Properties

Label 6422.2.a.bc.1.3
Level $6422$
Weight $2$
Character 6422.1
Self dual yes
Analytic conductor $51.280$
Analytic rank $0$
Dimension $6$
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] = [6422,2,Mod(1,6422)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6422, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6422.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6422.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: \(51.2799281781\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.316645497.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 12x^{4} + 24x^{3} + 13x^{2} - 24x + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 494)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.260281\) of defining polynomial
Character \(\chi\) \(=\) 6422.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +0.260281 q^{3} +1.00000 q^{4} +4.36375 q^{5} -0.260281 q^{6} -0.775135 q^{7} -1.00000 q^{8} -2.93225 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +0.260281 q^{3} +1.00000 q^{4} +4.36375 q^{5} -0.260281 q^{6} -0.775135 q^{7} -1.00000 q^{8} -2.93225 q^{9} -4.36375 q^{10} +4.36375 q^{11} +0.260281 q^{12} +0.775135 q^{14} +1.13580 q^{15} +1.00000 q^{16} -6.93885 q^{17} +2.93225 q^{18} +1.00000 q^{19} +4.36375 q^{20} -0.201753 q^{21} -4.36375 q^{22} -4.08036 q^{23} -0.260281 q^{24} +14.0423 q^{25} -1.54405 q^{27} -0.775135 q^{28} +2.54367 q^{29} -1.13580 q^{30} -6.41220 q^{31} -1.00000 q^{32} +1.13580 q^{33} +6.93885 q^{34} -3.38249 q^{35} -2.93225 q^{36} +6.64765 q^{37} -1.00000 q^{38} -4.36375 q^{40} -1.20663 q^{41} +0.201753 q^{42} +8.33493 q^{43} +4.36375 q^{44} -12.7956 q^{45} +4.08036 q^{46} +1.36375 q^{47} +0.260281 q^{48} -6.39917 q^{49} -14.0423 q^{50} -1.80605 q^{51} +3.52493 q^{53} +1.54405 q^{54} +19.0423 q^{55} +0.775135 q^{56} +0.260281 q^{57} -2.54367 q^{58} +12.4214 q^{59} +1.13580 q^{60} +2.03105 q^{61} +6.41220 q^{62} +2.27289 q^{63} +1.00000 q^{64} -1.13580 q^{66} +11.5249 q^{67} -6.93885 q^{68} -1.06204 q^{69} +3.38249 q^{70} -12.1096 q^{71} +2.93225 q^{72} +2.30351 q^{73} -6.64765 q^{74} +3.65495 q^{75} +1.00000 q^{76} -3.38249 q^{77} +8.16942 q^{79} +4.36375 q^{80} +8.39487 q^{81} +1.20663 q^{82} -2.75361 q^{83} -0.201753 q^{84} -30.2794 q^{85} -8.33493 q^{86} +0.662069 q^{87} -4.36375 q^{88} +3.34501 q^{89} +12.7956 q^{90} -4.08036 q^{92} -1.66897 q^{93} -1.36375 q^{94} +4.36375 q^{95} -0.260281 q^{96} +2.12923 q^{97} +6.39917 q^{98} -12.7956 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} + 2 q^{3} + 6 q^{4} - 2 q^{5} - 2 q^{6} - q^{7} - 6 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{2} + 2 q^{3} + 6 q^{4} - 2 q^{5} - 2 q^{6} - q^{7} - 6 q^{8} + 10 q^{9} + 2 q^{10} - 2 q^{11} + 2 q^{12} + q^{14} + 9 q^{15} + 6 q^{16} - 2 q^{17} - 10 q^{18} + 6 q^{19} - 2 q^{20} - q^{21} + 2 q^{22} + 8 q^{23} - 2 q^{24} + 16 q^{25} - 4 q^{27} - q^{28} + 20 q^{29} - 9 q^{30} - 3 q^{31} - 6 q^{32} + 9 q^{33} + 2 q^{34} + 7 q^{35} + 10 q^{36} + 9 q^{37} - 6 q^{38} + 2 q^{40} - 3 q^{41} + q^{42} + 13 q^{43} - 2 q^{44} - 38 q^{45} - 8 q^{46} - 20 q^{47} + 2 q^{48} - 7 q^{49} - 16 q^{50} + 2 q^{51} + 25 q^{53} + 4 q^{54} + 46 q^{55} + q^{56} + 2 q^{57} - 20 q^{58} + 9 q^{60} + 6 q^{61} + 3 q^{62} - 46 q^{63} + 6 q^{64} - 9 q^{66} + 32 q^{67} - 2 q^{68} - 29 q^{69} - 7 q^{70} - 39 q^{71} - 10 q^{72} - 7 q^{73} - 9 q^{74} - 15 q^{75} + 6 q^{76} + 7 q^{77} + 18 q^{79} - 2 q^{80} + 54 q^{81} + 3 q^{82} + 7 q^{83} - q^{84} + 2 q^{85} - 13 q^{86} + 12 q^{87} + 2 q^{88} - 9 q^{89} + 38 q^{90} + 8 q^{92} + 47 q^{93} + 20 q^{94} - 2 q^{95} - 2 q^{96} + 4 q^{97} + 7 q^{98} - 38 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0.260281 0.150273 0.0751366 0.997173i \(-0.476061\pi\)
0.0751366 + 0.997173i \(0.476061\pi\)
\(4\) 1.00000 0.500000
\(5\) 4.36375 1.95153 0.975764 0.218824i \(-0.0702221\pi\)
0.975764 + 0.218824i \(0.0702221\pi\)
\(6\) −0.260281 −0.106259
\(7\) −0.775135 −0.292973 −0.146487 0.989213i \(-0.546797\pi\)
−0.146487 + 0.989213i \(0.546797\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.93225 −0.977418
\(10\) −4.36375 −1.37994
\(11\) 4.36375 1.31572 0.657860 0.753140i \(-0.271462\pi\)
0.657860 + 0.753140i \(0.271462\pi\)
\(12\) 0.260281 0.0751366
\(13\) 0 0
\(14\) 0.775135 0.207163
\(15\) 1.13580 0.293262
\(16\) 1.00000 0.250000
\(17\) −6.93885 −1.68292 −0.841459 0.540321i \(-0.818303\pi\)
−0.841459 + 0.540321i \(0.818303\pi\)
\(18\) 2.93225 0.691139
\(19\) 1.00000 0.229416
\(20\) 4.36375 0.975764
\(21\) −0.201753 −0.0440260
\(22\) −4.36375 −0.930355
\(23\) −4.08036 −0.850813 −0.425407 0.905002i \(-0.639869\pi\)
−0.425407 + 0.905002i \(0.639869\pi\)
\(24\) −0.260281 −0.0531296
\(25\) 14.0423 2.80846
\(26\) 0 0
\(27\) −1.54405 −0.297153
\(28\) −0.775135 −0.146487
\(29\) 2.54367 0.472348 0.236174 0.971711i \(-0.424106\pi\)
0.236174 + 0.971711i \(0.424106\pi\)
\(30\) −1.13580 −0.207368
\(31\) −6.41220 −1.15167 −0.575833 0.817567i \(-0.695322\pi\)
−0.575833 + 0.817567i \(0.695322\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.13580 0.197717
\(34\) 6.93885 1.19000
\(35\) −3.38249 −0.571746
\(36\) −2.93225 −0.488709
\(37\) 6.64765 1.09287 0.546434 0.837502i \(-0.315985\pi\)
0.546434 + 0.837502i \(0.315985\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) −4.36375 −0.689970
\(41\) −1.20663 −0.188444 −0.0942221 0.995551i \(-0.530036\pi\)
−0.0942221 + 0.995551i \(0.530036\pi\)
\(42\) 0.201753 0.0311311
\(43\) 8.33493 1.27106 0.635532 0.772074i \(-0.280781\pi\)
0.635532 + 0.772074i \(0.280781\pi\)
\(44\) 4.36375 0.657860
\(45\) −12.7956 −1.90746
\(46\) 4.08036 0.601616
\(47\) 1.36375 0.198923 0.0994617 0.995041i \(-0.468288\pi\)
0.0994617 + 0.995041i \(0.468288\pi\)
\(48\) 0.260281 0.0375683
\(49\) −6.39917 −0.914167
\(50\) −14.0423 −1.98588
\(51\) −1.80605 −0.252897
\(52\) 0 0
\(53\) 3.52493 0.484186 0.242093 0.970253i \(-0.422166\pi\)
0.242093 + 0.970253i \(0.422166\pi\)
\(54\) 1.54405 0.210119
\(55\) 19.0423 2.56767
\(56\) 0.775135 0.103582
\(57\) 0.260281 0.0344750
\(58\) −2.54367 −0.334001
\(59\) 12.4214 1.61712 0.808562 0.588410i \(-0.200246\pi\)
0.808562 + 0.588410i \(0.200246\pi\)
\(60\) 1.13580 0.146631
\(61\) 2.03105 0.260049 0.130024 0.991511i \(-0.458494\pi\)
0.130024 + 0.991511i \(0.458494\pi\)
\(62\) 6.41220 0.814351
\(63\) 2.27289 0.286357
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.13580 −0.139807
\(67\) 11.5249 1.40799 0.703995 0.710205i \(-0.251398\pi\)
0.703995 + 0.710205i \(0.251398\pi\)
\(68\) −6.93885 −0.841459
\(69\) −1.06204 −0.127854
\(70\) 3.38249 0.404285
\(71\) −12.1096 −1.43714 −0.718570 0.695455i \(-0.755203\pi\)
−0.718570 + 0.695455i \(0.755203\pi\)
\(72\) 2.93225 0.345569
\(73\) 2.30351 0.269605 0.134802 0.990872i \(-0.456960\pi\)
0.134802 + 0.990872i \(0.456960\pi\)
\(74\) −6.64765 −0.772774
\(75\) 3.65495 0.422037
\(76\) 1.00000 0.114708
\(77\) −3.38249 −0.385471
\(78\) 0 0
\(79\) 8.16942 0.919132 0.459566 0.888144i \(-0.348005\pi\)
0.459566 + 0.888144i \(0.348005\pi\)
\(80\) 4.36375 0.487882
\(81\) 8.39487 0.932764
\(82\) 1.20663 0.133250
\(83\) −2.75361 −0.302248 −0.151124 0.988515i \(-0.548289\pi\)
−0.151124 + 0.988515i \(0.548289\pi\)
\(84\) −0.201753 −0.0220130
\(85\) −30.2794 −3.28426
\(86\) −8.33493 −0.898779
\(87\) 0.662069 0.0709813
\(88\) −4.36375 −0.465177
\(89\) 3.34501 0.354570 0.177285 0.984160i \(-0.443269\pi\)
0.177285 + 0.984160i \(0.443269\pi\)
\(90\) 12.7956 1.34878
\(91\) 0 0
\(92\) −4.08036 −0.425407
\(93\) −1.66897 −0.173064
\(94\) −1.36375 −0.140660
\(95\) 4.36375 0.447711
\(96\) −0.260281 −0.0265648
\(97\) 2.12923 0.216190 0.108095 0.994141i \(-0.465525\pi\)
0.108095 + 0.994141i \(0.465525\pi\)
\(98\) 6.39917 0.646413
\(99\) −12.7956 −1.28601
\(100\) 14.0423 1.40423
\(101\) 11.5122 1.14550 0.572751 0.819729i \(-0.305876\pi\)
0.572751 + 0.819729i \(0.305876\pi\)
\(102\) 1.80605 0.178826
\(103\) 8.59033 0.846431 0.423215 0.906029i \(-0.360901\pi\)
0.423215 + 0.906029i \(0.360901\pi\)
\(104\) 0 0
\(105\) −0.880398 −0.0859181
\(106\) −3.52493 −0.342371
\(107\) 12.2990 1.18899 0.594495 0.804099i \(-0.297352\pi\)
0.594495 + 0.804099i \(0.297352\pi\)
\(108\) −1.54405 −0.148576
\(109\) 8.17344 0.782874 0.391437 0.920205i \(-0.371978\pi\)
0.391437 + 0.920205i \(0.371978\pi\)
\(110\) −19.0423 −1.81561
\(111\) 1.73026 0.164229
\(112\) −0.775135 −0.0732433
\(113\) 4.05116 0.381101 0.190551 0.981677i \(-0.438973\pi\)
0.190551 + 0.981677i \(0.438973\pi\)
\(114\) −0.260281 −0.0243775
\(115\) −17.8057 −1.66039
\(116\) 2.54367 0.236174
\(117\) 0 0
\(118\) −12.4214 −1.14348
\(119\) 5.37854 0.493050
\(120\) −1.13580 −0.103684
\(121\) 8.04232 0.731120
\(122\) −2.03105 −0.183882
\(123\) −0.314063 −0.0283181
\(124\) −6.41220 −0.575833
\(125\) 39.4584 3.52927
\(126\) −2.27289 −0.202485
\(127\) 6.03228 0.535279 0.267639 0.963519i \(-0.413756\pi\)
0.267639 + 0.963519i \(0.413756\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 2.16942 0.191007
\(130\) 0 0
\(131\) 12.8272 1.12071 0.560357 0.828251i \(-0.310664\pi\)
0.560357 + 0.828251i \(0.310664\pi\)
\(132\) 1.13580 0.0988587
\(133\) −0.775135 −0.0672127
\(134\) −11.5249 −0.995599
\(135\) −6.73786 −0.579902
\(136\) 6.93885 0.595001
\(137\) −13.5152 −1.15468 −0.577339 0.816504i \(-0.695909\pi\)
−0.577339 + 0.816504i \(0.695909\pi\)
\(138\) 1.06204 0.0904067
\(139\) 2.17364 0.184365 0.0921827 0.995742i \(-0.470616\pi\)
0.0921827 + 0.995742i \(0.470616\pi\)
\(140\) −3.38249 −0.285873
\(141\) 0.354958 0.0298929
\(142\) 12.1096 1.01621
\(143\) 0 0
\(144\) −2.93225 −0.244354
\(145\) 11.1000 0.921801
\(146\) −2.30351 −0.190639
\(147\) −1.66558 −0.137375
\(148\) 6.64765 0.546434
\(149\) 10.4236 0.853935 0.426968 0.904267i \(-0.359582\pi\)
0.426968 + 0.904267i \(0.359582\pi\)
\(150\) −3.65495 −0.298425
\(151\) −7.19913 −0.585857 −0.292928 0.956134i \(-0.594630\pi\)
−0.292928 + 0.956134i \(0.594630\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 20.3465 1.64491
\(154\) 3.38249 0.272569
\(155\) −27.9813 −2.24751
\(156\) 0 0
\(157\) 20.5120 1.63704 0.818519 0.574479i \(-0.194795\pi\)
0.818519 + 0.574479i \(0.194795\pi\)
\(158\) −8.16942 −0.649924
\(159\) 0.917471 0.0727602
\(160\) −4.36375 −0.344985
\(161\) 3.16283 0.249266
\(162\) −8.39487 −0.659564
\(163\) 2.11182 0.165411 0.0827053 0.996574i \(-0.473644\pi\)
0.0827053 + 0.996574i \(0.473644\pi\)
\(164\) −1.20663 −0.0942221
\(165\) 4.95635 0.385851
\(166\) 2.75361 0.213722
\(167\) 20.3048 1.57123 0.785615 0.618715i \(-0.212346\pi\)
0.785615 + 0.618715i \(0.212346\pi\)
\(168\) 0.201753 0.0155656
\(169\) 0 0
\(170\) 30.2794 2.32232
\(171\) −2.93225 −0.224235
\(172\) 8.33493 0.635532
\(173\) 23.1299 1.75854 0.879268 0.476327i \(-0.158032\pi\)
0.879268 + 0.476327i \(0.158032\pi\)
\(174\) −0.662069 −0.0501914
\(175\) −10.8847 −0.822805
\(176\) 4.36375 0.328930
\(177\) 3.23304 0.243011
\(178\) −3.34501 −0.250719
\(179\) −13.1550 −0.983253 −0.491626 0.870806i \(-0.663597\pi\)
−0.491626 + 0.870806i \(0.663597\pi\)
\(180\) −12.7956 −0.953730
\(181\) −0.551157 −0.0409672 −0.0204836 0.999790i \(-0.506521\pi\)
−0.0204836 + 0.999790i \(0.506521\pi\)
\(182\) 0 0
\(183\) 0.528643 0.0390784
\(184\) 4.08036 0.300808
\(185\) 29.0087 2.13276
\(186\) 1.66897 0.122375
\(187\) −30.2794 −2.21425
\(188\) 1.36375 0.0994617
\(189\) 1.19685 0.0870579
\(190\) −4.36375 −0.316580
\(191\) −5.61275 −0.406124 −0.203062 0.979166i \(-0.565089\pi\)
−0.203062 + 0.979166i \(0.565089\pi\)
\(192\) 0.260281 0.0187841
\(193\) −7.88071 −0.567266 −0.283633 0.958933i \(-0.591540\pi\)
−0.283633 + 0.958933i \(0.591540\pi\)
\(194\) −2.12923 −0.152870
\(195\) 0 0
\(196\) −6.39917 −0.457083
\(197\) 6.81454 0.485516 0.242758 0.970087i \(-0.421948\pi\)
0.242758 + 0.970087i \(0.421948\pi\)
\(198\) 12.7956 0.909345
\(199\) −24.2470 −1.71882 −0.859411 0.511286i \(-0.829169\pi\)
−0.859411 + 0.511286i \(0.829169\pi\)
\(200\) −14.0423 −0.992942
\(201\) 2.99971 0.211583
\(202\) −11.5122 −0.809993
\(203\) −1.97169 −0.138386
\(204\) −1.80605 −0.126449
\(205\) −5.26544 −0.367754
\(206\) −8.59033 −0.598517
\(207\) 11.9646 0.831600
\(208\) 0 0
\(209\) 4.36375 0.301847
\(210\) 0.880398 0.0607533
\(211\) −5.05776 −0.348190 −0.174095 0.984729i \(-0.555700\pi\)
−0.174095 + 0.984729i \(0.555700\pi\)
\(212\) 3.52493 0.242093
\(213\) −3.15188 −0.215964
\(214\) −12.2990 −0.840743
\(215\) 36.3716 2.48052
\(216\) 1.54405 0.105059
\(217\) 4.97032 0.337407
\(218\) −8.17344 −0.553575
\(219\) 0.599558 0.0405144
\(220\) 19.0423 1.28383
\(221\) 0 0
\(222\) −1.73026 −0.116127
\(223\) −14.0800 −0.942864 −0.471432 0.881902i \(-0.656263\pi\)
−0.471432 + 0.881902i \(0.656263\pi\)
\(224\) 0.775135 0.0517909
\(225\) −41.1756 −2.74504
\(226\) −4.05116 −0.269479
\(227\) 18.6540 1.23811 0.619056 0.785347i \(-0.287515\pi\)
0.619056 + 0.785347i \(0.287515\pi\)
\(228\) 0.260281 0.0172375
\(229\) 5.65636 0.373783 0.186891 0.982381i \(-0.440159\pi\)
0.186891 + 0.982381i \(0.440159\pi\)
\(230\) 17.8057 1.17407
\(231\) −0.880398 −0.0579260
\(232\) −2.54367 −0.167000
\(233\) −19.9409 −1.30637 −0.653185 0.757198i \(-0.726568\pi\)
−0.653185 + 0.757198i \(0.726568\pi\)
\(234\) 0 0
\(235\) 5.95107 0.388205
\(236\) 12.4214 0.808562
\(237\) 2.12634 0.138121
\(238\) −5.37854 −0.348639
\(239\) −3.43781 −0.222373 −0.111187 0.993800i \(-0.535465\pi\)
−0.111187 + 0.993800i \(0.535465\pi\)
\(240\) 1.13580 0.0733156
\(241\) 5.43845 0.350321 0.175161 0.984540i \(-0.443956\pi\)
0.175161 + 0.984540i \(0.443956\pi\)
\(242\) −8.04232 −0.516980
\(243\) 6.81718 0.437322
\(244\) 2.03105 0.130024
\(245\) −27.9244 −1.78402
\(246\) 0.314063 0.0200239
\(247\) 0 0
\(248\) 6.41220 0.407175
\(249\) −0.716713 −0.0454198
\(250\) −39.4584 −2.49557
\(251\) −25.9262 −1.63644 −0.818222 0.574902i \(-0.805040\pi\)
−0.818222 + 0.574902i \(0.805040\pi\)
\(252\) 2.27289 0.143179
\(253\) −17.8057 −1.11943
\(254\) −6.03228 −0.378499
\(255\) −7.88115 −0.493537
\(256\) 1.00000 0.0625000
\(257\) −1.88939 −0.117857 −0.0589283 0.998262i \(-0.518768\pi\)
−0.0589283 + 0.998262i \(0.518768\pi\)
\(258\) −2.16942 −0.135062
\(259\) −5.15283 −0.320181
\(260\) 0 0
\(261\) −7.45870 −0.461682
\(262\) −12.8272 −0.792464
\(263\) −4.19046 −0.258395 −0.129197 0.991619i \(-0.541240\pi\)
−0.129197 + 0.991619i \(0.541240\pi\)
\(264\) −1.13580 −0.0699037
\(265\) 15.3819 0.944904
\(266\) 0.775135 0.0475266
\(267\) 0.870641 0.0532824
\(268\) 11.5249 0.703995
\(269\) 5.06288 0.308689 0.154345 0.988017i \(-0.450673\pi\)
0.154345 + 0.988017i \(0.450673\pi\)
\(270\) 6.73786 0.410053
\(271\) −23.4633 −1.42529 −0.712647 0.701522i \(-0.752504\pi\)
−0.712647 + 0.701522i \(0.752504\pi\)
\(272\) −6.93885 −0.420730
\(273\) 0 0
\(274\) 13.5152 0.816481
\(275\) 61.2772 3.69515
\(276\) −1.06204 −0.0639272
\(277\) 7.03193 0.422508 0.211254 0.977431i \(-0.432245\pi\)
0.211254 + 0.977431i \(0.432245\pi\)
\(278\) −2.17364 −0.130366
\(279\) 18.8022 1.12566
\(280\) 3.38249 0.202143
\(281\) −20.4814 −1.22182 −0.610910 0.791700i \(-0.709196\pi\)
−0.610910 + 0.791700i \(0.709196\pi\)
\(282\) −0.354958 −0.0211374
\(283\) −26.9345 −1.60109 −0.800544 0.599274i \(-0.795456\pi\)
−0.800544 + 0.599274i \(0.795456\pi\)
\(284\) −12.1096 −0.718570
\(285\) 1.13580 0.0672790
\(286\) 0 0
\(287\) 0.935302 0.0552091
\(288\) 2.93225 0.172785
\(289\) 31.1476 1.83221
\(290\) −11.1000 −0.651812
\(291\) 0.554197 0.0324876
\(292\) 2.30351 0.134802
\(293\) 9.68946 0.566064 0.283032 0.959110i \(-0.408660\pi\)
0.283032 + 0.959110i \(0.408660\pi\)
\(294\) 1.66558 0.0971386
\(295\) 54.2038 3.15587
\(296\) −6.64765 −0.386387
\(297\) −6.73786 −0.390970
\(298\) −10.4236 −0.603823
\(299\) 0 0
\(300\) 3.65495 0.211018
\(301\) −6.46069 −0.372388
\(302\) 7.19913 0.414263
\(303\) 2.99639 0.172138
\(304\) 1.00000 0.0573539
\(305\) 8.86298 0.507493
\(306\) −20.3465 −1.16313
\(307\) 12.5815 0.718064 0.359032 0.933325i \(-0.383107\pi\)
0.359032 + 0.933325i \(0.383107\pi\)
\(308\) −3.38249 −0.192736
\(309\) 2.23590 0.127196
\(310\) 27.9813 1.58923
\(311\) −16.7407 −0.949280 −0.474640 0.880180i \(-0.657422\pi\)
−0.474640 + 0.880180i \(0.657422\pi\)
\(312\) 0 0
\(313\) −11.5101 −0.650590 −0.325295 0.945613i \(-0.605464\pi\)
−0.325295 + 0.945613i \(0.605464\pi\)
\(314\) −20.5120 −1.15756
\(315\) 9.91833 0.558835
\(316\) 8.16942 0.459566
\(317\) −4.14425 −0.232764 −0.116382 0.993205i \(-0.537130\pi\)
−0.116382 + 0.993205i \(0.537130\pi\)
\(318\) −0.917471 −0.0514492
\(319\) 11.1000 0.621478
\(320\) 4.36375 0.243941
\(321\) 3.20119 0.178673
\(322\) −3.16283 −0.176257
\(323\) −6.93885 −0.386088
\(324\) 8.39487 0.466382
\(325\) 0 0
\(326\) −2.11182 −0.116963
\(327\) 2.12739 0.117645
\(328\) 1.20663 0.0666251
\(329\) −1.05709 −0.0582793
\(330\) −4.95635 −0.272838
\(331\) 0.316380 0.0173898 0.00869492 0.999962i \(-0.497232\pi\)
0.00869492 + 0.999962i \(0.497232\pi\)
\(332\) −2.75361 −0.151124
\(333\) −19.4926 −1.06819
\(334\) −20.3048 −1.11103
\(335\) 50.2917 2.74773
\(336\) −0.201753 −0.0110065
\(337\) 17.7103 0.964739 0.482370 0.875968i \(-0.339776\pi\)
0.482370 + 0.875968i \(0.339776\pi\)
\(338\) 0 0
\(339\) 1.05444 0.0572693
\(340\) −30.2794 −1.64213
\(341\) −27.9813 −1.51527
\(342\) 2.93225 0.158558
\(343\) 10.3862 0.560800
\(344\) −8.33493 −0.449389
\(345\) −4.63447 −0.249512
\(346\) −23.1299 −1.24347
\(347\) 22.4138 1.20324 0.601619 0.798783i \(-0.294523\pi\)
0.601619 + 0.798783i \(0.294523\pi\)
\(348\) 0.662069 0.0354906
\(349\) −20.0965 −1.07574 −0.537871 0.843027i \(-0.680771\pi\)
−0.537871 + 0.843027i \(0.680771\pi\)
\(350\) 10.8847 0.581811
\(351\) 0 0
\(352\) −4.36375 −0.232589
\(353\) −11.7682 −0.626360 −0.313180 0.949694i \(-0.601394\pi\)
−0.313180 + 0.949694i \(0.601394\pi\)
\(354\) −3.23304 −0.171834
\(355\) −52.8431 −2.80462
\(356\) 3.34501 0.177285
\(357\) 1.39993 0.0740922
\(358\) 13.1550 0.695265
\(359\) −7.47763 −0.394654 −0.197327 0.980338i \(-0.563226\pi\)
−0.197327 + 0.980338i \(0.563226\pi\)
\(360\) 12.7956 0.674389
\(361\) 1.00000 0.0526316
\(362\) 0.551157 0.0289682
\(363\) 2.09326 0.109868
\(364\) 0 0
\(365\) 10.0519 0.526142
\(366\) −0.528643 −0.0276326
\(367\) 16.4233 0.857289 0.428645 0.903473i \(-0.358991\pi\)
0.428645 + 0.903473i \(0.358991\pi\)
\(368\) −4.08036 −0.212703
\(369\) 3.53815 0.184189
\(370\) −29.0087 −1.50809
\(371\) −2.73230 −0.141854
\(372\) −1.66897 −0.0865322
\(373\) −12.8349 −0.664567 −0.332283 0.943180i \(-0.607819\pi\)
−0.332283 + 0.943180i \(0.607819\pi\)
\(374\) 30.2794 1.56571
\(375\) 10.2703 0.530354
\(376\) −1.36375 −0.0703301
\(377\) 0 0
\(378\) −1.19685 −0.0615592
\(379\) −36.0892 −1.85378 −0.926889 0.375334i \(-0.877528\pi\)
−0.926889 + 0.375334i \(0.877528\pi\)
\(380\) 4.36375 0.223856
\(381\) 1.57009 0.0804380
\(382\) 5.61275 0.287173
\(383\) −32.8134 −1.67669 −0.838344 0.545142i \(-0.816476\pi\)
−0.838344 + 0.545142i \(0.816476\pi\)
\(384\) −0.260281 −0.0132824
\(385\) −14.7604 −0.752258
\(386\) 7.88071 0.401117
\(387\) −24.4401 −1.24236
\(388\) 2.12923 0.108095
\(389\) −2.97899 −0.151041 −0.0755204 0.997144i \(-0.524062\pi\)
−0.0755204 + 0.997144i \(0.524062\pi\)
\(390\) 0 0
\(391\) 28.3130 1.43185
\(392\) 6.39917 0.323207
\(393\) 3.33866 0.168413
\(394\) −6.81454 −0.343312
\(395\) 35.6493 1.79371
\(396\) −12.7956 −0.643004
\(397\) −20.3722 −1.02245 −0.511225 0.859447i \(-0.670808\pi\)
−0.511225 + 0.859447i \(0.670808\pi\)
\(398\) 24.2470 1.21539
\(399\) −0.201753 −0.0101003
\(400\) 14.0423 0.702116
\(401\) 13.8861 0.693438 0.346719 0.937969i \(-0.387296\pi\)
0.346719 + 0.937969i \(0.387296\pi\)
\(402\) −2.99971 −0.149612
\(403\) 0 0
\(404\) 11.5122 0.572751
\(405\) 36.6331 1.82032
\(406\) 1.97169 0.0978533
\(407\) 29.0087 1.43791
\(408\) 1.80605 0.0894128
\(409\) −31.1280 −1.53918 −0.769590 0.638538i \(-0.779539\pi\)
−0.769590 + 0.638538i \(0.779539\pi\)
\(410\) 5.26544 0.260041
\(411\) −3.51774 −0.173517
\(412\) 8.59033 0.423215
\(413\) −9.62824 −0.473775
\(414\) −11.9646 −0.588030
\(415\) −12.0161 −0.589846
\(416\) 0 0
\(417\) 0.565756 0.0277052
\(418\) −4.36375 −0.213438
\(419\) −28.0907 −1.37232 −0.686160 0.727451i \(-0.740705\pi\)
−0.686160 + 0.727451i \(0.740705\pi\)
\(420\) −0.880398 −0.0429590
\(421\) −10.8442 −0.528512 −0.264256 0.964453i \(-0.585126\pi\)
−0.264256 + 0.964453i \(0.585126\pi\)
\(422\) 5.05776 0.246208
\(423\) −3.99886 −0.194431
\(424\) −3.52493 −0.171186
\(425\) −97.4375 −4.72641
\(426\) 3.15188 0.152709
\(427\) −1.57434 −0.0761874
\(428\) 12.2990 0.594495
\(429\) 0 0
\(430\) −36.3716 −1.75399
\(431\) 6.90864 0.332777 0.166389 0.986060i \(-0.446789\pi\)
0.166389 + 0.986060i \(0.446789\pi\)
\(432\) −1.54405 −0.0742882
\(433\) 39.3767 1.89232 0.946161 0.323697i \(-0.104926\pi\)
0.946161 + 0.323697i \(0.104926\pi\)
\(434\) −4.97032 −0.238583
\(435\) 2.88911 0.138522
\(436\) 8.17344 0.391437
\(437\) −4.08036 −0.195190
\(438\) −0.599558 −0.0286480
\(439\) −36.9430 −1.76319 −0.881597 0.472003i \(-0.843531\pi\)
−0.881597 + 0.472003i \(0.843531\pi\)
\(440\) −19.0423 −0.907807
\(441\) 18.7640 0.893523
\(442\) 0 0
\(443\) 10.0111 0.475642 0.237821 0.971309i \(-0.423567\pi\)
0.237821 + 0.971309i \(0.423567\pi\)
\(444\) 1.73026 0.0821143
\(445\) 14.5968 0.691953
\(446\) 14.0800 0.666706
\(447\) 2.71306 0.128324
\(448\) −0.775135 −0.0366217
\(449\) −19.3184 −0.911694 −0.455847 0.890058i \(-0.650664\pi\)
−0.455847 + 0.890058i \(0.650664\pi\)
\(450\) 41.1756 1.94104
\(451\) −5.26544 −0.247940
\(452\) 4.05116 0.190551
\(453\) −1.87380 −0.0880386
\(454\) −18.6540 −0.875478
\(455\) 0 0
\(456\) −0.260281 −0.0121888
\(457\) 8.71798 0.407810 0.203905 0.978991i \(-0.434637\pi\)
0.203905 + 0.978991i \(0.434637\pi\)
\(458\) −5.65636 −0.264304
\(459\) 10.7139 0.500084
\(460\) −17.8057 −0.830193
\(461\) 4.77897 0.222579 0.111290 0.993788i \(-0.464502\pi\)
0.111290 + 0.993788i \(0.464502\pi\)
\(462\) 0.880398 0.0409598
\(463\) 24.7123 1.14848 0.574239 0.818688i \(-0.305298\pi\)
0.574239 + 0.818688i \(0.305298\pi\)
\(464\) 2.54367 0.118087
\(465\) −7.28298 −0.337740
\(466\) 19.9409 0.923744
\(467\) −28.5590 −1.32155 −0.660776 0.750583i \(-0.729773\pi\)
−0.660776 + 0.750583i \(0.729773\pi\)
\(468\) 0 0
\(469\) −8.93334 −0.412503
\(470\) −5.95107 −0.274502
\(471\) 5.33889 0.246003
\(472\) −12.4214 −0.571740
\(473\) 36.3716 1.67237
\(474\) −2.12634 −0.0976662
\(475\) 14.0423 0.644306
\(476\) 5.37854 0.246525
\(477\) −10.3360 −0.473252
\(478\) 3.43781 0.157242
\(479\) −33.7512 −1.54213 −0.771066 0.636755i \(-0.780276\pi\)
−0.771066 + 0.636755i \(0.780276\pi\)
\(480\) −1.13580 −0.0518420
\(481\) 0 0
\(482\) −5.43845 −0.247715
\(483\) 0.823223 0.0374579
\(484\) 8.04232 0.365560
\(485\) 9.29141 0.421901
\(486\) −6.81718 −0.309234
\(487\) 5.50881 0.249628 0.124814 0.992180i \(-0.460167\pi\)
0.124814 + 0.992180i \(0.460167\pi\)
\(488\) −2.03105 −0.0919412
\(489\) 0.549666 0.0248568
\(490\) 27.9244 1.26149
\(491\) 22.4320 1.01234 0.506171 0.862433i \(-0.331060\pi\)
0.506171 + 0.862433i \(0.331060\pi\)
\(492\) −0.314063 −0.0141591
\(493\) −17.6502 −0.794924
\(494\) 0 0
\(495\) −55.8369 −2.50968
\(496\) −6.41220 −0.287916
\(497\) 9.38654 0.421044
\(498\) 0.716713 0.0321167
\(499\) 2.39076 0.107025 0.0535125 0.998567i \(-0.482958\pi\)
0.0535125 + 0.998567i \(0.482958\pi\)
\(500\) 39.4584 1.76463
\(501\) 5.28494 0.236114
\(502\) 25.9262 1.15714
\(503\) −22.0472 −0.983035 −0.491517 0.870868i \(-0.663558\pi\)
−0.491517 + 0.870868i \(0.663558\pi\)
\(504\) −2.27289 −0.101243
\(505\) 50.2362 2.23548
\(506\) 17.8057 0.791558
\(507\) 0 0
\(508\) 6.03228 0.267639
\(509\) 3.86985 0.171528 0.0857641 0.996315i \(-0.472667\pi\)
0.0857641 + 0.996315i \(0.472667\pi\)
\(510\) 7.88115 0.348983
\(511\) −1.78553 −0.0789871
\(512\) −1.00000 −0.0441942
\(513\) −1.54405 −0.0681715
\(514\) 1.88939 0.0833372
\(515\) 37.4861 1.65183
\(516\) 2.16942 0.0955035
\(517\) 5.95107 0.261728
\(518\) 5.15283 0.226402
\(519\) 6.02028 0.264261
\(520\) 0 0
\(521\) 41.7278 1.82813 0.914064 0.405570i \(-0.132927\pi\)
0.914064 + 0.405570i \(0.132927\pi\)
\(522\) 7.45870 0.326458
\(523\) −8.08529 −0.353545 −0.176773 0.984252i \(-0.556566\pi\)
−0.176773 + 0.984252i \(0.556566\pi\)
\(524\) 12.8272 0.560357
\(525\) −2.83308 −0.123646
\(526\) 4.19046 0.182713
\(527\) 44.4933 1.93816
\(528\) 1.13580 0.0494294
\(529\) −6.35068 −0.276117
\(530\) −15.3819 −0.668148
\(531\) −36.4226 −1.58061
\(532\) −0.775135 −0.0336064
\(533\) 0 0
\(534\) −0.870641 −0.0376763
\(535\) 53.6698 2.32035
\(536\) −11.5249 −0.497799
\(537\) −3.42400 −0.147757
\(538\) −5.06288 −0.218276
\(539\) −27.9244 −1.20279
\(540\) −6.73786 −0.289951
\(541\) 33.6272 1.44574 0.722872 0.690982i \(-0.242821\pi\)
0.722872 + 0.690982i \(0.242821\pi\)
\(542\) 23.4633 1.00784
\(543\) −0.143456 −0.00615627
\(544\) 6.93885 0.297501
\(545\) 35.6669 1.52780
\(546\) 0 0
\(547\) 24.6548 1.05417 0.527083 0.849814i \(-0.323286\pi\)
0.527083 + 0.849814i \(0.323286\pi\)
\(548\) −13.5152 −0.577339
\(549\) −5.95555 −0.254177
\(550\) −61.2772 −2.61287
\(551\) 2.54367 0.108364
\(552\) 1.06204 0.0452034
\(553\) −6.33240 −0.269281
\(554\) −7.03193 −0.298758
\(555\) 7.55041 0.320497
\(556\) 2.17364 0.0921827
\(557\) −8.69700 −0.368504 −0.184252 0.982879i \(-0.558986\pi\)
−0.184252 + 0.982879i \(0.558986\pi\)
\(558\) −18.8022 −0.795961
\(559\) 0 0
\(560\) −3.38249 −0.142936
\(561\) −7.88115 −0.332742
\(562\) 20.4814 0.863957
\(563\) 44.6483 1.88170 0.940851 0.338821i \(-0.110028\pi\)
0.940851 + 0.338821i \(0.110028\pi\)
\(564\) 0.354958 0.0149464
\(565\) 17.6783 0.743730
\(566\) 26.9345 1.13214
\(567\) −6.50716 −0.273275
\(568\) 12.1096 0.508106
\(569\) −41.8543 −1.75462 −0.877312 0.479920i \(-0.840666\pi\)
−0.877312 + 0.479920i \(0.840666\pi\)
\(570\) −1.13580 −0.0475734
\(571\) 1.94613 0.0814429 0.0407214 0.999171i \(-0.487034\pi\)
0.0407214 + 0.999171i \(0.487034\pi\)
\(572\) 0 0
\(573\) −1.46089 −0.0610296
\(574\) −0.935302 −0.0390387
\(575\) −57.2977 −2.38948
\(576\) −2.93225 −0.122177
\(577\) −17.3159 −0.720870 −0.360435 0.932784i \(-0.617372\pi\)
−0.360435 + 0.932784i \(0.617372\pi\)
\(578\) −31.1476 −1.29557
\(579\) −2.05120 −0.0852448
\(580\) 11.1000 0.460901
\(581\) 2.13442 0.0885507
\(582\) −0.554197 −0.0229722
\(583\) 15.3819 0.637054
\(584\) −2.30351 −0.0953197
\(585\) 0 0
\(586\) −9.68946 −0.400268
\(587\) −45.0778 −1.86056 −0.930279 0.366852i \(-0.880436\pi\)
−0.930279 + 0.366852i \(0.880436\pi\)
\(588\) −1.66558 −0.0686874
\(589\) −6.41220 −0.264210
\(590\) −54.2038 −2.23153
\(591\) 1.77369 0.0729600
\(592\) 6.64765 0.273217
\(593\) 39.5863 1.62561 0.812807 0.582533i \(-0.197938\pi\)
0.812807 + 0.582533i \(0.197938\pi\)
\(594\) 6.73786 0.276458
\(595\) 23.4706 0.962202
\(596\) 10.4236 0.426968
\(597\) −6.31102 −0.258293
\(598\) 0 0
\(599\) 15.9168 0.650341 0.325171 0.945655i \(-0.394578\pi\)
0.325171 + 0.945655i \(0.394578\pi\)
\(600\) −3.65495 −0.149213
\(601\) 21.8645 0.891870 0.445935 0.895065i \(-0.352871\pi\)
0.445935 + 0.895065i \(0.352871\pi\)
\(602\) 6.46069 0.263318
\(603\) −33.7939 −1.37619
\(604\) −7.19913 −0.292928
\(605\) 35.0947 1.42680
\(606\) −2.99639 −0.121720
\(607\) −5.59196 −0.226971 −0.113485 0.993540i \(-0.536201\pi\)
−0.113485 + 0.993540i \(0.536201\pi\)
\(608\) −1.00000 −0.0405554
\(609\) −0.513193 −0.0207956
\(610\) −8.86298 −0.358852
\(611\) 0 0
\(612\) 20.3465 0.822457
\(613\) −45.9423 −1.85559 −0.927796 0.373088i \(-0.878299\pi\)
−0.927796 + 0.373088i \(0.878299\pi\)
\(614\) −12.5815 −0.507748
\(615\) −1.37049 −0.0552636
\(616\) 3.38249 0.136285
\(617\) 10.6161 0.427389 0.213695 0.976900i \(-0.431450\pi\)
0.213695 + 0.976900i \(0.431450\pi\)
\(618\) −2.23590 −0.0899410
\(619\) 2.47379 0.0994299 0.0497149 0.998763i \(-0.484169\pi\)
0.0497149 + 0.998763i \(0.484169\pi\)
\(620\) −27.9813 −1.12375
\(621\) 6.30028 0.252822
\(622\) 16.7407 0.671242
\(623\) −2.59283 −0.103880
\(624\) 0 0
\(625\) 101.975 4.07901
\(626\) 11.5101 0.460037
\(627\) 1.13580 0.0453595
\(628\) 20.5120 0.818519
\(629\) −46.1271 −1.83921
\(630\) −9.91833 −0.395156
\(631\) 28.5339 1.13592 0.567958 0.823058i \(-0.307734\pi\)
0.567958 + 0.823058i \(0.307734\pi\)
\(632\) −8.16942 −0.324962
\(633\) −1.31644 −0.0523236
\(634\) 4.14425 0.164589
\(635\) 26.3234 1.04461
\(636\) 0.917471 0.0363801
\(637\) 0 0
\(638\) −11.1000 −0.439452
\(639\) 35.5083 1.40469
\(640\) −4.36375 −0.172492
\(641\) 22.3729 0.883678 0.441839 0.897094i \(-0.354326\pi\)
0.441839 + 0.897094i \(0.354326\pi\)
\(642\) −3.20119 −0.126341
\(643\) 11.1412 0.439365 0.219682 0.975571i \(-0.429498\pi\)
0.219682 + 0.975571i \(0.429498\pi\)
\(644\) 3.16283 0.124633
\(645\) 9.46682 0.372756
\(646\) 6.93885 0.273005
\(647\) −9.46200 −0.371990 −0.185995 0.982551i \(-0.559551\pi\)
−0.185995 + 0.982551i \(0.559551\pi\)
\(648\) −8.39487 −0.329782
\(649\) 54.2038 2.12768
\(650\) 0 0
\(651\) 1.29368 0.0507033
\(652\) 2.11182 0.0827053
\(653\) 9.39067 0.367485 0.183743 0.982974i \(-0.441179\pi\)
0.183743 + 0.982974i \(0.441179\pi\)
\(654\) −2.12739 −0.0831875
\(655\) 55.9745 2.18710
\(656\) −1.20663 −0.0471110
\(657\) −6.75446 −0.263517
\(658\) 1.05709 0.0412097
\(659\) −17.9190 −0.698027 −0.349013 0.937118i \(-0.613483\pi\)
−0.349013 + 0.937118i \(0.613483\pi\)
\(660\) 4.95635 0.192926
\(661\) −0.722654 −0.0281080 −0.0140540 0.999901i \(-0.504474\pi\)
−0.0140540 + 0.999901i \(0.504474\pi\)
\(662\) −0.316380 −0.0122965
\(663\) 0 0
\(664\) 2.75361 0.106861
\(665\) −3.38249 −0.131168
\(666\) 19.4926 0.755323
\(667\) −10.3791 −0.401880
\(668\) 20.3048 0.785615
\(669\) −3.66475 −0.141687
\(670\) −50.2917 −1.94294
\(671\) 8.86298 0.342152
\(672\) 0.201753 0.00778278
\(673\) 40.2647 1.55209 0.776045 0.630677i \(-0.217223\pi\)
0.776045 + 0.630677i \(0.217223\pi\)
\(674\) −17.7103 −0.682174
\(675\) −21.6821 −0.834543
\(676\) 0 0
\(677\) −19.1338 −0.735373 −0.367686 0.929950i \(-0.619850\pi\)
−0.367686 + 0.929950i \(0.619850\pi\)
\(678\) −1.05444 −0.0404955
\(679\) −1.65044 −0.0633380
\(680\) 30.2794 1.16116
\(681\) 4.85529 0.186055
\(682\) 27.9813 1.07146
\(683\) −34.3981 −1.31621 −0.658103 0.752928i \(-0.728641\pi\)
−0.658103 + 0.752928i \(0.728641\pi\)
\(684\) −2.93225 −0.112118
\(685\) −58.9768 −2.25339
\(686\) −10.3862 −0.396545
\(687\) 1.47224 0.0561695
\(688\) 8.33493 0.317766
\(689\) 0 0
\(690\) 4.63447 0.176431
\(691\) 41.8318 1.59136 0.795678 0.605719i \(-0.207115\pi\)
0.795678 + 0.605719i \(0.207115\pi\)
\(692\) 23.1299 0.879268
\(693\) 9.91833 0.376766
\(694\) −22.4138 −0.850818
\(695\) 9.48521 0.359794
\(696\) −0.662069 −0.0250957
\(697\) 8.37263 0.317136
\(698\) 20.0965 0.760665
\(699\) −5.19023 −0.196312
\(700\) −10.8847 −0.411403
\(701\) 9.76756 0.368916 0.184458 0.982840i \(-0.440947\pi\)
0.184458 + 0.982840i \(0.440947\pi\)
\(702\) 0 0
\(703\) 6.64765 0.250721
\(704\) 4.36375 0.164465
\(705\) 1.54895 0.0583368
\(706\) 11.7682 0.442903
\(707\) −8.92348 −0.335602
\(708\) 3.23304 0.121505
\(709\) 2.81576 0.105748 0.0528740 0.998601i \(-0.483162\pi\)
0.0528740 + 0.998601i \(0.483162\pi\)
\(710\) 52.8431 1.98317
\(711\) −23.9548 −0.898376
\(712\) −3.34501 −0.125359
\(713\) 26.1641 0.979852
\(714\) −1.39993 −0.0523911
\(715\) 0 0
\(716\) −13.1550 −0.491626
\(717\) −0.894795 −0.0334167
\(718\) 7.47763 0.279063
\(719\) 11.0765 0.413083 0.206542 0.978438i \(-0.433779\pi\)
0.206542 + 0.978438i \(0.433779\pi\)
\(720\) −12.7956 −0.476865
\(721\) −6.65867 −0.247982
\(722\) −1.00000 −0.0372161
\(723\) 1.41552 0.0526439
\(724\) −0.551157 −0.0204836
\(725\) 35.7191 1.32657
\(726\) −2.09326 −0.0776882
\(727\) 0.407640 0.0151185 0.00755927 0.999971i \(-0.497594\pi\)
0.00755927 + 0.999971i \(0.497594\pi\)
\(728\) 0 0
\(729\) −23.4102 −0.867046
\(730\) −10.0519 −0.372038
\(731\) −57.8348 −2.13910
\(732\) 0.528643 0.0195392
\(733\) 32.2976 1.19294 0.596470 0.802636i \(-0.296570\pi\)
0.596470 + 0.802636i \(0.296570\pi\)
\(734\) −16.4233 −0.606195
\(735\) −7.26818 −0.268091
\(736\) 4.08036 0.150404
\(737\) 50.2917 1.85252
\(738\) −3.53815 −0.130241
\(739\) 3.62754 0.133441 0.0667206 0.997772i \(-0.478746\pi\)
0.0667206 + 0.997772i \(0.478746\pi\)
\(740\) 29.0087 1.06638
\(741\) 0 0
\(742\) 2.73230 0.100306
\(743\) 8.04553 0.295162 0.147581 0.989050i \(-0.452851\pi\)
0.147581 + 0.989050i \(0.452851\pi\)
\(744\) 1.66897 0.0611875
\(745\) 45.4860 1.66648
\(746\) 12.8349 0.469920
\(747\) 8.07429 0.295423
\(748\) −30.2794 −1.10712
\(749\) −9.53339 −0.348342
\(750\) −10.2703 −0.375017
\(751\) 18.8879 0.689230 0.344615 0.938744i \(-0.388009\pi\)
0.344615 + 0.938744i \(0.388009\pi\)
\(752\) 1.36375 0.0497309
\(753\) −6.74808 −0.245914
\(754\) 0 0
\(755\) −31.4152 −1.14332
\(756\) 1.19685 0.0435289
\(757\) −21.6822 −0.788051 −0.394026 0.919099i \(-0.628918\pi\)
−0.394026 + 0.919099i \(0.628918\pi\)
\(758\) 36.0892 1.31082
\(759\) −4.63447 −0.168221
\(760\) −4.36375 −0.158290
\(761\) −1.28461 −0.0465670 −0.0232835 0.999729i \(-0.507412\pi\)
−0.0232835 + 0.999729i \(0.507412\pi\)
\(762\) −1.57009 −0.0568783
\(763\) −6.33552 −0.229361
\(764\) −5.61275 −0.203062
\(765\) 88.7869 3.21010
\(766\) 32.8134 1.18560
\(767\) 0 0
\(768\) 0.260281 0.00939207
\(769\) 11.4195 0.411796 0.205898 0.978573i \(-0.433988\pi\)
0.205898 + 0.978573i \(0.433988\pi\)
\(770\) 14.7604 0.531927
\(771\) −0.491771 −0.0177107
\(772\) −7.88071 −0.283633
\(773\) 35.6963 1.28391 0.641953 0.766744i \(-0.278124\pi\)
0.641953 + 0.766744i \(0.278124\pi\)
\(774\) 24.4401 0.878482
\(775\) −90.0422 −3.23441
\(776\) −2.12923 −0.0764348
\(777\) −1.34118 −0.0481146
\(778\) 2.97899 0.106802
\(779\) −1.20663 −0.0432321
\(780\) 0 0
\(781\) −52.8431 −1.89087
\(782\) −28.3130 −1.01247
\(783\) −3.92756 −0.140360
\(784\) −6.39917 −0.228542
\(785\) 89.5094 3.19473
\(786\) −3.33866 −0.119086
\(787\) 47.2502 1.68429 0.842144 0.539253i \(-0.181293\pi\)
0.842144 + 0.539253i \(0.181293\pi\)
\(788\) 6.81454 0.242758
\(789\) −1.09070 −0.0388298
\(790\) −35.6493 −1.26835
\(791\) −3.14019 −0.111652
\(792\) 12.7956 0.454673
\(793\) 0 0
\(794\) 20.3722 0.722981
\(795\) 4.00362 0.141994
\(796\) −24.2470 −0.859411
\(797\) 45.8247 1.62319 0.811597 0.584218i \(-0.198599\pi\)
0.811597 + 0.584218i \(0.198599\pi\)
\(798\) 0.201753 0.00714197
\(799\) −9.46286 −0.334772
\(800\) −14.0423 −0.496471
\(801\) −9.80841 −0.346563
\(802\) −13.8861 −0.490335
\(803\) 10.0519 0.354725
\(804\) 2.99971 0.105792
\(805\) 13.8018 0.486449
\(806\) 0 0
\(807\) 1.31777 0.0463877
\(808\) −11.5122 −0.404996
\(809\) 16.5686 0.582522 0.291261 0.956644i \(-0.405925\pi\)
0.291261 + 0.956644i \(0.405925\pi\)
\(810\) −36.6331 −1.28716
\(811\) −15.5277 −0.545252 −0.272626 0.962120i \(-0.587892\pi\)
−0.272626 + 0.962120i \(0.587892\pi\)
\(812\) −1.97169 −0.0691928
\(813\) −6.10705 −0.214184
\(814\) −29.0087 −1.01675
\(815\) 9.21546 0.322804
\(816\) −1.80605 −0.0632244
\(817\) 8.33493 0.291602
\(818\) 31.1280 1.08836
\(819\) 0 0
\(820\) −5.26544 −0.183877
\(821\) 2.24388 0.0783120 0.0391560 0.999233i \(-0.487533\pi\)
0.0391560 + 0.999233i \(0.487533\pi\)
\(822\) 3.51774 0.122695
\(823\) −16.5355 −0.576391 −0.288195 0.957572i \(-0.593055\pi\)
−0.288195 + 0.957572i \(0.593055\pi\)
\(824\) −8.59033 −0.299258
\(825\) 15.9493 0.555282
\(826\) 9.62824 0.335009
\(827\) 28.8538 1.00334 0.501672 0.865058i \(-0.332718\pi\)
0.501672 + 0.865058i \(0.332718\pi\)
\(828\) 11.9646 0.415800
\(829\) −16.8384 −0.584823 −0.292411 0.956293i \(-0.594458\pi\)
−0.292411 + 0.956293i \(0.594458\pi\)
\(830\) 12.0161 0.417084
\(831\) 1.83028 0.0634916
\(832\) 0 0
\(833\) 44.4028 1.53847
\(834\) −0.565756 −0.0195905
\(835\) 88.6050 3.06630
\(836\) 4.36375 0.150923
\(837\) 9.90077 0.342221
\(838\) 28.0907 0.970376
\(839\) −34.0552 −1.17572 −0.587859 0.808964i \(-0.700029\pi\)
−0.587859 + 0.808964i \(0.700029\pi\)
\(840\) 0.880398 0.0303766
\(841\) −22.5297 −0.776887
\(842\) 10.8442 0.373714
\(843\) −5.33092 −0.183607
\(844\) −5.05776 −0.174095
\(845\) 0 0
\(846\) 3.99886 0.137484
\(847\) −6.23388 −0.214199
\(848\) 3.52493 0.121047
\(849\) −7.01052 −0.240601
\(850\) 97.4375 3.34208
\(851\) −27.1248 −0.929826
\(852\) −3.15188 −0.107982
\(853\) −5.48844 −0.187921 −0.0939603 0.995576i \(-0.529953\pi\)
−0.0939603 + 0.995576i \(0.529953\pi\)
\(854\) 1.57434 0.0538726
\(855\) −12.7956 −0.437601
\(856\) −12.2990 −0.420371
\(857\) −12.5021 −0.427064 −0.213532 0.976936i \(-0.568497\pi\)
−0.213532 + 0.976936i \(0.568497\pi\)
\(858\) 0 0
\(859\) −37.7834 −1.28915 −0.644577 0.764540i \(-0.722966\pi\)
−0.644577 + 0.764540i \(0.722966\pi\)
\(860\) 36.3716 1.24026
\(861\) 0.243441 0.00829645
\(862\) −6.90864 −0.235309
\(863\) −35.2458 −1.19978 −0.599891 0.800082i \(-0.704789\pi\)
−0.599891 + 0.800082i \(0.704789\pi\)
\(864\) 1.54405 0.0525297
\(865\) 100.933 3.43183
\(866\) −39.3767 −1.33807
\(867\) 8.10713 0.275333
\(868\) 4.97032 0.168704
\(869\) 35.6493 1.20932
\(870\) −2.88911 −0.0979499
\(871\) 0 0
\(872\) −8.17344 −0.276788
\(873\) −6.24343 −0.211308
\(874\) 4.08036 0.138020
\(875\) −30.5856 −1.03398
\(876\) 0.599558 0.0202572
\(877\) 36.9518 1.24777 0.623887 0.781515i \(-0.285553\pi\)
0.623887 + 0.781515i \(0.285553\pi\)
\(878\) 36.9430 1.24677
\(879\) 2.52198 0.0850643
\(880\) 19.0423 0.641916
\(881\) 1.25780 0.0423764 0.0211882 0.999776i \(-0.493255\pi\)
0.0211882 + 0.999776i \(0.493255\pi\)
\(882\) −18.7640 −0.631816
\(883\) 7.95258 0.267626 0.133813 0.991007i \(-0.457278\pi\)
0.133813 + 0.991007i \(0.457278\pi\)
\(884\) 0 0
\(885\) 14.1082 0.474242
\(886\) −10.0111 −0.336330
\(887\) −6.38803 −0.214489 −0.107245 0.994233i \(-0.534203\pi\)
−0.107245 + 0.994233i \(0.534203\pi\)
\(888\) −1.73026 −0.0580636
\(889\) −4.67583 −0.156822
\(890\) −14.5968 −0.489285
\(891\) 36.6331 1.22726
\(892\) −14.0800 −0.471432
\(893\) 1.36375 0.0456362
\(894\) −2.71306 −0.0907385
\(895\) −57.4053 −1.91885
\(896\) 0.775135 0.0258954
\(897\) 0 0
\(898\) 19.3184 0.644665
\(899\) −16.3106 −0.543987
\(900\) −41.1756 −1.37252
\(901\) −24.4590 −0.814846
\(902\) 5.26544 0.175320
\(903\) −1.68159 −0.0559600
\(904\) −4.05116 −0.134740
\(905\) −2.40511 −0.0799487
\(906\) 1.87380 0.0622527
\(907\) −10.0113 −0.332420 −0.166210 0.986090i \(-0.553153\pi\)
−0.166210 + 0.986090i \(0.553153\pi\)
\(908\) 18.6540 0.619056
\(909\) −33.7566 −1.11964
\(910\) 0 0
\(911\) −31.2277 −1.03462 −0.517309 0.855798i \(-0.673066\pi\)
−0.517309 + 0.855798i \(0.673066\pi\)
\(912\) 0.260281 0.00861876
\(913\) −12.0161 −0.397674
\(914\) −8.71798 −0.288365
\(915\) 2.30686 0.0762626
\(916\) 5.65636 0.186891
\(917\) −9.94277 −0.328339
\(918\) −10.7139 −0.353613
\(919\) −12.4072 −0.409277 −0.204638 0.978838i \(-0.565602\pi\)
−0.204638 + 0.978838i \(0.565602\pi\)
\(920\) 17.8057 0.587035
\(921\) 3.27472 0.107906
\(922\) −4.77897 −0.157387
\(923\) 0 0
\(924\) −0.880398 −0.0289630
\(925\) 93.3485 3.06928
\(926\) −24.7123 −0.812097
\(927\) −25.1890 −0.827316
\(928\) −2.54367 −0.0835002
\(929\) 51.4106 1.68673 0.843363 0.537344i \(-0.180572\pi\)
0.843363 + 0.537344i \(0.180572\pi\)
\(930\) 7.28298 0.238818
\(931\) −6.39917 −0.209724
\(932\) −19.9409 −0.653185
\(933\) −4.35729 −0.142651
\(934\) 28.5590 0.934479
\(935\) −132.132 −4.32117
\(936\) 0 0
\(937\) −2.03140 −0.0663628 −0.0331814 0.999449i \(-0.510564\pi\)
−0.0331814 + 0.999449i \(0.510564\pi\)
\(938\) 8.93334 0.291684
\(939\) −2.99586 −0.0977663
\(940\) 5.95107 0.194102
\(941\) 42.4246 1.38300 0.691501 0.722376i \(-0.256950\pi\)
0.691501 + 0.722376i \(0.256950\pi\)
\(942\) −5.33889 −0.173950
\(943\) 4.92349 0.160331
\(944\) 12.4214 0.404281
\(945\) 5.22275 0.169896
\(946\) −36.3716 −1.18254
\(947\) 7.29296 0.236989 0.118495 0.992955i \(-0.462193\pi\)
0.118495 + 0.992955i \(0.462193\pi\)
\(948\) 2.12634 0.0690604
\(949\) 0 0
\(950\) −14.0423 −0.455593
\(951\) −1.07867 −0.0349782
\(952\) −5.37854 −0.174320
\(953\) −32.7289 −1.06019 −0.530096 0.847938i \(-0.677844\pi\)
−0.530096 + 0.847938i \(0.677844\pi\)
\(954\) 10.3360 0.334640
\(955\) −24.4926 −0.792563
\(956\) −3.43781 −0.111187
\(957\) 2.88911 0.0933915
\(958\) 33.7512 1.09045
\(959\) 10.4761 0.338290
\(960\) 1.13580 0.0366578
\(961\) 10.1163 0.326334
\(962\) 0 0
\(963\) −36.0638 −1.16214
\(964\) 5.43845 0.175161
\(965\) −34.3894 −1.10704
\(966\) −0.823223 −0.0264868
\(967\) 2.05333 0.0660307 0.0330154 0.999455i \(-0.489489\pi\)
0.0330154 + 0.999455i \(0.489489\pi\)
\(968\) −8.04232 −0.258490
\(969\) −1.80605 −0.0580187
\(970\) −9.29141 −0.298329
\(971\) −41.6592 −1.33691 −0.668454 0.743754i \(-0.733044\pi\)
−0.668454 + 0.743754i \(0.733044\pi\)
\(972\) 6.81718 0.218661
\(973\) −1.68486 −0.0540142
\(974\) −5.50881 −0.176514
\(975\) 0 0
\(976\) 2.03105 0.0650122
\(977\) −56.4520 −1.80606 −0.903030 0.429576i \(-0.858663\pi\)
−0.903030 + 0.429576i \(0.858663\pi\)
\(978\) −0.549666 −0.0175764
\(979\) 14.5968 0.466515
\(980\) −27.9244 −0.892011
\(981\) −23.9666 −0.765195
\(982\) −22.4320 −0.715835
\(983\) −32.1452 −1.02527 −0.512636 0.858606i \(-0.671331\pi\)
−0.512636 + 0.858606i \(0.671331\pi\)
\(984\) 0.314063 0.0100120
\(985\) 29.7370 0.947498
\(986\) 17.6502 0.562096
\(987\) −0.275140 −0.00875781
\(988\) 0 0
\(989\) −34.0095 −1.08144
\(990\) 55.8369 1.77461
\(991\) 35.1925 1.11793 0.558964 0.829192i \(-0.311199\pi\)
0.558964 + 0.829192i \(0.311199\pi\)
\(992\) 6.41220 0.203588
\(993\) 0.0823477 0.00261323
\(994\) −9.38654 −0.297723
\(995\) −105.808 −3.35433
\(996\) −0.716713 −0.0227099
\(997\) −29.2648 −0.926827 −0.463413 0.886142i \(-0.653375\pi\)
−0.463413 + 0.886142i \(0.653375\pi\)
\(998\) −2.39076 −0.0756781
\(999\) −10.2643 −0.324749
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 6422.2.a.bc.1.3 6
13.4 even 6 494.2.g.f.419.4 yes 12
13.10 even 6 494.2.g.f.191.4 12
13.12 even 2 6422.2.a.bd.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
494.2.g.f.191.4 12 13.10 even 6
494.2.g.f.419.4 yes 12 13.4 even 6
6422.2.a.bc.1.3 6 1.1 even 1 trivial
6422.2.a.bd.1.3 6 13.12 even 2