\(511\) 2.20321 0.0974641
\(512\) −14.8017 −0.654148
\(513\) 8.06309 0.355994
\(514\) −8.10770 −0.357615
\(515\) 4.95404 0.218301
\(516\) −17.7065 −0.779485
\(517\) 0 0
\(518\) −9.56201 −0.420131
\(519\) −18.3704 −0.806369
\(520\) 13.9759 0.612884
\(521\) −19.9483 −0.873953 −0.436977 0.899473i \(-0.643951\pi\)
−0.436977 + 0.899473i \(0.643951\pi\)
\(522\) 20.5023 0.897363
\(523\) 13.8834 0.607077 0.303538 0.952819i \(-0.401832\pi\)
0.303538 + 0.952819i \(0.401832\pi\)
\(524\) 11.6618 0.509448
\(525\) −2.73241 −0.119252
\(526\) 18.0573 0.787338
\(527\) 7.84763 0.341848
\(528\) 0 0
\(529\) 10.7547 0.467597
\(530\) 1.16851 0.0507568
\(531\) −2.40674 −0.104444
\(532\) 1.81616 0.0787407
\(533\) −28.9370 −1.25340
\(534\) −53.3184 −2.30731
\(535\) 4.85171 0.209758
\(536\) −22.1558 −0.956985
\(537\) −5.75376 −0.248293
\(538\) 7.89038 0.340178
\(539\) 0 0
\(540\) −3.61445 −0.155541
\(541\) 8.27453 0.355750 0.177875 0.984053i \(-0.443078\pi\)
0.177875 + 0.984053i \(0.443078\pi\)
\(542\) 8.40186 0.360891
\(543\) 9.05610 0.388635
\(544\) −7.60757 −0.326172
\(545\) −10.0461 −0.430329
\(546\) −13.1578 −0.563103
\(547\) −30.5425 −1.30590 −0.652952 0.757399i \(-0.726470\pi\)
−0.652952 + 0.757399i \(0.726470\pi\)
\(548\) 17.6760 0.755080
\(549\) −56.4682 −2.41000
\(550\) 0 0
\(551\) −8.81950 −0.375723
\(552\) −48.2722 −2.05460
\(553\) −14.5839 −0.620169
\(554\) 16.8434 0.715606
\(555\) −24.9374 −1.05853
\(556\) 2.38148 0.100997
\(557\) −25.2508 −1.06991 −0.534956 0.844880i \(-0.679672\pi\)
−0.534956 + 0.844880i \(0.679672\pi\)
\(558\) −22.3692 −0.946962
\(559\) −33.0092 −1.39614
\(560\) 1.38128 0.0583698
\(561\) 0 0
\(562\) −9.16520 −0.386611
\(563\) 0.489020 0.0206098 0.0103049 0.999947i \(-0.496720\pi\)
0.0103049 + 0.999947i \(0.496720\pi\)
\(564\) 14.6591 0.617261
\(565\) −9.97995 −0.419860
\(566\) 26.5060 1.11413
\(567\) −2.45255 −0.102997
\(568\) −1.76143 −0.0739081
\(569\) −7.77347 −0.325881 −0.162940 0.986636i \(-0.552098\pi\)
−0.162940 + 0.986636i \(0.552098\pi\)
\(570\) −5.76230 −0.241356
\(571\) 6.91474 0.289373 0.144686 0.989478i \(-0.453783\pi\)
0.144686 + 0.989478i \(0.453783\pi\)
\(572\) 0 0
\(573\) −61.7888 −2.58127
\(574\) −6.59631 −0.275325
\(575\) −5.80988 −0.242289
\(576\) 34.0226 1.41761
\(577\) 24.7118 1.02876 0.514382 0.857561i \(-0.328021\pi\)
0.514382 + 0.857561i \(0.328021\pi\)
\(578\) −14.9879 −0.623413
\(579\) −24.0598 −0.999892
\(580\) 3.95352 0.164161
\(581\) 12.9399 0.536839
\(582\) −54.8500 −2.27361
\(583\) 0 0
\(584\) −6.69946 −0.277226
\(585\) −20.5267 −0.848675
\(586\) 23.2637 0.961016
\(587\) 47.3790 1.95554 0.977771 0.209677i \(-0.0672414\pi\)
0.977771 + 0.209677i \(0.0672414\pi\)
\(588\) 2.46543 0.101673
\(589\) 9.62256 0.396490
\(590\) 0.564610 0.0232446
\(591\) −12.2262 −0.502918
\(592\) 12.6063 0.518116
\(593\) 4.36324 0.179177 0.0895884 0.995979i \(-0.471445\pi\)
0.0895884 + 0.995979i \(0.471445\pi\)
\(594\) 0 0
\(595\) 1.64156 0.0672972
\(596\) 7.68161 0.314651
\(597\) −44.2259 −1.81004
\(598\) −27.9773 −1.14408
\(599\) 27.1050 1.10748 0.553741 0.832689i \(-0.313200\pi\)
0.553741 + 0.832689i \(0.313200\pi\)
\(600\) 8.30864 0.339199
\(601\) 9.94004 0.405463 0.202731 0.979234i \(-0.435018\pi\)
0.202731 + 0.979234i \(0.435018\pi\)
\(602\) −7.52459 −0.306679
\(603\) 32.5407 1.32516
\(604\) 3.75212 0.152671
\(605\) 0 0
\(606\) −42.5894 −1.73008
\(607\) 21.9031 0.889020 0.444510 0.895774i \(-0.353378\pi\)
0.444510 + 0.895774i \(0.353378\pi\)
\(608\) −9.32819 −0.378308
\(609\) −11.9724 −0.485147
\(610\) 13.2472 0.536363
\(611\) 27.3282 1.10558
\(612\) 6.61496 0.267394
\(613\) 40.5461 1.63764 0.818821 0.574049i \(-0.194628\pi\)
0.818821 + 0.574049i \(0.194628\pi\)
\(614\) 18.9643 0.765339
\(615\) −17.2030 −0.693691
\(616\) 0 0
\(617\) −31.8520 −1.28231 −0.641156 0.767411i \(-0.721545\pi\)
−0.641156 + 0.767411i \(0.721545\pi\)
\(618\) 14.1823 0.570498
\(619\) −2.21449 −0.0890077 −0.0445039 0.999009i \(-0.514171\pi\)
−0.0445039 + 0.999009i \(0.514171\pi\)
\(620\) −4.31351 −0.173235
\(621\) 23.2735 0.933933
\(622\) −30.4528 −1.22105
\(623\) 18.6247 0.746182
\(624\) 17.3469 0.694434
\(625\) 1.00000 0.0400000
\(626\) 27.8862 1.11456
\(627\) 0 0
\(628\) −13.2790 −0.529890
\(629\) 14.9817 0.597361
\(630\) −4.67915 −0.186422
\(631\) −20.9442 −0.833776 −0.416888 0.908958i \(-0.636879\pi\)
−0.416888 + 0.908958i \(0.636879\pi\)
\(632\) 44.3463 1.76400
\(633\) −24.7323 −0.983019
\(634\) −31.6588 −1.25733
\(635\) −11.7630 −0.466800
\(636\) −2.74968 −0.109032
\(637\) 4.59617 0.182107
\(638\) 0 0
\(639\) 2.58705 0.102342
\(640\) 1.28718 0.0508802
\(641\) 31.3010 1.23631 0.618157 0.786055i \(-0.287880\pi\)
0.618157 + 0.786055i \(0.287880\pi\)
\(642\) 13.8894 0.548171
\(643\) 19.2816 0.760392 0.380196 0.924906i \(-0.375856\pi\)
0.380196 + 0.924906i \(0.375856\pi\)
\(644\) 5.24222 0.206572
\(645\) −19.6239 −0.772690
\(646\) 3.46184 0.136204
\(647\) −12.0653 −0.474337 −0.237168 0.971469i \(-0.576219\pi\)
−0.237168 + 0.971469i \(0.576219\pi\)
\(648\) 7.45764 0.292964
\(649\) 0 0
\(650\) 4.81547 0.188878
\(651\) 13.0626 0.511962
\(652\) 9.35214 0.366258
\(653\) 20.3108 0.794824 0.397412 0.917640i \(-0.369908\pi\)
0.397412 + 0.917640i \(0.369908\pi\)
\(654\) −28.7599 −1.12460
\(655\) 12.9246 0.505007
\(656\) 8.69641 0.339538
\(657\) 9.83963 0.383881
\(658\) 6.22957 0.242854
\(659\) −13.0845 −0.509699 −0.254850 0.966981i \(-0.582026\pi\)
−0.254850 + 0.966981i \(0.582026\pi\)
\(660\) 0 0
\(661\) −23.5447 −0.915782 −0.457891 0.889008i \(-0.651395\pi\)
−0.457891 + 0.889008i \(0.651395\pi\)
\(662\) 17.9720 0.698502
\(663\) 20.6156 0.800645
\(664\) −39.3474 −1.52698
\(665\) 2.01283 0.0780543
\(666\) −42.7044 −1.65476
\(667\) −25.4568 −0.985691
\(668\) 14.0986 0.545492
\(669\) −57.2552 −2.21362
\(670\) −7.63390 −0.294923
\(671\) 0 0
\(672\) −12.6630 −0.488485
\(673\) −2.04496 −0.0788276 −0.0394138 0.999223i \(-0.512549\pi\)
−0.0394138 + 0.999223i \(0.512549\pi\)
\(674\) 26.7753 1.03134
\(675\) −4.00585 −0.154185
\(676\) −7.33090 −0.281958
\(677\) 31.2868 1.20245 0.601225 0.799080i \(-0.294680\pi\)
0.601225 + 0.799080i \(0.294680\pi\)
\(678\) −28.5704 −1.09724
\(679\) 19.1597 0.735281
\(680\) −4.99161 −0.191419
\(681\) 38.2079 1.46413
\(682\) 0 0
\(683\) −10.8927 −0.416796 −0.208398 0.978044i \(-0.566825\pi\)
−0.208398 + 0.978044i \(0.566825\pi\)
\(684\) 8.11108 0.310135
\(685\) 19.5900 0.748497
\(686\) 1.04772 0.0400019
\(687\) −51.3678 −1.95981
\(688\) 9.92022 0.378205
\(689\) −5.12607 −0.195288
\(690\) −16.6324 −0.633186
\(691\) 26.5904 1.01155 0.505773 0.862667i \(-0.331207\pi\)
0.505773 + 0.862667i \(0.331207\pi\)
\(692\) −6.06624 −0.230604
\(693\) 0 0
\(694\) −36.3889 −1.38130
\(695\) 2.63936 0.100117
\(696\) 36.4054 1.37994
\(697\) 10.3351 0.391469
\(698\) −0.301418 −0.0114088
\(699\) 11.5488 0.436816
\(700\) −0.902293 −0.0341035
\(701\) 33.6145 1.26960 0.634801 0.772676i \(-0.281082\pi\)
0.634801 + 0.772676i \(0.281082\pi\)
\(702\) −19.2900 −0.728056
\(703\) 18.3702 0.692845
\(704\) 0 0
\(705\) 16.2465 0.611880
\(706\) 4.05043 0.152440
\(707\) 14.8769 0.559504
\(708\) −1.32861 −0.0499324
\(709\) 43.1845 1.62183 0.810913 0.585167i \(-0.198971\pi\)
0.810913 + 0.585167i \(0.198971\pi\)
\(710\) −0.606911 −0.0227770
\(711\) −65.1322 −2.44265
\(712\) −56.6335 −2.12243
\(713\) 27.7748 1.04017
\(714\) 4.69942 0.175871
\(715\) 0 0
\(716\) −1.90000 −0.0710064
\(717\) −43.5640 −1.62693
\(718\) −16.4330 −0.613274
\(719\) 8.80255 0.328280 0.164140 0.986437i \(-0.447515\pi\)
0.164140 + 0.986437i \(0.447515\pi\)
\(720\) 6.16887 0.229900
\(721\) −4.95404 −0.184498
\(722\) −15.6618 −0.582871
\(723\) −49.4714 −1.83986
\(724\) 2.99050 0.111141
\(725\) 4.38164 0.162730
\(726\) 0 0
\(727\) −31.3796 −1.16381 −0.581903 0.813258i \(-0.697692\pi\)
−0.581903 + 0.813258i \(0.697692\pi\)
\(728\) −13.9759 −0.517982
\(729\) −43.7900 −1.62185
\(730\) −2.30833 −0.0854352
\(731\) 11.7895 0.436050
\(732\) −31.1726 −1.15217
\(733\) 24.1821 0.893188 0.446594 0.894737i \(-0.352637\pi\)
0.446594 + 0.894737i \(0.352637\pi\)
\(734\) −9.06496 −0.334594
\(735\) 2.73241 0.100786
\(736\) −26.9251 −0.992473
\(737\) 0 0
\(738\) −29.4595 −1.08442
\(739\) 52.6385 1.93634 0.968170 0.250293i \(-0.0805270\pi\)
0.968170 + 0.250293i \(0.0805270\pi\)
\(740\) −8.23481 −0.302718
\(741\) 25.2783 0.928623
\(742\) −1.16851 −0.0428973
\(743\) −47.4589 −1.74110 −0.870549 0.492082i \(-0.836236\pi\)
−0.870549 + 0.492082i \(0.836236\pi\)
\(744\) −39.7203 −1.45622
\(745\) 8.51343 0.311908
\(746\) 33.0930 1.21162
\(747\) 57.7904 2.11444
\(748\) 0 0
\(749\) −4.85171 −0.177278
\(750\) 2.86278 0.104534
\(751\) −33.8636 −1.23570 −0.617851 0.786295i \(-0.711996\pi\)
−0.617851 + 0.786295i \(0.711996\pi\)
\(752\) −8.21291 −0.299494
\(753\) 3.73451 0.136093
\(754\) 21.0997 0.768404
\(755\) 4.15842 0.151340
\(756\) 3.61445 0.131456
\(757\) 35.8625 1.30345 0.651723 0.758457i \(-0.274046\pi\)
0.651723 + 0.758457i \(0.274046\pi\)
\(758\) −22.0556 −0.801094
\(759\) 0 0
\(760\) −6.12057 −0.222016
\(761\) −10.5657 −0.383007 −0.191503 0.981492i \(-0.561336\pi\)
−0.191503 + 0.981492i \(0.561336\pi\)
\(762\) −33.6749 −1.21991
\(763\) 10.0461 0.363695
\(764\) −20.4039 −0.738186
\(765\) 7.33127 0.265063
\(766\) −8.40141 −0.303555
\(767\) −2.47686 −0.0894342
\(768\) 45.3162 1.63521
\(769\) −42.2581 −1.52387 −0.761934 0.647655i \(-0.775750\pi\)
−0.761934 + 0.647655i \(0.775750\pi\)
\(770\) 0 0
\(771\) 21.1446 0.761505
\(772\) −7.94501 −0.285947
\(773\) −43.3141 −1.55790 −0.778950 0.627087i \(-0.784247\pi\)
−0.778950 + 0.627087i \(0.784247\pi\)
\(774\) −33.6052 −1.20791
\(775\) −4.78061 −0.171724
\(776\) −58.2603 −2.09142
\(777\) 24.9374 0.894625
\(778\) −8.28033 −0.296864
\(779\) 12.6726 0.454043
\(780\) −11.3315 −0.405734
\(781\) 0 0
\(782\) 9.99232 0.357325
\(783\) −17.5522 −0.627263
\(784\) −1.38128 −0.0493315
\(785\) −14.7170 −0.525271
\(786\) 37.0004 1.31976
\(787\) 47.5942 1.69655 0.848274 0.529557i \(-0.177642\pi\)
0.848274 + 0.529557i \(0.177642\pi\)
\(788\) −4.03732 −0.143824
\(789\) −47.0930 −1.67655
\(790\) 15.2797 0.543628
\(791\) 9.97995 0.354846
\(792\) 0 0
\(793\) −58.1134 −2.06367
\(794\) 34.3970 1.22070
\(795\) −3.04744 −0.108081
\(796\) −14.6042 −0.517633
\(797\) −14.4957 −0.513463 −0.256731 0.966483i \(-0.582646\pi\)
−0.256731 + 0.966483i \(0.582646\pi\)
\(798\) 5.76230 0.203983
\(799\) −9.76047 −0.345301
\(800\) 4.63436 0.163850
\(801\) 83.1787 2.93898
\(802\) −37.9041 −1.33844
\(803\) 0 0
\(804\) 17.9637 0.633532
\(805\) 5.80988 0.204771
\(806\) −23.0209 −0.810876
\(807\) −20.5779 −0.724375
\(808\) −45.2373 −1.59144
\(809\) −7.16254 −0.251822 −0.125911 0.992042i \(-0.540185\pi\)
−0.125911 + 0.992042i \(0.540185\pi\)
\(810\) 2.56957 0.0902855
\(811\) 22.6807 0.796427 0.398214 0.917293i \(-0.369630\pi\)
0.398214 + 0.917293i \(0.369630\pi\)
\(812\) −3.95352 −0.138741
\(813\) −21.9118 −0.768480
\(814\) 0 0
\(815\) 10.3649 0.363065
\(816\) −6.19560 −0.216889
\(817\) 14.4560 0.505750
\(818\) −11.6967 −0.408967
\(819\) 20.5267 0.717261
\(820\) −5.68075 −0.198380
\(821\) −4.52513 −0.157928 −0.0789640 0.996877i \(-0.525161\pi\)
−0.0789640 + 0.996877i \(0.525161\pi\)
\(822\) 56.0821 1.95609
\(823\) −31.6334 −1.10267 −0.551336 0.834283i \(-0.685882\pi\)
−0.551336 + 0.834283i \(0.685882\pi\)
\(824\) 15.0641 0.524784
\(825\) 0 0
\(826\) −0.564610 −0.0196453
\(827\) 20.8564 0.725247 0.362624 0.931936i \(-0.381881\pi\)
0.362624 + 0.931936i \(0.381881\pi\)
\(828\) 23.4120 0.813623
\(829\) −35.0216 −1.21635 −0.608176 0.793802i \(-0.708098\pi\)
−0.608176 + 0.793802i \(0.708098\pi\)
\(830\) −13.5574 −0.470583
\(831\) −43.9270 −1.52381
\(832\) 35.0138 1.21389
\(833\) −1.64156 −0.0568766
\(834\) 7.55592 0.261640
\(835\) 15.6253 0.540736
\(836\) 0 0
\(837\) 19.1504 0.661934
\(838\) 29.4618 1.01774
\(839\) −0.302452 −0.0104418 −0.00522089 0.999986i \(-0.501662\pi\)
−0.00522089 + 0.999986i \(0.501662\pi\)
\(840\) −8.30864 −0.286675
\(841\) −9.80126 −0.337974
\(842\) −0.713573 −0.0245914
\(843\) 23.9026 0.823248
\(844\) −8.16707 −0.281122
\(845\) −8.12474 −0.279500
\(846\) 27.8216 0.956526
\(847\) 0 0
\(848\) 1.54053 0.0529021
\(849\) −69.1267 −2.37242
\(850\) −1.71988 −0.0589915
\(851\) 53.0241 1.81764
\(852\) 1.42815 0.0489277
\(853\) 39.1069 1.33899 0.669497 0.742814i \(-0.266509\pi\)
0.669497 + 0.742814i \(0.266509\pi\)
\(854\) −13.2472 −0.453309
\(855\) 8.98941 0.307431
\(856\) 14.7530 0.504246
\(857\) −5.25170 −0.179395 −0.0896974 0.995969i \(-0.528590\pi\)
−0.0896974 + 0.995969i \(0.528590\pi\)
\(858\) 0 0
\(859\) −4.17391 −0.142412 −0.0712059 0.997462i \(-0.522685\pi\)
−0.0712059 + 0.997462i \(0.522685\pi\)
\(860\) −6.48018 −0.220972
\(861\) 17.2030 0.586276
\(862\) −30.6868 −1.04520
\(863\) −35.0583 −1.19340 −0.596699 0.802465i \(-0.703521\pi\)
−0.596699 + 0.802465i \(0.703521\pi\)
\(864\) −18.5646 −0.631579
\(865\) −6.72314 −0.228594
\(866\) −32.1353 −1.09200
\(867\) 39.0879 1.32749
\(868\) 4.31351 0.146410
\(869\) 0 0
\(870\) 12.5437 0.425271
\(871\) 33.4887 1.13472
\(872\) −30.5481 −1.03449
\(873\) 85.5681 2.89604
\(874\) 12.2523 0.414441
\(875\) −1.00000 −0.0338062
\(876\) 5.43186 0.183525
\(877\) 11.5470 0.389915 0.194958 0.980812i \(-0.437543\pi\)
0.194958 + 0.980812i \(0.437543\pi\)
\(878\) −14.0450 −0.473995
\(879\) −60.6711 −2.04638
\(880\) 0 0
\(881\) 20.6320 0.695108 0.347554 0.937660i \(-0.387012\pi\)
0.347554 + 0.937660i \(0.387012\pi\)
\(882\) 4.67915 0.157555
\(883\) −3.78173 −0.127265 −0.0636327 0.997973i \(-0.520269\pi\)
−0.0636327 + 0.997973i \(0.520269\pi\)
\(884\) 6.80768 0.228967
\(885\) −1.47249 −0.0494971
\(886\) 29.6429 0.995872
\(887\) −18.8031 −0.631347 −0.315673 0.948868i \(-0.602230\pi\)
−0.315673 + 0.948868i \(0.602230\pi\)
\(888\) −75.8291 −2.54466
\(889\) 11.7630 0.394518
\(890\) −19.5134 −0.654089
\(891\) 0 0
\(892\) −18.9068 −0.633046
\(893\) −11.9680 −0.400495
\(894\) 24.3721 0.815125
\(895\) −2.10575 −0.0703874
\(896\) −1.28718 −0.0430016
\(897\) 72.9640 2.43620
\(898\) 22.4209 0.748195
\(899\) −20.9469 −0.698618
\(900\) −4.02969 −0.134323
\(901\) 1.83082 0.0609934
\(902\) 0 0
\(903\) 19.6239 0.653042
\(904\) −30.3468 −1.00932
\(905\) 3.31433 0.110172
\(906\) 11.9047 0.395506
\(907\) 50.4063 1.67372 0.836858 0.547421i \(-0.184390\pi\)
0.836858 + 0.547421i \(0.184390\pi\)
\(908\) 12.6170 0.418709
\(909\) 66.4410 2.20371
\(910\) −4.81547 −0.159631
\(911\) 11.9079 0.394526 0.197263 0.980351i \(-0.436795\pi\)
0.197263 + 0.980351i \(0.436795\pi\)
\(912\) −7.59687 −0.251558
\(913\) 0 0
\(914\) 34.0557 1.12646
\(915\) −34.5482 −1.14213
\(916\) −16.9626 −0.560461
\(917\) −12.9246 −0.426809
\(918\) 6.88959 0.227390
\(919\) 44.0298 1.45241 0.726205 0.687478i \(-0.241282\pi\)
0.726205 + 0.687478i \(0.241282\pi\)
\(920\) −17.6665 −0.582449
\(921\) −49.4584 −1.62971
\(922\) 17.9964 0.592678
\(923\) 2.66243 0.0876348
\(924\) 0 0
\(925\) −9.12654 −0.300079
\(926\) 7.65753 0.251642
\(927\) −22.1250 −0.726680
\(928\) 20.3061 0.666580
\(929\) 7.64599 0.250857 0.125428 0.992103i \(-0.459969\pi\)
0.125428 + 0.992103i \(0.459969\pi\)
\(930\) −13.6858 −0.448777
\(931\) −2.01283 −0.0659679
\(932\) 3.81363 0.124920
\(933\) 79.4199 2.60009
\(934\) −23.8632 −0.780828
\(935\) 0 0
\(936\) −62.4171 −2.04017
\(937\) −60.6763 −1.98221 −0.991104 0.133092i \(-0.957509\pi\)
−0.991104 + 0.133092i \(0.957509\pi\)
\(938\) 7.63390 0.249256
\(939\) −72.7264 −2.37334
\(940\) 5.36491 0.174984
\(941\) −57.1546 −1.86319 −0.931594 0.363500i \(-0.881582\pi\)
−0.931594 + 0.363500i \(0.881582\pi\)
\(942\) −42.1315 −1.37272
\(943\) 36.5785 1.19116
\(944\) 0.744368 0.0242271
\(945\) 4.00585 0.130310
\(946\) 0 0
\(947\) 7.80815 0.253731 0.126865 0.991920i \(-0.459508\pi\)
0.126865 + 0.991920i \(0.459508\pi\)
\(948\) −35.9555 −1.16778
\(949\) 10.1263 0.328714
\(950\) −2.10887 −0.0684209
\(951\) 82.5652 2.67736
\(952\) 4.99161 0.161779
\(953\) −14.2043 −0.460122 −0.230061 0.973176i \(-0.573893\pi\)
−0.230061 + 0.973176i \(0.573893\pi\)
\(954\) −5.21862 −0.168959
\(955\) −22.6133 −0.731750
\(956\) −14.3857 −0.465266
\(957\) 0 0
\(958\) 2.39352 0.0773312
\(959\) −19.5900 −0.632595
\(960\) 20.8156 0.671821
\(961\) −8.14581 −0.262768
\(962\) −43.9486 −1.41696
\(963\) −21.6680 −0.698241
\(964\) −16.3364 −0.526160
\(965\) −8.80536 −0.283455
\(966\) 16.6324 0.535140
\(967\) −3.22512 −0.103713 −0.0518564 0.998655i \(-0.516514\pi\)
−0.0518564 + 0.998655i \(0.516514\pi\)
\(968\) 0 0
\(969\) −9.02836 −0.290033
\(970\) −20.0739 −0.644533
\(971\) −54.2365 −1.74053 −0.870266 0.492583i \(-0.836053\pi\)
−0.870266 + 0.492583i \(0.836053\pi\)
\(972\) −16.8899 −0.541745
\(973\) −2.63936 −0.0846140
\(974\) 42.9259 1.37543
\(975\) −12.5586 −0.402197
\(976\) 17.4647 0.559033
\(977\) 49.0331 1.56871 0.784355 0.620313i \(-0.212994\pi\)
0.784355 + 0.620313i \(0.212994\pi\)
\(978\) 29.6723 0.948817
\(979\) 0 0
\(980\) 0.902293 0.0288227
\(981\) 44.8666 1.43248
\(982\) −5.50225 −0.175584
\(983\) 20.3715 0.649750 0.324875 0.945757i \(-0.394678\pi\)
0.324875 + 0.945757i \(0.394678\pi\)
\(984\) −52.3104 −1.66759
\(985\) −4.47451 −0.142570
\(986\) −7.53590 −0.239992
\(987\) −16.2465 −0.517133
\(988\) 8.34739 0.265566
\(989\) 41.7260 1.32681
\(990\) 0 0
\(991\) −34.0420 −1.08138 −0.540689 0.841222i \(-0.681837\pi\)
−0.540689 + 0.841222i \(0.681837\pi\)
\(992\) −22.1551 −0.703424
\(993\) −46.8704 −1.48739
\(994\) 0.606911 0.0192500
\(995\) −16.1857 −0.513120
\(996\) 31.9025 1.01087
\(997\) 38.2505 1.21141 0.605703 0.795691i \(-0.292892\pi\)
0.605703 + 0.795691i \(0.292892\pi\)
\(998\) −18.2058 −0.576293
\(999\) 36.5595 1.15669
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 4235.2.a.bl.1.7 yes 10
11.10 odd 2 4235.2.a.bj.1.4 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4235.2.a.bj.1.4 10 11.10 odd 2
4235.2.a.bl.1.7 yes 10 1.1 even 1 trivial