Properties

Label 6422.2.a.r.1.2
Level $6422$
Weight $2$
Character 6422.1
Self dual yes
Analytic conductor $51.280$
Analytic rank $1$
Dimension $3$
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] = [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: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) 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 1.2
Root \(0.445042\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.35690 q^{3} +1.00000 q^{4} -2.69202 q^{5} -2.35690 q^{6} +2.49396 q^{7} +1.00000 q^{8} +2.55496 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.35690 q^{3} +1.00000 q^{4} -2.69202 q^{5} -2.35690 q^{6} +2.49396 q^{7} +1.00000 q^{8} +2.55496 q^{9} -2.69202 q^{10} +3.60388 q^{11} -2.35690 q^{12} +2.49396 q^{14} +6.34481 q^{15} +1.00000 q^{16} -4.54288 q^{17} +2.55496 q^{18} -1.00000 q^{19} -2.69202 q^{20} -5.87800 q^{21} +3.60388 q^{22} -2.21983 q^{23} -2.35690 q^{24} +2.24698 q^{25} +1.04892 q^{27} +2.49396 q^{28} -6.98792 q^{29} +6.34481 q^{30} +6.45473 q^{31} +1.00000 q^{32} -8.49396 q^{33} -4.54288 q^{34} -6.71379 q^{35} +2.55496 q^{36} -8.71379 q^{37} -1.00000 q^{38} -2.69202 q^{40} +9.70171 q^{41} -5.87800 q^{42} -4.09783 q^{43} +3.60388 q^{44} -6.87800 q^{45} -2.21983 q^{46} -2.35690 q^{48} -0.780167 q^{49} +2.24698 q^{50} +10.7071 q^{51} -0.176292 q^{53} +1.04892 q^{54} -9.70171 q^{55} +2.49396 q^{56} +2.35690 q^{57} -6.98792 q^{58} +7.28382 q^{59} +6.34481 q^{60} +8.51573 q^{61} +6.45473 q^{62} +6.37196 q^{63} +1.00000 q^{64} -8.49396 q^{66} +8.05861 q^{67} -4.54288 q^{68} +5.23191 q^{69} -6.71379 q^{70} +1.28083 q^{71} +2.55496 q^{72} -13.9705 q^{73} -8.71379 q^{74} -5.29590 q^{75} -1.00000 q^{76} +8.98792 q^{77} +2.17092 q^{79} -2.69202 q^{80} -10.1371 q^{81} +9.70171 q^{82} +15.7017 q^{83} -5.87800 q^{84} +12.2295 q^{85} -4.09783 q^{86} +16.4698 q^{87} +3.60388 q^{88} -12.1957 q^{89} -6.87800 q^{90} -2.21983 q^{92} -15.2131 q^{93} +2.69202 q^{95} -2.35690 q^{96} -6.89008 q^{97} -0.780167 q^{98} +9.20775 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} - 2 q^{7} + 3 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} - 2 q^{7} + 3 q^{8} + 8 q^{9} - 3 q^{10} + 2 q^{11} - 3 q^{12} - 2 q^{14} - 4 q^{15} + 3 q^{16} + 5 q^{17} + 8 q^{18} - 3 q^{19} - 3 q^{20} + 2 q^{21} + 2 q^{22} - 8 q^{23} - 3 q^{24} + 2 q^{25} - 6 q^{27} - 2 q^{28} - 2 q^{29} - 4 q^{30} - 3 q^{31} + 3 q^{32} - 16 q^{33} + 5 q^{34} - 12 q^{35} + 8 q^{36} - 18 q^{37} - 3 q^{38} - 3 q^{40} + 2 q^{41} + 2 q^{42} + 6 q^{43} + 2 q^{44} - q^{45} - 8 q^{46} - 3 q^{48} - q^{49} + 2 q^{50} + 2 q^{51} - 8 q^{53} - 6 q^{54} - 2 q^{55} - 2 q^{56} + 3 q^{57} - 2 q^{58} - 11 q^{59} - 4 q^{60} + 13 q^{61} - 3 q^{62} - 10 q^{63} + 3 q^{64} - 16 q^{66} - 7 q^{67} + 5 q^{68} + 36 q^{69} - 12 q^{70} + 15 q^{71} + 8 q^{72} - 7 q^{73} - 18 q^{74} - 2 q^{75} - 3 q^{76} + 8 q^{77} + 17 q^{79} - 3 q^{80} - 25 q^{81} + 2 q^{82} + 20 q^{83} + 2 q^{84} + 16 q^{85} + 6 q^{86} + 2 q^{87} + 2 q^{88} - q^{90} - 8 q^{92} - 25 q^{93} + 3 q^{95} - 3 q^{96} - 20 q^{97} - q^{98} + 10 q^{99}+O(q^{100}) \) 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
\(3\) −2.35690 −1.36075 −0.680377 0.732862i \(-0.738184\pi\)
−0.680377 + 0.732862i \(0.738184\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.69202 −1.20391 −0.601954 0.798531i \(-0.705611\pi\)
−0.601954 + 0.798531i \(0.705611\pi\)
\(6\) −2.35690 −0.962199
\(7\) 2.49396 0.942628 0.471314 0.881965i \(-0.343780\pi\)
0.471314 + 0.881965i \(0.343780\pi\)
\(8\) 1.00000 0.353553
\(9\) 2.55496 0.851653
\(10\) −2.69202 −0.851292
\(11\) 3.60388 1.08661 0.543305 0.839536i \(-0.317173\pi\)
0.543305 + 0.839536i \(0.317173\pi\)
\(12\) −2.35690 −0.680377
\(13\) 0 0
\(14\) 2.49396 0.666539
\(15\) 6.34481 1.63822
\(16\) 1.00000 0.250000
\(17\) −4.54288 −1.10181 −0.550905 0.834568i \(-0.685717\pi\)
−0.550905 + 0.834568i \(0.685717\pi\)
\(18\) 2.55496 0.602209
\(19\) −1.00000 −0.229416
\(20\) −2.69202 −0.601954
\(21\) −5.87800 −1.28269
\(22\) 3.60388 0.768349
\(23\) −2.21983 −0.462867 −0.231434 0.972851i \(-0.574342\pi\)
−0.231434 + 0.972851i \(0.574342\pi\)
\(24\) −2.35690 −0.481099
\(25\) 2.24698 0.449396
\(26\) 0 0
\(27\) 1.04892 0.201864
\(28\) 2.49396 0.471314
\(29\) −6.98792 −1.29762 −0.648812 0.760949i \(-0.724734\pi\)
−0.648812 + 0.760949i \(0.724734\pi\)
\(30\) 6.34481 1.15840
\(31\) 6.45473 1.15930 0.579652 0.814864i \(-0.303189\pi\)
0.579652 + 0.814864i \(0.303189\pi\)
\(32\) 1.00000 0.176777
\(33\) −8.49396 −1.47861
\(34\) −4.54288 −0.779097
\(35\) −6.71379 −1.13484
\(36\) 2.55496 0.425826
\(37\) −8.71379 −1.43254 −0.716269 0.697824i \(-0.754152\pi\)
−0.716269 + 0.697824i \(0.754152\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) −2.69202 −0.425646
\(41\) 9.70171 1.51515 0.757576 0.652747i \(-0.226383\pi\)
0.757576 + 0.652747i \(0.226383\pi\)
\(42\) −5.87800 −0.906995
\(43\) −4.09783 −0.624914 −0.312457 0.949932i \(-0.601152\pi\)
−0.312457 + 0.949932i \(0.601152\pi\)
\(44\) 3.60388 0.543305
\(45\) −6.87800 −1.02531
\(46\) −2.21983 −0.327296
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −2.35690 −0.340189
\(49\) −0.780167 −0.111452
\(50\) 2.24698 0.317771
\(51\) 10.7071 1.49929
\(52\) 0 0
\(53\) −0.176292 −0.0242156 −0.0121078 0.999927i \(-0.503854\pi\)
−0.0121078 + 0.999927i \(0.503854\pi\)
\(54\) 1.04892 0.142740
\(55\) −9.70171 −1.30818
\(56\) 2.49396 0.333269
\(57\) 2.35690 0.312178
\(58\) −6.98792 −0.917559
\(59\) 7.28382 0.948272 0.474136 0.880452i \(-0.342761\pi\)
0.474136 + 0.880452i \(0.342761\pi\)
\(60\) 6.34481 0.819112
\(61\) 8.51573 1.09033 0.545164 0.838330i \(-0.316467\pi\)
0.545164 + 0.838330i \(0.316467\pi\)
\(62\) 6.45473 0.819752
\(63\) 6.37196 0.802792
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −8.49396 −1.04553
\(67\) 8.05861 0.984515 0.492258 0.870450i \(-0.336172\pi\)
0.492258 + 0.870450i \(0.336172\pi\)
\(68\) −4.54288 −0.550905
\(69\) 5.23191 0.629848
\(70\) −6.71379 −0.802452
\(71\) 1.28083 0.152007 0.0760034 0.997108i \(-0.475784\pi\)
0.0760034 + 0.997108i \(0.475784\pi\)
\(72\) 2.55496 0.301105
\(73\) −13.9705 −1.63512 −0.817559 0.575844i \(-0.804673\pi\)
−0.817559 + 0.575844i \(0.804673\pi\)
\(74\) −8.71379 −1.01296
\(75\) −5.29590 −0.611518
\(76\) −1.00000 −0.114708
\(77\) 8.98792 1.02427
\(78\) 0 0
\(79\) 2.17092 0.244247 0.122124 0.992515i \(-0.461030\pi\)
0.122124 + 0.992515i \(0.461030\pi\)
\(80\) −2.69202 −0.300977
\(81\) −10.1371 −1.12634
\(82\) 9.70171 1.07137
\(83\) 15.7017 1.72349 0.861743 0.507345i \(-0.169373\pi\)
0.861743 + 0.507345i \(0.169373\pi\)
\(84\) −5.87800 −0.641343
\(85\) 12.2295 1.32648
\(86\) −4.09783 −0.441881
\(87\) 16.4698 1.76575
\(88\) 3.60388 0.384174
\(89\) −12.1957 −1.29274 −0.646369 0.763025i \(-0.723713\pi\)
−0.646369 + 0.763025i \(0.723713\pi\)
\(90\) −6.87800 −0.725005
\(91\) 0 0
\(92\) −2.21983 −0.231434
\(93\) −15.2131 −1.57753
\(94\) 0 0
\(95\) 2.69202 0.276196
\(96\) −2.35690 −0.240550
\(97\) −6.89008 −0.699582 −0.349791 0.936828i \(-0.613747\pi\)
−0.349791 + 0.936828i \(0.613747\pi\)
\(98\) −0.780167 −0.0788088
\(99\) 9.20775 0.925414
\(100\) 2.24698 0.224698
\(101\) 5.46011 0.543301 0.271650 0.962396i \(-0.412431\pi\)
0.271650 + 0.962396i \(0.412431\pi\)
\(102\) 10.7071 1.06016
\(103\) 8.17390 0.805398 0.402699 0.915332i \(-0.368072\pi\)
0.402699 + 0.915332i \(0.368072\pi\)
\(104\) 0 0
\(105\) 15.8237 1.54424
\(106\) −0.176292 −0.0171230
\(107\) 5.25667 0.508181 0.254091 0.967180i \(-0.418224\pi\)
0.254091 + 0.967180i \(0.418224\pi\)
\(108\) 1.04892 0.100932
\(109\) −7.70171 −0.737690 −0.368845 0.929491i \(-0.620247\pi\)
−0.368845 + 0.929491i \(0.620247\pi\)
\(110\) −9.70171 −0.925022
\(111\) 20.5375 1.94933
\(112\) 2.49396 0.235657
\(113\) −6.09783 −0.573636 −0.286818 0.957985i \(-0.592598\pi\)
−0.286818 + 0.957985i \(0.592598\pi\)
\(114\) 2.35690 0.220744
\(115\) 5.97584 0.557250
\(116\) −6.98792 −0.648812
\(117\) 0 0
\(118\) 7.28382 0.670530
\(119\) −11.3297 −1.03860
\(120\) 6.34481 0.579200
\(121\) 1.98792 0.180720
\(122\) 8.51573 0.770978
\(123\) −22.8659 −2.06175
\(124\) 6.45473 0.579652
\(125\) 7.41119 0.662877
\(126\) 6.37196 0.567659
\(127\) −21.6625 −1.92223 −0.961117 0.276141i \(-0.910944\pi\)
−0.961117 + 0.276141i \(0.910944\pi\)
\(128\) 1.00000 0.0883883
\(129\) 9.65817 0.850354
\(130\) 0 0
\(131\) 3.72587 0.325531 0.162766 0.986665i \(-0.447959\pi\)
0.162766 + 0.986665i \(0.447959\pi\)
\(132\) −8.49396 −0.739304
\(133\) −2.49396 −0.216254
\(134\) 8.05861 0.696158
\(135\) −2.82371 −0.243026
\(136\) −4.54288 −0.389548
\(137\) −10.7463 −0.918120 −0.459060 0.888405i \(-0.651814\pi\)
−0.459060 + 0.888405i \(0.651814\pi\)
\(138\) 5.23191 0.445370
\(139\) 8.68963 0.737045 0.368522 0.929619i \(-0.379864\pi\)
0.368522 + 0.929619i \(0.379864\pi\)
\(140\) −6.71379 −0.567419
\(141\) 0 0
\(142\) 1.28083 0.107485
\(143\) 0 0
\(144\) 2.55496 0.212913
\(145\) 18.8116 1.56222
\(146\) −13.9705 −1.15620
\(147\) 1.83877 0.151659
\(148\) −8.71379 −0.716269
\(149\) 6.64071 0.544028 0.272014 0.962293i \(-0.412310\pi\)
0.272014 + 0.962293i \(0.412310\pi\)
\(150\) −5.29590 −0.432408
\(151\) −20.2446 −1.64748 −0.823741 0.566967i \(-0.808117\pi\)
−0.823741 + 0.566967i \(0.808117\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −11.6069 −0.938359
\(154\) 8.98792 0.724267
\(155\) −17.3763 −1.39570
\(156\) 0 0
\(157\) −19.5308 −1.55873 −0.779364 0.626572i \(-0.784457\pi\)
−0.779364 + 0.626572i \(0.784457\pi\)
\(158\) 2.17092 0.172709
\(159\) 0.415502 0.0329514
\(160\) −2.69202 −0.212823
\(161\) −5.53617 −0.436311
\(162\) −10.1371 −0.796443
\(163\) 0.944378 0.0739694 0.0369847 0.999316i \(-0.488225\pi\)
0.0369847 + 0.999316i \(0.488225\pi\)
\(164\) 9.70171 0.757576
\(165\) 22.8659 1.78011
\(166\) 15.7017 1.21869
\(167\) −9.54527 −0.738635 −0.369318 0.929303i \(-0.620409\pi\)
−0.369318 + 0.929303i \(0.620409\pi\)
\(168\) −5.87800 −0.453498
\(169\) 0 0
\(170\) 12.2295 0.937961
\(171\) −2.55496 −0.195383
\(172\) −4.09783 −0.312457
\(173\) 16.7681 1.27485 0.637427 0.770511i \(-0.279999\pi\)
0.637427 + 0.770511i \(0.279999\pi\)
\(174\) 16.4698 1.24857
\(175\) 5.60388 0.423613
\(176\) 3.60388 0.271652
\(177\) −17.1672 −1.29037
\(178\) −12.1957 −0.914104
\(179\) −7.37867 −0.551507 −0.275754 0.961228i \(-0.588927\pi\)
−0.275754 + 0.961228i \(0.588927\pi\)
\(180\) −6.87800 −0.512656
\(181\) −22.7875 −1.69378 −0.846889 0.531769i \(-0.821527\pi\)
−0.846889 + 0.531769i \(0.821527\pi\)
\(182\) 0 0
\(183\) −20.0707 −1.48367
\(184\) −2.21983 −0.163648
\(185\) 23.4577 1.72465
\(186\) −15.2131 −1.11548
\(187\) −16.3720 −1.19724
\(188\) 0 0
\(189\) 2.61596 0.190283
\(190\) 2.69202 0.195300
\(191\) −3.54958 −0.256839 −0.128419 0.991720i \(-0.540990\pi\)
−0.128419 + 0.991720i \(0.540990\pi\)
\(192\) −2.35690 −0.170094
\(193\) −26.6896 −1.92116 −0.960581 0.278001i \(-0.910328\pi\)
−0.960581 + 0.278001i \(0.910328\pi\)
\(194\) −6.89008 −0.494679
\(195\) 0 0
\(196\) −0.780167 −0.0557262
\(197\) 10.1032 0.719824 0.359912 0.932986i \(-0.382807\pi\)
0.359912 + 0.932986i \(0.382807\pi\)
\(198\) 9.20775 0.654366
\(199\) −4.39612 −0.311633 −0.155817 0.987786i \(-0.549801\pi\)
−0.155817 + 0.987786i \(0.549801\pi\)
\(200\) 2.24698 0.158885
\(201\) −18.9933 −1.33968
\(202\) 5.46011 0.384172
\(203\) −17.4276 −1.22318
\(204\) 10.7071 0.749646
\(205\) −26.1172 −1.82411
\(206\) 8.17390 0.569503
\(207\) −5.67158 −0.394202
\(208\) 0 0
\(209\) −3.60388 −0.249285
\(210\) 15.8237 1.09194
\(211\) −9.12498 −0.628190 −0.314095 0.949392i \(-0.601701\pi\)
−0.314095 + 0.949392i \(0.601701\pi\)
\(212\) −0.176292 −0.0121078
\(213\) −3.01879 −0.206844
\(214\) 5.25667 0.359338
\(215\) 11.0315 0.752339
\(216\) 1.04892 0.0713698
\(217\) 16.0978 1.09279
\(218\) −7.70171 −0.521626
\(219\) 32.9269 2.22500
\(220\) −9.70171 −0.654089
\(221\) 0 0
\(222\) 20.5375 1.37839
\(223\) −7.70709 −0.516105 −0.258052 0.966131i \(-0.583081\pi\)
−0.258052 + 0.966131i \(0.583081\pi\)
\(224\) 2.49396 0.166635
\(225\) 5.74094 0.382729
\(226\) −6.09783 −0.405622
\(227\) 3.86294 0.256392 0.128196 0.991749i \(-0.459081\pi\)
0.128196 + 0.991749i \(0.459081\pi\)
\(228\) 2.35690 0.156089
\(229\) 12.1196 0.800886 0.400443 0.916322i \(-0.368856\pi\)
0.400443 + 0.916322i \(0.368856\pi\)
\(230\) 5.97584 0.394035
\(231\) −21.1836 −1.39378
\(232\) −6.98792 −0.458779
\(233\) −17.8019 −1.16624 −0.583122 0.812385i \(-0.698169\pi\)
−0.583122 + 0.812385i \(0.698169\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 7.28382 0.474136
\(237\) −5.11662 −0.332360
\(238\) −11.3297 −0.734399
\(239\) 6.78746 0.439044 0.219522 0.975607i \(-0.429550\pi\)
0.219522 + 0.975607i \(0.429550\pi\)
\(240\) 6.34481 0.409556
\(241\) 10.8116 0.696438 0.348219 0.937413i \(-0.386787\pi\)
0.348219 + 0.937413i \(0.386787\pi\)
\(242\) 1.98792 0.127788
\(243\) 20.7453 1.33081
\(244\) 8.51573 0.545164
\(245\) 2.10023 0.134179
\(246\) −22.8659 −1.45788
\(247\) 0 0
\(248\) 6.45473 0.409876
\(249\) −37.0073 −2.34524
\(250\) 7.41119 0.468725
\(251\) −5.50604 −0.347538 −0.173769 0.984786i \(-0.555595\pi\)
−0.173769 + 0.984786i \(0.555595\pi\)
\(252\) 6.37196 0.401396
\(253\) −8.00000 −0.502956
\(254\) −21.6625 −1.35922
\(255\) −28.8237 −1.80501
\(256\) 1.00000 0.0625000
\(257\) 14.9879 0.934921 0.467460 0.884014i \(-0.345169\pi\)
0.467460 + 0.884014i \(0.345169\pi\)
\(258\) 9.65817 0.601291
\(259\) −21.7318 −1.35035
\(260\) 0 0
\(261\) −17.8538 −1.10512
\(262\) 3.72587 0.230185
\(263\) 11.4276 0.704655 0.352327 0.935877i \(-0.385390\pi\)
0.352327 + 0.935877i \(0.385390\pi\)
\(264\) −8.49396 −0.522767
\(265\) 0.474582 0.0291533
\(266\) −2.49396 −0.152914
\(267\) 28.7439 1.75910
\(268\) 8.05861 0.492258
\(269\) −22.4155 −1.36670 −0.683349 0.730092i \(-0.739477\pi\)
−0.683349 + 0.730092i \(0.739477\pi\)
\(270\) −2.82371 −0.171845
\(271\) 4.39612 0.267046 0.133523 0.991046i \(-0.457371\pi\)
0.133523 + 0.991046i \(0.457371\pi\)
\(272\) −4.54288 −0.275452
\(273\) 0 0
\(274\) −10.7463 −0.649209
\(275\) 8.09783 0.488318
\(276\) 5.23191 0.314924
\(277\) −8.71140 −0.523417 −0.261709 0.965147i \(-0.584286\pi\)
−0.261709 + 0.965147i \(0.584286\pi\)
\(278\) 8.68963 0.521169
\(279\) 16.4916 0.987324
\(280\) −6.71379 −0.401226
\(281\) −22.3913 −1.33576 −0.667878 0.744271i \(-0.732797\pi\)
−0.667878 + 0.744271i \(0.732797\pi\)
\(282\) 0 0
\(283\) −16.7332 −0.994684 −0.497342 0.867555i \(-0.665691\pi\)
−0.497342 + 0.867555i \(0.665691\pi\)
\(284\) 1.28083 0.0760034
\(285\) −6.34481 −0.375834
\(286\) 0 0
\(287\) 24.1957 1.42823
\(288\) 2.55496 0.150552
\(289\) 3.63773 0.213984
\(290\) 18.8116 1.10466
\(291\) 16.2392 0.951959
\(292\) −13.9705 −0.817559
\(293\) −1.84787 −0.107954 −0.0539769 0.998542i \(-0.517190\pi\)
−0.0539769 + 0.998542i \(0.517190\pi\)
\(294\) 1.83877 0.107239
\(295\) −19.6082 −1.14163
\(296\) −8.71379 −0.506479
\(297\) 3.78017 0.219348
\(298\) 6.64071 0.384686
\(299\) 0 0
\(300\) −5.29590 −0.305759
\(301\) −10.2198 −0.589061
\(302\) −20.2446 −1.16495
\(303\) −12.8689 −0.739299
\(304\) −1.00000 −0.0573539
\(305\) −22.9245 −1.31265
\(306\) −11.6069 −0.663520
\(307\) −32.4198 −1.85030 −0.925148 0.379606i \(-0.876060\pi\)
−0.925148 + 0.379606i \(0.876060\pi\)
\(308\) 8.98792 0.512134
\(309\) −19.2650 −1.09595
\(310\) −17.3763 −0.986906
\(311\) 1.37329 0.0778721 0.0389360 0.999242i \(-0.487603\pi\)
0.0389360 + 0.999242i \(0.487603\pi\)
\(312\) 0 0
\(313\) −31.2295 −1.76520 −0.882598 0.470128i \(-0.844208\pi\)
−0.882598 + 0.470128i \(0.844208\pi\)
\(314\) −19.5308 −1.10219
\(315\) −17.1535 −0.966488
\(316\) 2.17092 0.122124
\(317\) −12.6353 −0.709671 −0.354836 0.934929i \(-0.615463\pi\)
−0.354836 + 0.934929i \(0.615463\pi\)
\(318\) 0.415502 0.0233002
\(319\) −25.1836 −1.41001
\(320\) −2.69202 −0.150489
\(321\) −12.3894 −0.691510
\(322\) −5.53617 −0.308519
\(323\) 4.54288 0.252772
\(324\) −10.1371 −0.563170
\(325\) 0 0
\(326\) 0.944378 0.0523043
\(327\) 18.1521 1.00382
\(328\) 9.70171 0.535687
\(329\) 0 0
\(330\) 22.8659 1.25873
\(331\) 25.6819 1.41160 0.705801 0.708410i \(-0.250587\pi\)
0.705801 + 0.708410i \(0.250587\pi\)
\(332\) 15.7017 0.861743
\(333\) −22.2634 −1.22003
\(334\) −9.54527 −0.522294
\(335\) −21.6939 −1.18527
\(336\) −5.87800 −0.320671
\(337\) 20.6703 1.12598 0.562990 0.826464i \(-0.309651\pi\)
0.562990 + 0.826464i \(0.309651\pi\)
\(338\) 0 0
\(339\) 14.3720 0.780578
\(340\) 12.2295 0.663239
\(341\) 23.2620 1.25971
\(342\) −2.55496 −0.138156
\(343\) −19.4034 −1.04769
\(344\) −4.09783 −0.220940
\(345\) −14.0844 −0.758280
\(346\) 16.7681 0.901458
\(347\) 5.75600 0.308999 0.154499 0.987993i \(-0.450624\pi\)
0.154499 + 0.987993i \(0.450624\pi\)
\(348\) 16.4698 0.882874
\(349\) −25.0224 −1.33942 −0.669708 0.742624i \(-0.733581\pi\)
−0.669708 + 0.742624i \(0.733581\pi\)
\(350\) 5.60388 0.299540
\(351\) 0 0
\(352\) 3.60388 0.192087
\(353\) −27.3099 −1.45356 −0.726780 0.686871i \(-0.758984\pi\)
−0.726780 + 0.686871i \(0.758984\pi\)
\(354\) −17.1672 −0.912426
\(355\) −3.44803 −0.183002
\(356\) −12.1957 −0.646369
\(357\) 26.7030 1.41327
\(358\) −7.37867 −0.389975
\(359\) 22.7922 1.20293 0.601464 0.798900i \(-0.294584\pi\)
0.601464 + 0.798900i \(0.294584\pi\)
\(360\) −6.87800 −0.362503
\(361\) 1.00000 0.0526316
\(362\) −22.7875 −1.19768
\(363\) −4.68532 −0.245915
\(364\) 0 0
\(365\) 37.6088 1.96853
\(366\) −20.0707 −1.04911
\(367\) −14.8358 −0.774422 −0.387211 0.921991i \(-0.626561\pi\)
−0.387211 + 0.921991i \(0.626561\pi\)
\(368\) −2.21983 −0.115717
\(369\) 24.7875 1.29038
\(370\) 23.4577 1.21951
\(371\) −0.439665 −0.0228263
\(372\) −15.2131 −0.788764
\(373\) 22.0978 1.14418 0.572091 0.820190i \(-0.306132\pi\)
0.572091 + 0.820190i \(0.306132\pi\)
\(374\) −16.3720 −0.846574
\(375\) −17.4674 −0.902013
\(376\) 0 0
\(377\) 0 0
\(378\) 2.61596 0.134550
\(379\) −27.2078 −1.39757 −0.698784 0.715333i \(-0.746275\pi\)
−0.698784 + 0.715333i \(0.746275\pi\)
\(380\) 2.69202 0.138098
\(381\) 51.0562 2.61569
\(382\) −3.54958 −0.181612
\(383\) −10.0411 −0.513079 −0.256539 0.966534i \(-0.582582\pi\)
−0.256539 + 0.966534i \(0.582582\pi\)
\(384\) −2.35690 −0.120275
\(385\) −24.1957 −1.23313
\(386\) −26.6896 −1.35847
\(387\) −10.4698 −0.532210
\(388\) −6.89008 −0.349791
\(389\) 20.3672 1.03266 0.516328 0.856391i \(-0.327298\pi\)
0.516328 + 0.856391i \(0.327298\pi\)
\(390\) 0 0
\(391\) 10.0844 0.509991
\(392\) −0.780167 −0.0394044
\(393\) −8.78150 −0.442968
\(394\) 10.1032 0.508992
\(395\) −5.84415 −0.294051
\(396\) 9.20775 0.462707
\(397\) −38.8442 −1.94953 −0.974766 0.223229i \(-0.928340\pi\)
−0.974766 + 0.223229i \(0.928340\pi\)
\(398\) −4.39612 −0.220358
\(399\) 5.87800 0.294268
\(400\) 2.24698 0.112349
\(401\) −26.8552 −1.34108 −0.670542 0.741872i \(-0.733938\pi\)
−0.670542 + 0.741872i \(0.733938\pi\)
\(402\) −18.9933 −0.947299
\(403\) 0 0
\(404\) 5.46011 0.271650
\(405\) 27.2892 1.35601
\(406\) −17.4276 −0.864916
\(407\) −31.4034 −1.55661
\(408\) 10.7071 0.530080
\(409\) 38.1909 1.88842 0.944209 0.329347i \(-0.106829\pi\)
0.944209 + 0.329347i \(0.106829\pi\)
\(410\) −26.1172 −1.28984
\(411\) 25.3279 1.24934
\(412\) 8.17390 0.402699
\(413\) 18.1655 0.893868
\(414\) −5.67158 −0.278743
\(415\) −42.2693 −2.07492
\(416\) 0 0
\(417\) −20.4805 −1.00294
\(418\) −3.60388 −0.176271
\(419\) −0.968541 −0.0473163 −0.0236582 0.999720i \(-0.507531\pi\)
−0.0236582 + 0.999720i \(0.507531\pi\)
\(420\) 15.8237 0.772118
\(421\) −5.59312 −0.272592 −0.136296 0.990668i \(-0.543520\pi\)
−0.136296 + 0.990668i \(0.543520\pi\)
\(422\) −9.12498 −0.444197
\(423\) 0 0
\(424\) −0.176292 −0.00856150
\(425\) −10.2078 −0.495149
\(426\) −3.01879 −0.146261
\(427\) 21.2379 1.02777
\(428\) 5.25667 0.254091
\(429\) 0 0
\(430\) 11.0315 0.531984
\(431\) 18.7289 0.902137 0.451069 0.892489i \(-0.351043\pi\)
0.451069 + 0.892489i \(0.351043\pi\)
\(432\) 1.04892 0.0504661
\(433\) −4.18705 −0.201217 −0.100608 0.994926i \(-0.532079\pi\)
−0.100608 + 0.994926i \(0.532079\pi\)
\(434\) 16.0978 0.772721
\(435\) −44.3370 −2.12580
\(436\) −7.70171 −0.368845
\(437\) 2.21983 0.106189
\(438\) 32.9269 1.57331
\(439\) 40.9487 1.95437 0.977187 0.212380i \(-0.0681214\pi\)
0.977187 + 0.212380i \(0.0681214\pi\)
\(440\) −9.70171 −0.462511
\(441\) −1.99330 −0.0949188
\(442\) 0 0
\(443\) −35.2707 −1.67576 −0.837880 0.545854i \(-0.816205\pi\)
−0.837880 + 0.545854i \(0.816205\pi\)
\(444\) 20.5375 0.974667
\(445\) 32.8310 1.55634
\(446\) −7.70709 −0.364941
\(447\) −15.6515 −0.740289
\(448\) 2.49396 0.117828
\(449\) −17.1099 −0.807467 −0.403733 0.914877i \(-0.632288\pi\)
−0.403733 + 0.914877i \(0.632288\pi\)
\(450\) 5.74094 0.270630
\(451\) 34.9638 1.64638
\(452\) −6.09783 −0.286818
\(453\) 47.7144 2.24182
\(454\) 3.86294 0.181297
\(455\) 0 0
\(456\) 2.35690 0.110372
\(457\) −18.7899 −0.878952 −0.439476 0.898254i \(-0.644836\pi\)
−0.439476 + 0.898254i \(0.644836\pi\)
\(458\) 12.1196 0.566312
\(459\) −4.76510 −0.222416
\(460\) 5.97584 0.278625
\(461\) 40.1769 1.87122 0.935612 0.353030i \(-0.114849\pi\)
0.935612 + 0.353030i \(0.114849\pi\)
\(462\) −21.1836 −0.985550
\(463\) 6.13275 0.285013 0.142507 0.989794i \(-0.454484\pi\)
0.142507 + 0.989794i \(0.454484\pi\)
\(464\) −6.98792 −0.324406
\(465\) 40.9541 1.89920
\(466\) −17.8019 −0.824659
\(467\) −3.35391 −0.155201 −0.0776003 0.996985i \(-0.524726\pi\)
−0.0776003 + 0.996985i \(0.524726\pi\)
\(468\) 0 0
\(469\) 20.0978 0.928032
\(470\) 0 0
\(471\) 46.0320 2.12104
\(472\) 7.28382 0.335265
\(473\) −14.7681 −0.679037
\(474\) −5.11662 −0.235014
\(475\) −2.24698 −0.103098
\(476\) −11.3297 −0.519298
\(477\) −0.450419 −0.0206233
\(478\) 6.78746 0.310451
\(479\) −38.7767 −1.77175 −0.885877 0.463921i \(-0.846442\pi\)
−0.885877 + 0.463921i \(0.846442\pi\)
\(480\) 6.34481 0.289600
\(481\) 0 0
\(482\) 10.8116 0.492456
\(483\) 13.0482 0.593713
\(484\) 1.98792 0.0903599
\(485\) 18.5483 0.842233
\(486\) 20.7453 0.941024
\(487\) −4.12631 −0.186981 −0.0934905 0.995620i \(-0.529802\pi\)
−0.0934905 + 0.995620i \(0.529802\pi\)
\(488\) 8.51573 0.385489
\(489\) −2.22580 −0.100654
\(490\) 2.10023 0.0948786
\(491\) 3.31037 0.149395 0.0746975 0.997206i \(-0.476201\pi\)
0.0746975 + 0.997206i \(0.476201\pi\)
\(492\) −22.8659 −1.03088
\(493\) 31.7453 1.42973
\(494\) 0 0
\(495\) −24.7875 −1.11411
\(496\) 6.45473 0.289826
\(497\) 3.19434 0.143286
\(498\) −37.0073 −1.65834
\(499\) 4.85517 0.217347 0.108674 0.994077i \(-0.465340\pi\)
0.108674 + 0.994077i \(0.465340\pi\)
\(500\) 7.41119 0.331438
\(501\) 22.4972 1.00510
\(502\) −5.50604 −0.245747
\(503\) 15.0121 0.669356 0.334678 0.942333i \(-0.391373\pi\)
0.334678 + 0.942333i \(0.391373\pi\)
\(504\) 6.37196 0.283830
\(505\) −14.6987 −0.654085
\(506\) −8.00000 −0.355643
\(507\) 0 0
\(508\) −21.6625 −0.961117
\(509\) −19.6775 −0.872192 −0.436096 0.899900i \(-0.643639\pi\)
−0.436096 + 0.899900i \(0.643639\pi\)
\(510\) −28.8237 −1.27634
\(511\) −34.8418 −1.54131
\(512\) 1.00000 0.0441942
\(513\) −1.04892 −0.0463108
\(514\) 14.9879 0.661089
\(515\) −22.0043 −0.969626
\(516\) 9.65817 0.425177
\(517\) 0 0
\(518\) −21.7318 −0.954842
\(519\) −39.5206 −1.73476
\(520\) 0 0
\(521\) 1.24267 0.0544423 0.0272211 0.999629i \(-0.491334\pi\)
0.0272211 + 0.999629i \(0.491334\pi\)
\(522\) −17.8538 −0.781441
\(523\) 24.0000 1.04945 0.524723 0.851273i \(-0.324169\pi\)
0.524723 + 0.851273i \(0.324169\pi\)
\(524\) 3.72587 0.162766
\(525\) −13.2078 −0.576434
\(526\) 11.4276 0.498266
\(527\) −29.3230 −1.27733
\(528\) −8.49396 −0.369652
\(529\) −18.0723 −0.785754
\(530\) 0.474582 0.0206145
\(531\) 18.6098 0.807598
\(532\) −2.49396 −0.108127
\(533\) 0 0
\(534\) 28.7439 1.24387
\(535\) −14.1511 −0.611804
\(536\) 8.05861 0.348079
\(537\) 17.3907 0.750466
\(538\) −22.4155 −0.966401
\(539\) −2.81163 −0.121105
\(540\) −2.82371 −0.121513
\(541\) 22.2543 0.956786 0.478393 0.878146i \(-0.341219\pi\)
0.478393 + 0.878146i \(0.341219\pi\)
\(542\) 4.39612 0.188830
\(543\) 53.7077 2.30482
\(544\) −4.54288 −0.194774
\(545\) 20.7332 0.888111
\(546\) 0 0
\(547\) 29.5502 1.26347 0.631737 0.775183i \(-0.282342\pi\)
0.631737 + 0.775183i \(0.282342\pi\)
\(548\) −10.7463 −0.459060
\(549\) 21.7573 0.928580
\(550\) 8.09783 0.345293
\(551\) 6.98792 0.297695
\(552\) 5.23191 0.222685
\(553\) 5.41417 0.230234
\(554\) −8.71140 −0.370112
\(555\) −55.2874 −2.34682
\(556\) 8.68963 0.368522
\(557\) −41.8950 −1.77515 −0.887574 0.460666i \(-0.847611\pi\)
−0.887574 + 0.460666i \(0.847611\pi\)
\(558\) 16.4916 0.698144
\(559\) 0 0
\(560\) −6.71379 −0.283709
\(561\) 38.5870 1.62914
\(562\) −22.3913 −0.944522
\(563\) 27.6974 1.16731 0.583653 0.812003i \(-0.301623\pi\)
0.583653 + 0.812003i \(0.301623\pi\)
\(564\) 0 0
\(565\) 16.4155 0.690605
\(566\) −16.7332 −0.703348
\(567\) −25.2814 −1.06172
\(568\) 1.28083 0.0537425
\(569\) −9.50604 −0.398514 −0.199257 0.979947i \(-0.563853\pi\)
−0.199257 + 0.979947i \(0.563853\pi\)
\(570\) −6.34481 −0.265755
\(571\) −26.6025 −1.11328 −0.556641 0.830753i \(-0.687910\pi\)
−0.556641 + 0.830753i \(0.687910\pi\)
\(572\) 0 0
\(573\) 8.36599 0.349494
\(574\) 24.1957 1.00991
\(575\) −4.98792 −0.208011
\(576\) 2.55496 0.106457
\(577\) 12.5633 0.523018 0.261509 0.965201i \(-0.415780\pi\)
0.261509 + 0.965201i \(0.415780\pi\)
\(578\) 3.63773 0.151310
\(579\) 62.9047 2.61423
\(580\) 18.8116 0.781110
\(581\) 39.1594 1.62461
\(582\) 16.2392 0.673137
\(583\) −0.635334 −0.0263129
\(584\) −13.9705 −0.578102
\(585\) 0 0
\(586\) −1.84787 −0.0763349
\(587\) −46.4215 −1.91602 −0.958010 0.286736i \(-0.907430\pi\)
−0.958010 + 0.286736i \(0.907430\pi\)
\(588\) 1.83877 0.0758297
\(589\) −6.45473 −0.265963
\(590\) −19.6082 −0.807256
\(591\) −23.8122 −0.979504
\(592\) −8.71379 −0.358135
\(593\) 31.1836 1.28056 0.640278 0.768143i \(-0.278819\pi\)
0.640278 + 0.768143i \(0.278819\pi\)
\(594\) 3.78017 0.155102
\(595\) 30.4999 1.25038
\(596\) 6.64071 0.272014
\(597\) 10.3612 0.424056
\(598\) 0 0
\(599\) −16.8702 −0.689299 −0.344650 0.938731i \(-0.612002\pi\)
−0.344650 + 0.938731i \(0.612002\pi\)
\(600\) −5.29590 −0.216204
\(601\) 15.2185 0.620776 0.310388 0.950610i \(-0.399541\pi\)
0.310388 + 0.950610i \(0.399541\pi\)
\(602\) −10.2198 −0.416529
\(603\) 20.5894 0.838465
\(604\) −20.2446 −0.823741
\(605\) −5.35152 −0.217570
\(606\) −12.8689 −0.522764
\(607\) 43.5123 1.76611 0.883054 0.469271i \(-0.155483\pi\)
0.883054 + 0.469271i \(0.155483\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 41.0750 1.66444
\(610\) −22.9245 −0.928187
\(611\) 0 0
\(612\) −11.6069 −0.469179
\(613\) −22.4112 −0.905179 −0.452590 0.891719i \(-0.649500\pi\)
−0.452590 + 0.891719i \(0.649500\pi\)
\(614\) −32.4198 −1.30836
\(615\) 61.5555 2.48216
\(616\) 8.98792 0.362134
\(617\) 6.13813 0.247112 0.123556 0.992338i \(-0.460570\pi\)
0.123556 + 0.992338i \(0.460570\pi\)
\(618\) −19.2650 −0.774953
\(619\) −48.6848 −1.95681 −0.978405 0.206699i \(-0.933728\pi\)
−0.978405 + 0.206699i \(0.933728\pi\)
\(620\) −17.3763 −0.697848
\(621\) −2.32842 −0.0934363
\(622\) 1.37329 0.0550639
\(623\) −30.4155 −1.21857
\(624\) 0 0
\(625\) −31.1860 −1.24744
\(626\) −31.2295 −1.24818
\(627\) 8.49396 0.339216
\(628\) −19.5308 −0.779364
\(629\) 39.5857 1.57838
\(630\) −17.1535 −0.683410
\(631\) 9.66679 0.384829 0.192414 0.981314i \(-0.438368\pi\)
0.192414 + 0.981314i \(0.438368\pi\)
\(632\) 2.17092 0.0863544
\(633\) 21.5066 0.854812
\(634\) −12.6353 −0.501813
\(635\) 58.3159 2.31419
\(636\) 0.415502 0.0164757
\(637\) 0 0
\(638\) −25.1836 −0.997028
\(639\) 3.27247 0.129457
\(640\) −2.69202 −0.106411
\(641\) 0.733169 0.0289584 0.0144792 0.999895i \(-0.495391\pi\)
0.0144792 + 0.999895i \(0.495391\pi\)
\(642\) −12.3894 −0.488971
\(643\) −39.8840 −1.57287 −0.786435 0.617673i \(-0.788076\pi\)
−0.786435 + 0.617673i \(0.788076\pi\)
\(644\) −5.53617 −0.218156
\(645\) −26.0000 −1.02375
\(646\) 4.54288 0.178737
\(647\) −29.9275 −1.17657 −0.588286 0.808653i \(-0.700197\pi\)
−0.588286 + 0.808653i \(0.700197\pi\)
\(648\) −10.1371 −0.398221
\(649\) 26.2500 1.03040
\(650\) 0 0
\(651\) −37.9409 −1.48702
\(652\) 0.944378 0.0369847
\(653\) 0.469205 0.0183614 0.00918071 0.999958i \(-0.497078\pi\)
0.00918071 + 0.999958i \(0.497078\pi\)
\(654\) 18.1521 0.709804
\(655\) −10.0301 −0.391910
\(656\) 9.70171 0.378788
\(657\) −35.6939 −1.39255
\(658\) 0 0
\(659\) −26.2911 −1.02416 −0.512078 0.858939i \(-0.671124\pi\)
−0.512078 + 0.858939i \(0.671124\pi\)
\(660\) 22.8659 0.890055
\(661\) 17.2707 0.671751 0.335876 0.941906i \(-0.390968\pi\)
0.335876 + 0.941906i \(0.390968\pi\)
\(662\) 25.6819 0.998154
\(663\) 0 0
\(664\) 15.7017 0.609345
\(665\) 6.71379 0.260350
\(666\) −22.2634 −0.862688
\(667\) 15.5120 0.600627
\(668\) −9.54527 −0.369318
\(669\) 18.1648 0.702292
\(670\) −21.6939 −0.838110
\(671\) 30.6896 1.18476
\(672\) −5.87800 −0.226749
\(673\) 13.7345 0.529426 0.264713 0.964327i \(-0.414723\pi\)
0.264713 + 0.964327i \(0.414723\pi\)
\(674\) 20.6703 0.796188
\(675\) 2.35690 0.0907170
\(676\) 0 0
\(677\) 29.9323 1.15039 0.575196 0.818016i \(-0.304926\pi\)
0.575196 + 0.818016i \(0.304926\pi\)
\(678\) 14.3720 0.551952
\(679\) −17.1836 −0.659446
\(680\) 12.2295 0.468981
\(681\) −9.10454 −0.348887
\(682\) 23.2620 0.890750
\(683\) 43.5445 1.66619 0.833093 0.553134i \(-0.186568\pi\)
0.833093 + 0.553134i \(0.186568\pi\)
\(684\) −2.55496 −0.0976913
\(685\) 28.9293 1.10533
\(686\) −19.4034 −0.740826
\(687\) −28.5646 −1.08981
\(688\) −4.09783 −0.156228
\(689\) 0 0
\(690\) −14.0844 −0.536185
\(691\) 7.48666 0.284806 0.142403 0.989809i \(-0.454517\pi\)
0.142403 + 0.989809i \(0.454517\pi\)
\(692\) 16.7681 0.637427
\(693\) 22.9638 0.872321
\(694\) 5.75600 0.218495
\(695\) −23.3927 −0.887334
\(696\) 16.4698 0.624286
\(697\) −44.0737 −1.66941
\(698\) −25.0224 −0.947110
\(699\) 41.9573 1.58697
\(700\) 5.60388 0.211807
\(701\) 50.4198 1.90433 0.952165 0.305584i \(-0.0988518\pi\)
0.952165 + 0.305584i \(0.0988518\pi\)
\(702\) 0 0
\(703\) 8.71379 0.328647
\(704\) 3.60388 0.135826
\(705\) 0 0
\(706\) −27.3099 −1.02782
\(707\) 13.6173 0.512131
\(708\) −17.1672 −0.645183
\(709\) −6.69979 −0.251616 −0.125808 0.992055i \(-0.540152\pi\)
−0.125808 + 0.992055i \(0.540152\pi\)
\(710\) −3.44803 −0.129402
\(711\) 5.54660 0.208014
\(712\) −12.1957 −0.457052
\(713\) −14.3284 −0.536604
\(714\) 26.7030 0.999336
\(715\) 0 0
\(716\) −7.37867 −0.275754
\(717\) −15.9973 −0.597432
\(718\) 22.7922 0.850599
\(719\) −14.2306 −0.530711 −0.265356 0.964151i \(-0.585489\pi\)
−0.265356 + 0.964151i \(0.585489\pi\)
\(720\) −6.87800 −0.256328
\(721\) 20.3854 0.759191
\(722\) 1.00000 0.0372161
\(723\) −25.4819 −0.947681
\(724\) −22.7875 −0.846889
\(725\) −15.7017 −0.583147
\(726\) −4.68532 −0.173888
\(727\) −24.8552 −0.921827 −0.460914 0.887445i \(-0.652478\pi\)
−0.460914 + 0.887445i \(0.652478\pi\)
\(728\) 0 0
\(729\) −18.4832 −0.684563
\(730\) 37.6088 1.39196
\(731\) 18.6160 0.688536
\(732\) −20.0707 −0.741834
\(733\) 11.8834 0.438923 0.219461 0.975621i \(-0.429570\pi\)
0.219461 + 0.975621i \(0.429570\pi\)
\(734\) −14.8358 −0.547599
\(735\) −4.95002 −0.182584
\(736\) −2.21983 −0.0818241
\(737\) 29.0422 1.06978
\(738\) 24.7875 0.912439
\(739\) −17.8974 −0.658366 −0.329183 0.944266i \(-0.606773\pi\)
−0.329183 + 0.944266i \(0.606773\pi\)
\(740\) 23.4577 0.862323
\(741\) 0 0
\(742\) −0.439665 −0.0161406
\(743\) 33.5453 1.23066 0.615328 0.788271i \(-0.289023\pi\)
0.615328 + 0.788271i \(0.289023\pi\)
\(744\) −15.2131 −0.557740
\(745\) −17.8769 −0.654960
\(746\) 22.0978 0.809059
\(747\) 40.1172 1.46781
\(748\) −16.3720 −0.598618
\(749\) 13.1099 0.479026
\(750\) −17.4674 −0.637819
\(751\) 1.88876 0.0689217 0.0344608 0.999406i \(-0.489029\pi\)
0.0344608 + 0.999406i \(0.489029\pi\)
\(752\) 0 0
\(753\) 12.9772 0.472914
\(754\) 0 0
\(755\) 54.4989 1.98342
\(756\) 2.61596 0.0951414
\(757\) −4.86353 −0.176768 −0.0883840 0.996086i \(-0.528170\pi\)
−0.0883840 + 0.996086i \(0.528170\pi\)
\(758\) −27.2078 −0.988230
\(759\) 18.8552 0.684399
\(760\) 2.69202 0.0976499
\(761\) −7.75063 −0.280960 −0.140480 0.990084i \(-0.544865\pi\)
−0.140480 + 0.990084i \(0.544865\pi\)
\(762\) 51.0562 1.84957
\(763\) −19.2078 −0.695367
\(764\) −3.54958 −0.128419
\(765\) 31.2459 1.12970
\(766\) −10.0411 −0.362801
\(767\) 0 0
\(768\) −2.35690 −0.0850472
\(769\) 0.369568 0.0133270 0.00666349 0.999978i \(-0.497879\pi\)
0.00666349 + 0.999978i \(0.497879\pi\)
\(770\) −24.1957 −0.871951
\(771\) −35.3250 −1.27220
\(772\) −26.6896 −0.960581
\(773\) −38.9724 −1.40174 −0.700870 0.713290i \(-0.747204\pi\)
−0.700870 + 0.713290i \(0.747204\pi\)
\(774\) −10.4698 −0.376329
\(775\) 14.5036 0.520986
\(776\) −6.89008 −0.247340
\(777\) 51.2197 1.83750
\(778\) 20.3672 0.730199
\(779\) −9.70171 −0.347600
\(780\) 0 0
\(781\) 4.61596 0.165172
\(782\) 10.0844 0.360618
\(783\) −7.32975 −0.261944
\(784\) −0.780167 −0.0278631
\(785\) 52.5773 1.87656
\(786\) −8.78150 −0.313226
\(787\) 37.1075 1.32274 0.661370 0.750060i \(-0.269975\pi\)
0.661370 + 0.750060i \(0.269975\pi\)
\(788\) 10.1032 0.359912
\(789\) −26.9336 −0.958862
\(790\) −5.84415 −0.207926
\(791\) −15.2078 −0.540725
\(792\) 9.20775 0.327183
\(793\) 0 0
\(794\) −38.8442 −1.37853
\(795\) −1.11854 −0.0396705
\(796\) −4.39612 −0.155817
\(797\) 38.2150 1.35365 0.676823 0.736146i \(-0.263356\pi\)
0.676823 + 0.736146i \(0.263356\pi\)
\(798\) 5.87800 0.208079
\(799\) 0 0
\(800\) 2.24698 0.0794427
\(801\) −31.1594 −1.10096
\(802\) −26.8552 −0.948289
\(803\) −50.3478 −1.77674
\(804\) −18.9933 −0.669842
\(805\) 14.9035 0.525279
\(806\) 0 0
\(807\) 52.8310 1.85974
\(808\) 5.46011 0.192086
\(809\) 11.5539 0.406213 0.203107 0.979157i \(-0.434896\pi\)
0.203107 + 0.979157i \(0.434896\pi\)
\(810\) 27.2892 0.958844
\(811\) −1.61224 −0.0566133 −0.0283066 0.999599i \(-0.509011\pi\)
−0.0283066 + 0.999599i \(0.509011\pi\)
\(812\) −17.4276 −0.611588
\(813\) −10.3612 −0.363383
\(814\) −31.4034 −1.10069
\(815\) −2.54229 −0.0890524
\(816\) 10.7071 0.374823
\(817\) 4.09783 0.143365
\(818\) 38.1909 1.33531
\(819\) 0 0
\(820\) −26.1172 −0.912053
\(821\) −24.8200 −0.866224 −0.433112 0.901340i \(-0.642584\pi\)
−0.433112 + 0.901340i \(0.642584\pi\)
\(822\) 25.3279 0.883414
\(823\) −2.52888 −0.0881511 −0.0440755 0.999028i \(-0.514034\pi\)
−0.0440755 + 0.999028i \(0.514034\pi\)
\(824\) 8.17390 0.284751
\(825\) −19.0858 −0.664481
\(826\) 18.1655 0.632060
\(827\) 21.0127 0.730682 0.365341 0.930874i \(-0.380952\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(828\) −5.67158 −0.197101
\(829\) −41.7271 −1.44924 −0.724620 0.689148i \(-0.757985\pi\)
−0.724620 + 0.689148i \(0.757985\pi\)
\(830\) −42.2693 −1.46719
\(831\) 20.5319 0.712242
\(832\) 0 0
\(833\) 3.54420 0.122799
\(834\) −20.4805 −0.709183
\(835\) 25.6961 0.889249
\(836\) −3.60388 −0.124643
\(837\) 6.77048 0.234022
\(838\) −0.968541 −0.0334577
\(839\) −18.8549 −0.650944 −0.325472 0.945552i \(-0.605523\pi\)
−0.325472 + 0.945552i \(0.605523\pi\)
\(840\) 15.8237 0.545970
\(841\) 19.8310 0.683828
\(842\) −5.59312 −0.192752
\(843\) 52.7741 1.81763
\(844\) −9.12498 −0.314095
\(845\) 0 0
\(846\) 0 0
\(847\) 4.95779 0.170352
\(848\) −0.176292 −0.00605389
\(849\) 39.4383 1.35352
\(850\) −10.2078 −0.350123
\(851\) 19.3432 0.663075
\(852\) −3.01879 −0.103422
\(853\) 23.2838 0.797223 0.398611 0.917120i \(-0.369492\pi\)
0.398611 + 0.917120i \(0.369492\pi\)
\(854\) 21.2379 0.726745
\(855\) 6.87800 0.235223
\(856\) 5.25667 0.179669
\(857\) 36.6025 1.25032 0.625160 0.780497i \(-0.285034\pi\)
0.625160 + 0.780497i \(0.285034\pi\)
\(858\) 0 0
\(859\) 40.7741 1.39119 0.695596 0.718433i \(-0.255140\pi\)
0.695596 + 0.718433i \(0.255140\pi\)
\(860\) 11.0315 0.376170
\(861\) −57.0267 −1.94346
\(862\) 18.7289 0.637907
\(863\) −53.7318 −1.82905 −0.914527 0.404526i \(-0.867437\pi\)
−0.914527 + 0.404526i \(0.867437\pi\)
\(864\) 1.04892 0.0356849
\(865\) −45.1400 −1.53481
\(866\) −4.18705 −0.142282
\(867\) −8.57374 −0.291180
\(868\) 16.0978 0.546396
\(869\) 7.82371 0.265401
\(870\) −44.3370 −1.50317
\(871\) 0 0
\(872\) −7.70171 −0.260813
\(873\) −17.6039 −0.595801
\(874\) 2.21983 0.0750870
\(875\) 18.4832 0.624846
\(876\) 32.9269 1.11250
\(877\) −9.19700 −0.310561 −0.155280 0.987870i \(-0.549628\pi\)
−0.155280 + 0.987870i \(0.549628\pi\)
\(878\) 40.9487 1.38195
\(879\) 4.35524 0.146899
\(880\) −9.70171 −0.327045
\(881\) −52.7590 −1.77750 −0.888748 0.458397i \(-0.848424\pi\)
−0.888748 + 0.458397i \(0.848424\pi\)
\(882\) −1.99330 −0.0671177
\(883\) 53.0810 1.78632 0.893158 0.449742i \(-0.148484\pi\)
0.893158 + 0.449742i \(0.148484\pi\)
\(884\) 0 0
\(885\) 46.2145 1.55348
\(886\) −35.2707 −1.18494
\(887\) 17.5754 0.590124 0.295062 0.955478i \(-0.404660\pi\)
0.295062 + 0.955478i \(0.404660\pi\)
\(888\) 20.5375 0.689193
\(889\) −54.0253 −1.81195
\(890\) 32.8310 1.10050
\(891\) −36.5327 −1.22389
\(892\) −7.70709 −0.258052
\(893\) 0 0
\(894\) −15.6515 −0.523463
\(895\) 19.8635 0.663965
\(896\) 2.49396 0.0833173
\(897\) 0 0
\(898\) −17.1099 −0.570965
\(899\) −45.1051 −1.50434
\(900\) 5.74094 0.191365
\(901\) 0.800873 0.0266809
\(902\) 34.9638 1.16417
\(903\) 24.0871 0.801568
\(904\) −6.09783 −0.202811
\(905\) 61.3443 2.03915
\(906\) 47.7144 1.58520
\(907\) 30.3672 1.00833 0.504163 0.863609i \(-0.331801\pi\)
0.504163 + 0.863609i \(0.331801\pi\)
\(908\) 3.86294 0.128196
\(909\) 13.9503 0.462704
\(910\) 0 0
\(911\) −26.3067 −0.871578 −0.435789 0.900049i \(-0.643531\pi\)
−0.435789 + 0.900049i \(0.643531\pi\)
\(912\) 2.35690 0.0780446
\(913\) 56.5870 1.87276
\(914\) −18.7899 −0.621513
\(915\) 54.0307 1.78620
\(916\) 12.1196 0.400443
\(917\) 9.29218 0.306855
\(918\) −4.76510 −0.157272
\(919\) −13.8780 −0.457793 −0.228897 0.973451i \(-0.573512\pi\)
−0.228897 + 0.973451i \(0.573512\pi\)
\(920\) 5.97584 0.197018
\(921\) 76.4101 2.51780
\(922\) 40.1769 1.32316
\(923\) 0 0
\(924\) −21.1836 −0.696889
\(925\) −19.5797 −0.643777
\(926\) 6.13275 0.201535
\(927\) 20.8840 0.685920
\(928\) −6.98792 −0.229390
\(929\) 3.66248 0.120162 0.0600811 0.998194i \(-0.480864\pi\)
0.0600811 + 0.998194i \(0.480864\pi\)
\(930\) 40.9541 1.34294
\(931\) 0.780167 0.0255690
\(932\) −17.8019 −0.583122
\(933\) −3.23670 −0.105965
\(934\) −3.35391 −0.109743
\(935\) 44.0737 1.44136
\(936\) 0 0
\(937\) 32.5332 1.06281 0.531406 0.847117i \(-0.321664\pi\)
0.531406 + 0.847117i \(0.321664\pi\)
\(938\) 20.0978 0.656218
\(939\) 73.6047 2.40200
\(940\) 0 0
\(941\) 19.5931 0.638718 0.319359 0.947634i \(-0.396533\pi\)
0.319359 + 0.947634i \(0.396533\pi\)
\(942\) 46.0320 1.49981
\(943\) −21.5362 −0.701314
\(944\) 7.28382 0.237068
\(945\) −7.04221 −0.229083
\(946\) −14.7681 −0.480152
\(947\) 31.8189 1.03398 0.516988 0.855993i \(-0.327053\pi\)
0.516988 + 0.855993i \(0.327053\pi\)
\(948\) −5.11662 −0.166180
\(949\) 0 0
\(950\) −2.24698 −0.0729016
\(951\) 29.7802 0.965688
\(952\) −11.3297 −0.367199
\(953\) 44.6762 1.44720 0.723602 0.690217i \(-0.242485\pi\)
0.723602 + 0.690217i \(0.242485\pi\)
\(954\) −0.450419 −0.0145828
\(955\) 9.55555 0.309210
\(956\) 6.78746 0.219522
\(957\) 59.3551 1.91868
\(958\) −38.7767 −1.25282
\(959\) −26.8009 −0.865445
\(960\) 6.34481 0.204778
\(961\) 10.6635 0.343985
\(962\) 0 0
\(963\) 13.4306 0.432794
\(964\) 10.8116 0.348219
\(965\) 71.8491 2.31290
\(966\) 13.0482 0.419818
\(967\) −24.1026 −0.775088 −0.387544 0.921851i \(-0.626676\pi\)
−0.387544 + 0.921851i \(0.626676\pi\)
\(968\) 1.98792 0.0638941
\(969\) −10.7071 −0.343961
\(970\) 18.5483 0.595549
\(971\) −31.4233 −1.00842 −0.504210 0.863581i \(-0.668216\pi\)
−0.504210 + 0.863581i \(0.668216\pi\)
\(972\) 20.7453 0.665404
\(973\) 21.6716 0.694759
\(974\) −4.12631 −0.132215
\(975\) 0 0
\(976\) 8.51573 0.272582
\(977\) −5.51083 −0.176307 −0.0881535 0.996107i \(-0.528097\pi\)
−0.0881535 + 0.996107i \(0.528097\pi\)
\(978\) −2.22580 −0.0711732
\(979\) −43.9517 −1.40470
\(980\) 2.10023 0.0670893
\(981\) −19.6775 −0.628256
\(982\) 3.31037 0.105638
\(983\) 29.9269 0.954520 0.477260 0.878762i \(-0.341630\pi\)
0.477260 + 0.878762i \(0.341630\pi\)
\(984\) −22.8659 −0.728939
\(985\) −27.1981 −0.866602
\(986\) 31.7453 1.01097
\(987\) 0 0
\(988\) 0 0
\(989\) 9.09651 0.289252
\(990\) −24.7875 −0.787797
\(991\) −46.4975 −1.47704 −0.738521 0.674230i \(-0.764476\pi\)
−0.738521 + 0.674230i \(0.764476\pi\)
\(992\) 6.45473 0.204938
\(993\) −60.5295 −1.92084
\(994\) 3.19434 0.101318
\(995\) 11.8345 0.375178
\(996\) −37.0073 −1.17262
\(997\) 18.3811 0.582134 0.291067 0.956703i \(-0.405990\pi\)
0.291067 + 0.956703i \(0.405990\pi\)
\(998\) 4.85517 0.153688
\(999\) −9.14005 −0.289178
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.r.1.2 yes 3
13.12 even 2 6422.2.a.l.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6422.2.a.l.1.2 3 13.12 even 2
6422.2.a.r.1.2 yes 3 1.1 even 1 trivial