Properties

Label 6046.2.a.f.1.6
Level $6046$
Weight $2$
Character 6046.1
Self dual yes
Analytic conductor $48.278$
Analytic rank $0$
Dimension $67$
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] = [6046,2,Mod(1,6046)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6046, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6046.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6046 = 2 \cdot 3023 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6046.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: \(48.2775530621\)
Analytic rank: \(0\)
Dimension: \(67\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Character \(\chi\) \(=\) 6046.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.74369 q^{3} +1.00000 q^{4} +0.0184852 q^{5} -2.74369 q^{6} +0.788378 q^{7} +1.00000 q^{8} +4.52782 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.74369 q^{3} +1.00000 q^{4} +0.0184852 q^{5} -2.74369 q^{6} +0.788378 q^{7} +1.00000 q^{8} +4.52782 q^{9} +0.0184852 q^{10} -2.48763 q^{11} -2.74369 q^{12} +1.53915 q^{13} +0.788378 q^{14} -0.0507177 q^{15} +1.00000 q^{16} +4.82077 q^{17} +4.52782 q^{18} +1.27873 q^{19} +0.0184852 q^{20} -2.16306 q^{21} -2.48763 q^{22} +8.09086 q^{23} -2.74369 q^{24} -4.99966 q^{25} +1.53915 q^{26} -4.19186 q^{27} +0.788378 q^{28} -3.75967 q^{29} -0.0507177 q^{30} -0.0360307 q^{31} +1.00000 q^{32} +6.82528 q^{33} +4.82077 q^{34} +0.0145734 q^{35} +4.52782 q^{36} -1.48010 q^{37} +1.27873 q^{38} -4.22296 q^{39} +0.0184852 q^{40} -2.01682 q^{41} -2.16306 q^{42} +6.31722 q^{43} -2.48763 q^{44} +0.0836978 q^{45} +8.09086 q^{46} -4.15993 q^{47} -2.74369 q^{48} -6.37846 q^{49} -4.99966 q^{50} -13.2267 q^{51} +1.53915 q^{52} +11.3059 q^{53} -4.19186 q^{54} -0.0459844 q^{55} +0.788378 q^{56} -3.50843 q^{57} -3.75967 q^{58} +2.97362 q^{59} -0.0507177 q^{60} -10.5034 q^{61} -0.0360307 q^{62} +3.56963 q^{63} +1.00000 q^{64} +0.0284516 q^{65} +6.82528 q^{66} +4.34181 q^{67} +4.82077 q^{68} -22.1988 q^{69} +0.0145734 q^{70} -7.99641 q^{71} +4.52782 q^{72} +7.78804 q^{73} -1.48010 q^{74} +13.7175 q^{75} +1.27873 q^{76} -1.96119 q^{77} -4.22296 q^{78} +2.65372 q^{79} +0.0184852 q^{80} -2.08230 q^{81} -2.01682 q^{82} +1.27427 q^{83} -2.16306 q^{84} +0.0891131 q^{85} +6.31722 q^{86} +10.3154 q^{87} -2.48763 q^{88} -15.1053 q^{89} +0.0836978 q^{90} +1.21344 q^{91} +8.09086 q^{92} +0.0988570 q^{93} -4.15993 q^{94} +0.0236376 q^{95} -2.74369 q^{96} +2.52670 q^{97} -6.37846 q^{98} -11.2635 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 67 q + 67 q^{2} + 21 q^{3} + 67 q^{4} + 21 q^{5} + 21 q^{6} + 38 q^{7} + 67 q^{8} + 90 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 67 q + 67 q^{2} + 21 q^{3} + 67 q^{4} + 21 q^{5} + 21 q^{6} + 38 q^{7} + 67 q^{8} + 90 q^{9} + 21 q^{10} + 56 q^{11} + 21 q^{12} + 33 q^{13} + 38 q^{14} + 25 q^{15} + 67 q^{16} + 30 q^{17} + 90 q^{18} + 36 q^{19} + 21 q^{20} + 20 q^{21} + 56 q^{22} + 65 q^{23} + 21 q^{24} + 72 q^{25} + 33 q^{26} + 57 q^{27} + 38 q^{28} + 84 q^{29} + 25 q^{30} + 52 q^{31} + 67 q^{32} - 9 q^{33} + 30 q^{34} + 30 q^{35} + 90 q^{36} + 52 q^{37} + 36 q^{38} + 41 q^{39} + 21 q^{40} + 46 q^{41} + 20 q^{42} + 61 q^{43} + 56 q^{44} + 23 q^{45} + 65 q^{46} + 51 q^{47} + 21 q^{48} + 81 q^{49} + 72 q^{50} + 33 q^{51} + 33 q^{52} + 72 q^{53} + 57 q^{54} + 14 q^{55} + 38 q^{56} - 26 q^{57} + 84 q^{58} + 71 q^{59} + 25 q^{60} + 42 q^{61} + 52 q^{62} + 63 q^{63} + 67 q^{64} - 2 q^{65} - 9 q^{66} + 70 q^{67} + 30 q^{68} + 21 q^{69} + 30 q^{70} + 104 q^{71} + 90 q^{72} - 31 q^{73} + 52 q^{74} + 69 q^{75} + 36 q^{76} + 48 q^{77} + 41 q^{78} + 79 q^{79} + 21 q^{80} + 123 q^{81} + 46 q^{82} + 41 q^{83} + 20 q^{84} + 6 q^{85} + 61 q^{86} + 19 q^{87} + 56 q^{88} + 58 q^{89} + 23 q^{90} + 31 q^{91} + 65 q^{92} + 13 q^{93} + 51 q^{94} + 77 q^{95} + 21 q^{96} - 8 q^{97} + 81 q^{98} + 129 q^{99}+O(q^{100}) \) 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
\(3\) −2.74369 −1.58407 −0.792034 0.610477i \(-0.790978\pi\)
−0.792034 + 0.610477i \(0.790978\pi\)
\(4\) 1.00000 0.500000
\(5\) 0.0184852 0.00826685 0.00413342 0.999991i \(-0.498684\pi\)
0.00413342 + 0.999991i \(0.498684\pi\)
\(6\) −2.74369 −1.12011
\(7\) 0.788378 0.297979 0.148989 0.988839i \(-0.452398\pi\)
0.148989 + 0.988839i \(0.452398\pi\)
\(8\) 1.00000 0.353553
\(9\) 4.52782 1.50927
\(10\) 0.0184852 0.00584554
\(11\) −2.48763 −0.750048 −0.375024 0.927015i \(-0.622366\pi\)
−0.375024 + 0.927015i \(0.622366\pi\)
\(12\) −2.74369 −0.792034
\(13\) 1.53915 0.426885 0.213442 0.976956i \(-0.431532\pi\)
0.213442 + 0.976956i \(0.431532\pi\)
\(14\) 0.788378 0.210703
\(15\) −0.0507177 −0.0130953
\(16\) 1.00000 0.250000
\(17\) 4.82077 1.16921 0.584604 0.811318i \(-0.301250\pi\)
0.584604 + 0.811318i \(0.301250\pi\)
\(18\) 4.52782 1.06722
\(19\) 1.27873 0.293360 0.146680 0.989184i \(-0.453141\pi\)
0.146680 + 0.989184i \(0.453141\pi\)
\(20\) 0.0184852 0.00413342
\(21\) −2.16306 −0.472019
\(22\) −2.48763 −0.530364
\(23\) 8.09086 1.68706 0.843531 0.537081i \(-0.180473\pi\)
0.843531 + 0.537081i \(0.180473\pi\)
\(24\) −2.74369 −0.560053
\(25\) −4.99966 −0.999932
\(26\) 1.53915 0.301853
\(27\) −4.19186 −0.806724
\(28\) 0.788378 0.148989
\(29\) −3.75967 −0.698153 −0.349076 0.937094i \(-0.613505\pi\)
−0.349076 + 0.937094i \(0.613505\pi\)
\(30\) −0.0507177 −0.00925974
\(31\) −0.0360307 −0.00647130 −0.00323565 0.999995i \(-0.501030\pi\)
−0.00323565 + 0.999995i \(0.501030\pi\)
\(32\) 1.00000 0.176777
\(33\) 6.82528 1.18813
\(34\) 4.82077 0.826755
\(35\) 0.0145734 0.00246335
\(36\) 4.52782 0.754637
\(37\) −1.48010 −0.243326 −0.121663 0.992571i \(-0.538823\pi\)
−0.121663 + 0.992571i \(0.538823\pi\)
\(38\) 1.27873 0.207437
\(39\) −4.22296 −0.676214
\(40\) 0.0184852 0.00292277
\(41\) −2.01682 −0.314975 −0.157487 0.987521i \(-0.550339\pi\)
−0.157487 + 0.987521i \(0.550339\pi\)
\(42\) −2.16306 −0.333768
\(43\) 6.31722 0.963366 0.481683 0.876345i \(-0.340026\pi\)
0.481683 + 0.876345i \(0.340026\pi\)
\(44\) −2.48763 −0.375024
\(45\) 0.0836978 0.0124769
\(46\) 8.09086 1.19293
\(47\) −4.15993 −0.606788 −0.303394 0.952865i \(-0.598120\pi\)
−0.303394 + 0.952865i \(0.598120\pi\)
\(48\) −2.74369 −0.396017
\(49\) −6.37846 −0.911209
\(50\) −4.99966 −0.707058
\(51\) −13.2267 −1.85211
\(52\) 1.53915 0.213442
\(53\) 11.3059 1.55299 0.776494 0.630125i \(-0.216996\pi\)
0.776494 + 0.630125i \(0.216996\pi\)
\(54\) −4.19186 −0.570440
\(55\) −0.0459844 −0.00620053
\(56\) 0.788378 0.105351
\(57\) −3.50843 −0.464703
\(58\) −3.75967 −0.493669
\(59\) 2.97362 0.387133 0.193566 0.981087i \(-0.437995\pi\)
0.193566 + 0.981087i \(0.437995\pi\)
\(60\) −0.0507177 −0.00654763
\(61\) −10.5034 −1.34482 −0.672410 0.740179i \(-0.734741\pi\)
−0.672410 + 0.740179i \(0.734741\pi\)
\(62\) −0.0360307 −0.00457590
\(63\) 3.56963 0.449732
\(64\) 1.00000 0.125000
\(65\) 0.0284516 0.00352899
\(66\) 6.82528 0.840133
\(67\) 4.34181 0.530437 0.265218 0.964188i \(-0.414556\pi\)
0.265218 + 0.964188i \(0.414556\pi\)
\(68\) 4.82077 0.584604
\(69\) −22.1988 −2.67242
\(70\) 0.0145734 0.00174185
\(71\) −7.99641 −0.949000 −0.474500 0.880256i \(-0.657371\pi\)
−0.474500 + 0.880256i \(0.657371\pi\)
\(72\) 4.52782 0.533609
\(73\) 7.78804 0.911521 0.455760 0.890102i \(-0.349367\pi\)
0.455760 + 0.890102i \(0.349367\pi\)
\(74\) −1.48010 −0.172058
\(75\) 13.7175 1.58396
\(76\) 1.27873 0.146680
\(77\) −1.96119 −0.223499
\(78\) −4.22296 −0.478156
\(79\) 2.65372 0.298566 0.149283 0.988794i \(-0.452303\pi\)
0.149283 + 0.988794i \(0.452303\pi\)
\(80\) 0.0184852 0.00206671
\(81\) −2.08230 −0.231367
\(82\) −2.01682 −0.222721
\(83\) 1.27427 0.139870 0.0699349 0.997552i \(-0.477721\pi\)
0.0699349 + 0.997552i \(0.477721\pi\)
\(84\) −2.16306 −0.236010
\(85\) 0.0891131 0.00966567
\(86\) 6.31722 0.681203
\(87\) 10.3154 1.10592
\(88\) −2.48763 −0.265182
\(89\) −15.1053 −1.60116 −0.800579 0.599227i \(-0.795475\pi\)
−0.800579 + 0.599227i \(0.795475\pi\)
\(90\) 0.0836978 0.00882252
\(91\) 1.21344 0.127203
\(92\) 8.09086 0.843531
\(93\) 0.0988570 0.0102510
\(94\) −4.15993 −0.429064
\(95\) 0.0236376 0.00242516
\(96\) −2.74369 −0.280026
\(97\) 2.52670 0.256548 0.128274 0.991739i \(-0.459056\pi\)
0.128274 + 0.991739i \(0.459056\pi\)
\(98\) −6.37846 −0.644322
\(99\) −11.2635 −1.13203
\(100\) −4.99966 −0.499966
\(101\) 7.37414 0.733755 0.366877 0.930269i \(-0.380427\pi\)
0.366877 + 0.930269i \(0.380427\pi\)
\(102\) −13.2267 −1.30964
\(103\) 12.0562 1.18793 0.593965 0.804491i \(-0.297562\pi\)
0.593965 + 0.804491i \(0.297562\pi\)
\(104\) 1.53915 0.150926
\(105\) −0.0399847 −0.00390211
\(106\) 11.3059 1.09813
\(107\) 8.88659 0.859099 0.429549 0.903043i \(-0.358672\pi\)
0.429549 + 0.903043i \(0.358672\pi\)
\(108\) −4.19186 −0.403362
\(109\) 16.8778 1.61660 0.808300 0.588771i \(-0.200388\pi\)
0.808300 + 0.588771i \(0.200388\pi\)
\(110\) −0.0459844 −0.00438444
\(111\) 4.06092 0.385446
\(112\) 0.788378 0.0744947
\(113\) 4.93298 0.464056 0.232028 0.972709i \(-0.425464\pi\)
0.232028 + 0.972709i \(0.425464\pi\)
\(114\) −3.50843 −0.328594
\(115\) 0.149561 0.0139467
\(116\) −3.75967 −0.349076
\(117\) 6.96901 0.644286
\(118\) 2.97362 0.273744
\(119\) 3.80059 0.348400
\(120\) −0.0507177 −0.00462987
\(121\) −4.81170 −0.437428
\(122\) −10.5034 −0.950931
\(123\) 5.53353 0.498942
\(124\) −0.0360307 −0.00323565
\(125\) −0.184846 −0.0165331
\(126\) 3.56963 0.318008
\(127\) −12.1590 −1.07894 −0.539469 0.842006i \(-0.681375\pi\)
−0.539469 + 0.842006i \(0.681375\pi\)
\(128\) 1.00000 0.0883883
\(129\) −17.3325 −1.52604
\(130\) 0.0284516 0.00249537
\(131\) 0.824810 0.0720640 0.0360320 0.999351i \(-0.488528\pi\)
0.0360320 + 0.999351i \(0.488528\pi\)
\(132\) 6.82528 0.594064
\(133\) 1.00812 0.0874151
\(134\) 4.34181 0.375076
\(135\) −0.0774875 −0.00666907
\(136\) 4.82077 0.413378
\(137\) −23.0088 −1.96578 −0.982889 0.184201i \(-0.941030\pi\)
−0.982889 + 0.184201i \(0.941030\pi\)
\(138\) −22.1988 −1.88969
\(139\) 22.1663 1.88012 0.940059 0.341011i \(-0.110769\pi\)
0.940059 + 0.341011i \(0.110769\pi\)
\(140\) 0.0145734 0.00123167
\(141\) 11.4135 0.961194
\(142\) −7.99641 −0.671044
\(143\) −3.82884 −0.320184
\(144\) 4.52782 0.377318
\(145\) −0.0694983 −0.00577152
\(146\) 7.78804 0.644543
\(147\) 17.5005 1.44342
\(148\) −1.48010 −0.121663
\(149\) −4.23945 −0.347310 −0.173655 0.984807i \(-0.555558\pi\)
−0.173655 + 0.984807i \(0.555558\pi\)
\(150\) 13.7175 1.12003
\(151\) −12.3953 −1.00871 −0.504356 0.863496i \(-0.668270\pi\)
−0.504356 + 0.863496i \(0.668270\pi\)
\(152\) 1.27873 0.103718
\(153\) 21.8276 1.76466
\(154\) −1.96119 −0.158037
\(155\) −0.000666036 0 −5.34973e−5 0
\(156\) −4.22296 −0.338107
\(157\) 14.4011 1.14933 0.574666 0.818388i \(-0.305132\pi\)
0.574666 + 0.818388i \(0.305132\pi\)
\(158\) 2.65372 0.211118
\(159\) −31.0199 −2.46004
\(160\) 0.0184852 0.00146139
\(161\) 6.37866 0.502709
\(162\) −2.08230 −0.163601
\(163\) 12.9831 1.01691 0.508456 0.861088i \(-0.330216\pi\)
0.508456 + 0.861088i \(0.330216\pi\)
\(164\) −2.01682 −0.157487
\(165\) 0.126167 0.00982207
\(166\) 1.27427 0.0989029
\(167\) 6.79378 0.525719 0.262859 0.964834i \(-0.415334\pi\)
0.262859 + 0.964834i \(0.415334\pi\)
\(168\) −2.16306 −0.166884
\(169\) −10.6310 −0.817770
\(170\) 0.0891131 0.00683466
\(171\) 5.78985 0.442761
\(172\) 6.31722 0.481683
\(173\) −14.6890 −1.11679 −0.558394 0.829576i \(-0.688582\pi\)
−0.558394 + 0.829576i \(0.688582\pi\)
\(174\) 10.3154 0.782005
\(175\) −3.94162 −0.297959
\(176\) −2.48763 −0.187512
\(177\) −8.15869 −0.613245
\(178\) −15.1053 −1.13219
\(179\) 10.5999 0.792273 0.396137 0.918192i \(-0.370351\pi\)
0.396137 + 0.918192i \(0.370351\pi\)
\(180\) 0.0836978 0.00623847
\(181\) −11.7055 −0.870060 −0.435030 0.900416i \(-0.643262\pi\)
−0.435030 + 0.900416i \(0.643262\pi\)
\(182\) 1.21344 0.0899458
\(183\) 28.8180 2.13029
\(184\) 8.09086 0.596466
\(185\) −0.0273599 −0.00201154
\(186\) 0.0988570 0.00724854
\(187\) −11.9923 −0.876963
\(188\) −4.15993 −0.303394
\(189\) −3.30477 −0.240387
\(190\) 0.0236376 0.00171485
\(191\) 22.1834 1.60513 0.802566 0.596563i \(-0.203468\pi\)
0.802566 + 0.596563i \(0.203468\pi\)
\(192\) −2.74369 −0.198009
\(193\) 4.63212 0.333427 0.166714 0.986005i \(-0.446685\pi\)
0.166714 + 0.986005i \(0.446685\pi\)
\(194\) 2.52670 0.181407
\(195\) −0.0780624 −0.00559016
\(196\) −6.37846 −0.455604
\(197\) 3.87619 0.276167 0.138084 0.990421i \(-0.455906\pi\)
0.138084 + 0.990421i \(0.455906\pi\)
\(198\) −11.2635 −0.800465
\(199\) −2.00985 −0.142475 −0.0712374 0.997459i \(-0.522695\pi\)
−0.0712374 + 0.997459i \(0.522695\pi\)
\(200\) −4.99966 −0.353529
\(201\) −11.9126 −0.840249
\(202\) 7.37414 0.518843
\(203\) −2.96404 −0.208035
\(204\) −13.2267 −0.926054
\(205\) −0.0372814 −0.00260385
\(206\) 12.0562 0.839993
\(207\) 36.6340 2.54624
\(208\) 1.53915 0.106721
\(209\) −3.18100 −0.220034
\(210\) −0.0399847 −0.00275921
\(211\) 25.9771 1.78834 0.894168 0.447731i \(-0.147768\pi\)
0.894168 + 0.447731i \(0.147768\pi\)
\(212\) 11.3059 0.776494
\(213\) 21.9397 1.50328
\(214\) 8.88659 0.607475
\(215\) 0.116775 0.00796400
\(216\) −4.19186 −0.285220
\(217\) −0.0284058 −0.00192831
\(218\) 16.8778 1.14311
\(219\) −21.3679 −1.44391
\(220\) −0.0459844 −0.00310027
\(221\) 7.41991 0.499117
\(222\) 4.06092 0.272551
\(223\) 13.6461 0.913811 0.456906 0.889515i \(-0.348958\pi\)
0.456906 + 0.889515i \(0.348958\pi\)
\(224\) 0.788378 0.0526757
\(225\) −22.6376 −1.50917
\(226\) 4.93298 0.328137
\(227\) −20.5680 −1.36515 −0.682575 0.730816i \(-0.739140\pi\)
−0.682575 + 0.730816i \(0.739140\pi\)
\(228\) −3.50843 −0.232351
\(229\) 3.11471 0.205826 0.102913 0.994690i \(-0.467184\pi\)
0.102913 + 0.994690i \(0.467184\pi\)
\(230\) 0.149561 0.00986179
\(231\) 5.38090 0.354037
\(232\) −3.75967 −0.246834
\(233\) −3.85654 −0.252651 −0.126325 0.991989i \(-0.540318\pi\)
−0.126325 + 0.991989i \(0.540318\pi\)
\(234\) 6.96901 0.455579
\(235\) −0.0768972 −0.00501622
\(236\) 2.97362 0.193566
\(237\) −7.28097 −0.472950
\(238\) 3.80059 0.246356
\(239\) 17.2722 1.11724 0.558622 0.829422i \(-0.311330\pi\)
0.558622 + 0.829422i \(0.311330\pi\)
\(240\) −0.0507177 −0.00327381
\(241\) 25.5288 1.64446 0.822229 0.569157i \(-0.192730\pi\)
0.822229 + 0.569157i \(0.192730\pi\)
\(242\) −4.81170 −0.309308
\(243\) 18.2888 1.17323
\(244\) −10.5034 −0.672410
\(245\) −0.117907 −0.00753282
\(246\) 5.53353 0.352805
\(247\) 1.96816 0.125231
\(248\) −0.0360307 −0.00228795
\(249\) −3.49621 −0.221563
\(250\) −0.184846 −0.0116907
\(251\) 11.9515 0.754371 0.377186 0.926138i \(-0.376892\pi\)
0.377186 + 0.926138i \(0.376892\pi\)
\(252\) 3.56963 0.224866
\(253\) −20.1271 −1.26538
\(254\) −12.1590 −0.762924
\(255\) −0.244498 −0.0153111
\(256\) 1.00000 0.0625000
\(257\) 14.0935 0.879128 0.439564 0.898211i \(-0.355133\pi\)
0.439564 + 0.898211i \(0.355133\pi\)
\(258\) −17.3325 −1.07907
\(259\) −1.16688 −0.0725061
\(260\) 0.0284516 0.00176449
\(261\) −17.0231 −1.05370
\(262\) 0.824810 0.0509569
\(263\) 18.4311 1.13651 0.568256 0.822851i \(-0.307618\pi\)
0.568256 + 0.822851i \(0.307618\pi\)
\(264\) 6.82528 0.420067
\(265\) 0.208993 0.0128383
\(266\) 1.00812 0.0618118
\(267\) 41.4442 2.53635
\(268\) 4.34181 0.265218
\(269\) 0.809822 0.0493757 0.0246879 0.999695i \(-0.492141\pi\)
0.0246879 + 0.999695i \(0.492141\pi\)
\(270\) −0.0774875 −0.00471574
\(271\) −8.62338 −0.523833 −0.261917 0.965091i \(-0.584355\pi\)
−0.261917 + 0.965091i \(0.584355\pi\)
\(272\) 4.82077 0.292302
\(273\) −3.32929 −0.201498
\(274\) −23.0088 −1.39001
\(275\) 12.4373 0.749997
\(276\) −22.1988 −1.33621
\(277\) 32.1168 1.92971 0.964856 0.262779i \(-0.0846390\pi\)
0.964856 + 0.262779i \(0.0846390\pi\)
\(278\) 22.1663 1.32944
\(279\) −0.163141 −0.00976697
\(280\) 0.0145734 0.000870924 0
\(281\) 7.35716 0.438892 0.219446 0.975625i \(-0.429575\pi\)
0.219446 + 0.975625i \(0.429575\pi\)
\(282\) 11.4135 0.679667
\(283\) −15.1597 −0.901148 −0.450574 0.892739i \(-0.648781\pi\)
−0.450574 + 0.892739i \(0.648781\pi\)
\(284\) −7.99641 −0.474500
\(285\) −0.0648541 −0.00384162
\(286\) −3.82884 −0.226404
\(287\) −1.59002 −0.0938558
\(288\) 4.52782 0.266804
\(289\) 6.23984 0.367049
\(290\) −0.0694983 −0.00408108
\(291\) −6.93248 −0.406389
\(292\) 7.78804 0.455760
\(293\) 29.0382 1.69643 0.848214 0.529653i \(-0.177678\pi\)
0.848214 + 0.529653i \(0.177678\pi\)
\(294\) 17.5005 1.02065
\(295\) 0.0549681 0.00320037
\(296\) −1.48010 −0.0860288
\(297\) 10.4278 0.605082
\(298\) −4.23945 −0.245585
\(299\) 12.4531 0.720181
\(300\) 13.7175 0.791980
\(301\) 4.98036 0.287063
\(302\) −12.3953 −0.713267
\(303\) −20.2323 −1.16232
\(304\) 1.27873 0.0733400
\(305\) −0.194157 −0.0111174
\(306\) 21.8276 1.24780
\(307\) 9.33714 0.532899 0.266449 0.963849i \(-0.414149\pi\)
0.266449 + 0.963849i \(0.414149\pi\)
\(308\) −1.96119 −0.111749
\(309\) −33.0784 −1.88176
\(310\) −0.000666036 0 −3.78283e−5 0
\(311\) −13.9792 −0.792689 −0.396344 0.918102i \(-0.629721\pi\)
−0.396344 + 0.918102i \(0.629721\pi\)
\(312\) −4.22296 −0.239078
\(313\) 5.19525 0.293653 0.146826 0.989162i \(-0.453094\pi\)
0.146826 + 0.989162i \(0.453094\pi\)
\(314\) 14.4011 0.812700
\(315\) 0.0659855 0.00371786
\(316\) 2.65372 0.149283
\(317\) −7.35207 −0.412934 −0.206467 0.978454i \(-0.566197\pi\)
−0.206467 + 0.978454i \(0.566197\pi\)
\(318\) −31.0199 −1.73951
\(319\) 9.35266 0.523648
\(320\) 0.0184852 0.00103336
\(321\) −24.3820 −1.36087
\(322\) 6.37866 0.355469
\(323\) 6.16445 0.342999
\(324\) −2.08230 −0.115683
\(325\) −7.69525 −0.426855
\(326\) 12.9831 0.719066
\(327\) −46.3074 −2.56081
\(328\) −2.01682 −0.111360
\(329\) −3.27960 −0.180810
\(330\) 0.126167 0.00694525
\(331\) −2.03960 −0.112106 −0.0560532 0.998428i \(-0.517852\pi\)
−0.0560532 + 0.998428i \(0.517852\pi\)
\(332\) 1.27427 0.0699349
\(333\) −6.70161 −0.367246
\(334\) 6.79378 0.371739
\(335\) 0.0802594 0.00438504
\(336\) −2.16306 −0.118005
\(337\) 15.4529 0.841772 0.420886 0.907114i \(-0.361719\pi\)
0.420886 + 0.907114i \(0.361719\pi\)
\(338\) −10.6310 −0.578250
\(339\) −13.5346 −0.735096
\(340\) 0.0891131 0.00483283
\(341\) 0.0896310 0.00485379
\(342\) 5.78985 0.313079
\(343\) −10.5473 −0.569500
\(344\) 6.31722 0.340601
\(345\) −0.410350 −0.0220925
\(346\) −14.6890 −0.789688
\(347\) 6.31059 0.338770 0.169385 0.985550i \(-0.445822\pi\)
0.169385 + 0.985550i \(0.445822\pi\)
\(348\) 10.3154 0.552961
\(349\) 14.4514 0.773568 0.386784 0.922170i \(-0.373586\pi\)
0.386784 + 0.922170i \(0.373586\pi\)
\(350\) −3.94162 −0.210689
\(351\) −6.45192 −0.344378
\(352\) −2.48763 −0.132591
\(353\) −11.9084 −0.633822 −0.316911 0.948455i \(-0.602646\pi\)
−0.316911 + 0.948455i \(0.602646\pi\)
\(354\) −8.15869 −0.433629
\(355\) −0.147816 −0.00784524
\(356\) −15.1053 −0.800579
\(357\) −10.4276 −0.551889
\(358\) 10.5999 0.560222
\(359\) −1.32615 −0.0699917 −0.0349958 0.999387i \(-0.511142\pi\)
−0.0349958 + 0.999387i \(0.511142\pi\)
\(360\) 0.0836978 0.00441126
\(361\) −17.3649 −0.913940
\(362\) −11.7055 −0.615225
\(363\) 13.2018 0.692915
\(364\) 1.21344 0.0636013
\(365\) 0.143964 0.00753540
\(366\) 28.8180 1.50634
\(367\) −6.64307 −0.346765 −0.173383 0.984855i \(-0.555470\pi\)
−0.173383 + 0.984855i \(0.555470\pi\)
\(368\) 8.09086 0.421765
\(369\) −9.13181 −0.475383
\(370\) −0.0273599 −0.00142237
\(371\) 8.91334 0.462758
\(372\) 0.0988570 0.00512550
\(373\) 21.9878 1.13848 0.569241 0.822170i \(-0.307237\pi\)
0.569241 + 0.822170i \(0.307237\pi\)
\(374\) −11.9923 −0.620107
\(375\) 0.507160 0.0261896
\(376\) −4.15993 −0.214532
\(377\) −5.78671 −0.298031
\(378\) −3.30477 −0.169979
\(379\) −12.0688 −0.619933 −0.309967 0.950747i \(-0.600318\pi\)
−0.309967 + 0.950747i \(0.600318\pi\)
\(380\) 0.0236376 0.00121258
\(381\) 33.3605 1.70911
\(382\) 22.1834 1.13500
\(383\) −28.9615 −1.47986 −0.739931 0.672683i \(-0.765142\pi\)
−0.739931 + 0.672683i \(0.765142\pi\)
\(384\) −2.74369 −0.140013
\(385\) −0.0362531 −0.00184763
\(386\) 4.63212 0.235768
\(387\) 28.6032 1.45398
\(388\) 2.52670 0.128274
\(389\) −3.08570 −0.156451 −0.0782255 0.996936i \(-0.524925\pi\)
−0.0782255 + 0.996936i \(0.524925\pi\)
\(390\) −0.0780624 −0.00395284
\(391\) 39.0042 1.97253
\(392\) −6.37846 −0.322161
\(393\) −2.26302 −0.114154
\(394\) 3.87619 0.195280
\(395\) 0.0490546 0.00246820
\(396\) −11.2635 −0.566014
\(397\) 7.76995 0.389963 0.194981 0.980807i \(-0.437535\pi\)
0.194981 + 0.980807i \(0.437535\pi\)
\(398\) −2.00985 −0.100745
\(399\) −2.76597 −0.138472
\(400\) −4.99966 −0.249983
\(401\) 34.9117 1.74341 0.871705 0.490032i \(-0.163015\pi\)
0.871705 + 0.490032i \(0.163015\pi\)
\(402\) −11.9126 −0.594146
\(403\) −0.0554568 −0.00276250
\(404\) 7.37414 0.366877
\(405\) −0.0384918 −0.00191267
\(406\) −2.96404 −0.147103
\(407\) 3.68193 0.182506
\(408\) −13.2267 −0.654819
\(409\) −34.9325 −1.72730 −0.863650 0.504092i \(-0.831827\pi\)
−0.863650 + 0.504092i \(0.831827\pi\)
\(410\) −0.0372814 −0.00184120
\(411\) 63.1290 3.11393
\(412\) 12.0562 0.593965
\(413\) 2.34434 0.115357
\(414\) 36.6340 1.80046
\(415\) 0.0235553 0.00115628
\(416\) 1.53915 0.0754632
\(417\) −60.8173 −2.97824
\(418\) −3.18100 −0.155588
\(419\) 30.3730 1.48382 0.741909 0.670501i \(-0.233921\pi\)
0.741909 + 0.670501i \(0.233921\pi\)
\(420\) −0.0399847 −0.00195105
\(421\) 23.0913 1.12540 0.562700 0.826661i \(-0.309763\pi\)
0.562700 + 0.826661i \(0.309763\pi\)
\(422\) 25.9771 1.26454
\(423\) −18.8354 −0.915809
\(424\) 11.3059 0.549064
\(425\) −24.1022 −1.16913
\(426\) 21.9397 1.06298
\(427\) −8.28064 −0.400728
\(428\) 8.88659 0.429549
\(429\) 10.5052 0.507194
\(430\) 0.116775 0.00563140
\(431\) 10.3071 0.496478 0.248239 0.968699i \(-0.420148\pi\)
0.248239 + 0.968699i \(0.420148\pi\)
\(432\) −4.19186 −0.201681
\(433\) −39.1152 −1.87976 −0.939878 0.341509i \(-0.889062\pi\)
−0.939878 + 0.341509i \(0.889062\pi\)
\(434\) −0.0284058 −0.00136352
\(435\) 0.190682 0.00914249
\(436\) 16.8778 0.808300
\(437\) 10.3460 0.494917
\(438\) −21.3679 −1.02100
\(439\) 6.05495 0.288987 0.144493 0.989506i \(-0.453845\pi\)
0.144493 + 0.989506i \(0.453845\pi\)
\(440\) −0.0459844 −0.00219222
\(441\) −28.8805 −1.37526
\(442\) 7.41991 0.352929
\(443\) 31.4493 1.49420 0.747100 0.664712i \(-0.231446\pi\)
0.747100 + 0.664712i \(0.231446\pi\)
\(444\) 4.06092 0.192723
\(445\) −0.279225 −0.0132365
\(446\) 13.6461 0.646162
\(447\) 11.6317 0.550162
\(448\) 0.788378 0.0372474
\(449\) 14.3148 0.675557 0.337778 0.941226i \(-0.390325\pi\)
0.337778 + 0.941226i \(0.390325\pi\)
\(450\) −22.6376 −1.06714
\(451\) 5.01711 0.236246
\(452\) 4.93298 0.232028
\(453\) 34.0087 1.59787
\(454\) −20.5680 −0.965306
\(455\) 0.0224306 0.00105156
\(456\) −3.50843 −0.164297
\(457\) −12.6993 −0.594050 −0.297025 0.954870i \(-0.595994\pi\)
−0.297025 + 0.954870i \(0.595994\pi\)
\(458\) 3.11471 0.145541
\(459\) −20.2080 −0.943229
\(460\) 0.149561 0.00697334
\(461\) 14.6325 0.681503 0.340751 0.940153i \(-0.389319\pi\)
0.340751 + 0.940153i \(0.389319\pi\)
\(462\) 5.38090 0.250342
\(463\) −3.91048 −0.181735 −0.0908676 0.995863i \(-0.528964\pi\)
−0.0908676 + 0.995863i \(0.528964\pi\)
\(464\) −3.75967 −0.174538
\(465\) 0.00182739 8.47434e−5 0
\(466\) −3.85654 −0.178651
\(467\) 37.7693 1.74775 0.873877 0.486148i \(-0.161598\pi\)
0.873877 + 0.486148i \(0.161598\pi\)
\(468\) 6.96901 0.322143
\(469\) 3.42299 0.158059
\(470\) −0.0768972 −0.00354700
\(471\) −39.5121 −1.82062
\(472\) 2.97362 0.136872
\(473\) −15.7149 −0.722571
\(474\) −7.28097 −0.334426
\(475\) −6.39320 −0.293340
\(476\) 3.80059 0.174200
\(477\) 51.1912 2.34388
\(478\) 17.2722 0.790011
\(479\) 2.74746 0.125535 0.0627673 0.998028i \(-0.480007\pi\)
0.0627673 + 0.998028i \(0.480007\pi\)
\(480\) −0.0507177 −0.00231494
\(481\) −2.27810 −0.103872
\(482\) 25.5288 1.16281
\(483\) −17.5010 −0.796325
\(484\) −4.81170 −0.218714
\(485\) 0.0467067 0.00212084
\(486\) 18.2888 0.829596
\(487\) −2.34055 −0.106061 −0.0530303 0.998593i \(-0.516888\pi\)
−0.0530303 + 0.998593i \(0.516888\pi\)
\(488\) −10.5034 −0.475466
\(489\) −35.6215 −1.61086
\(490\) −0.117907 −0.00532651
\(491\) 0.808608 0.0364920 0.0182460 0.999834i \(-0.494192\pi\)
0.0182460 + 0.999834i \(0.494192\pi\)
\(492\) 5.53353 0.249471
\(493\) −18.1245 −0.816286
\(494\) 1.96816 0.0885516
\(495\) −0.208209 −0.00935830
\(496\) −0.0360307 −0.00161783
\(497\) −6.30420 −0.282782
\(498\) −3.49621 −0.156669
\(499\) −4.23519 −0.189593 −0.0947967 0.995497i \(-0.530220\pi\)
−0.0947967 + 0.995497i \(0.530220\pi\)
\(500\) −0.184846 −0.00826656
\(501\) −18.6400 −0.832775
\(502\) 11.9515 0.533421
\(503\) 37.5604 1.67474 0.837369 0.546639i \(-0.184093\pi\)
0.837369 + 0.546639i \(0.184093\pi\)
\(504\) 3.56963 0.159004
\(505\) 0.136313 0.00606584
\(506\) −20.1271 −0.894757
\(507\) 29.1682 1.29540
\(508\) −12.1590 −0.539469
\(509\) −8.28309 −0.367142 −0.183571 0.983006i \(-0.558766\pi\)
−0.183571 + 0.983006i \(0.558766\pi\)
\(510\) −0.244498 −0.0108266
\(511\) 6.13992 0.271614
\(512\) 1.00000 0.0441942
\(513\) −5.36025 −0.236661
\(514\) 14.0935 0.621637
\(515\) 0.222861 0.00982043
\(516\) −17.3325 −0.763019
\(517\) 10.3484 0.455120
\(518\) −1.16688 −0.0512696
\(519\) 40.3021 1.76907
\(520\) 0.0284516 0.00124769
\(521\) −15.8685 −0.695210 −0.347605 0.937641i \(-0.613005\pi\)
−0.347605 + 0.937641i \(0.613005\pi\)
\(522\) −17.0231 −0.745081
\(523\) −14.0288 −0.613435 −0.306718 0.951801i \(-0.599231\pi\)
−0.306718 + 0.951801i \(0.599231\pi\)
\(524\) 0.824810 0.0360320
\(525\) 10.8146 0.471987
\(526\) 18.4311 0.803636
\(527\) −0.173696 −0.00756631
\(528\) 6.82528 0.297032
\(529\) 42.4621 1.84618
\(530\) 0.208993 0.00907806
\(531\) 13.4640 0.584289
\(532\) 1.00812 0.0437076
\(533\) −3.10420 −0.134458
\(534\) 41.4442 1.79347
\(535\) 0.164271 0.00710204
\(536\) 4.34181 0.187538
\(537\) −29.0828 −1.25502
\(538\) 0.809822 0.0349139
\(539\) 15.8672 0.683450
\(540\) −0.0774875 −0.00333453
\(541\) −6.00812 −0.258309 −0.129155 0.991624i \(-0.541226\pi\)
−0.129155 + 0.991624i \(0.541226\pi\)
\(542\) −8.62338 −0.370406
\(543\) 32.1161 1.37823
\(544\) 4.82077 0.206689
\(545\) 0.311990 0.0133642
\(546\) −3.32929 −0.142480
\(547\) −0.0147045 −0.000628719 0 −0.000314360 1.00000i \(-0.500100\pi\)
−0.000314360 1.00000i \(0.500100\pi\)
\(548\) −23.0088 −0.982889
\(549\) −47.5574 −2.02970
\(550\) 12.4373 0.530328
\(551\) −4.80759 −0.204810
\(552\) −22.1988 −0.944844
\(553\) 2.09213 0.0889665
\(554\) 32.1168 1.36451
\(555\) 0.0750671 0.00318642
\(556\) 22.1663 0.940059
\(557\) −13.7665 −0.583304 −0.291652 0.956524i \(-0.594205\pi\)
−0.291652 + 0.956524i \(0.594205\pi\)
\(558\) −0.163141 −0.00690629
\(559\) 9.72317 0.411246
\(560\) 0.0145734 0.000615837 0
\(561\) 32.9031 1.38917
\(562\) 7.35716 0.310343
\(563\) 39.9870 1.68525 0.842626 0.538500i \(-0.181009\pi\)
0.842626 + 0.538500i \(0.181009\pi\)
\(564\) 11.4135 0.480597
\(565\) 0.0911872 0.00383628
\(566\) −15.1597 −0.637208
\(567\) −1.64164 −0.0689424
\(568\) −7.99641 −0.335522
\(569\) 28.5605 1.19732 0.598659 0.801004i \(-0.295701\pi\)
0.598659 + 0.801004i \(0.295701\pi\)
\(570\) −0.0648541 −0.00271644
\(571\) 9.63595 0.403252 0.201626 0.979463i \(-0.435377\pi\)
0.201626 + 0.979463i \(0.435377\pi\)
\(572\) −3.82884 −0.160092
\(573\) −60.8642 −2.54264
\(574\) −1.59002 −0.0663661
\(575\) −40.4516 −1.68695
\(576\) 4.52782 0.188659
\(577\) −13.7790 −0.573627 −0.286813 0.957986i \(-0.592596\pi\)
−0.286813 + 0.957986i \(0.592596\pi\)
\(578\) 6.23984 0.259543
\(579\) −12.7091 −0.528171
\(580\) −0.0694983 −0.00288576
\(581\) 1.00461 0.0416782
\(582\) −6.93248 −0.287361
\(583\) −28.1249 −1.16482
\(584\) 7.78804 0.322271
\(585\) 0.128824 0.00532621
\(586\) 29.0382 1.19956
\(587\) 8.39657 0.346564 0.173282 0.984872i \(-0.444563\pi\)
0.173282 + 0.984872i \(0.444563\pi\)
\(588\) 17.5005 0.721708
\(589\) −0.0460734 −0.00189842
\(590\) 0.0549681 0.00226300
\(591\) −10.6351 −0.437468
\(592\) −1.48010 −0.0608316
\(593\) 0.968410 0.0397679 0.0198839 0.999802i \(-0.493670\pi\)
0.0198839 + 0.999802i \(0.493670\pi\)
\(594\) 10.4278 0.427858
\(595\) 0.0702548 0.00288017
\(596\) −4.23945 −0.173655
\(597\) 5.51441 0.225690
\(598\) 12.4531 0.509245
\(599\) 25.2843 1.03309 0.516544 0.856260i \(-0.327218\pi\)
0.516544 + 0.856260i \(0.327218\pi\)
\(600\) 13.7175 0.560015
\(601\) −17.2252 −0.702630 −0.351315 0.936257i \(-0.614265\pi\)
−0.351315 + 0.936257i \(0.614265\pi\)
\(602\) 4.98036 0.202984
\(603\) 19.6590 0.800575
\(604\) −12.3953 −0.504356
\(605\) −0.0889454 −0.00361615
\(606\) −20.2323 −0.821883
\(607\) −30.4396 −1.23551 −0.617753 0.786372i \(-0.711957\pi\)
−0.617753 + 0.786372i \(0.711957\pi\)
\(608\) 1.27873 0.0518592
\(609\) 8.13240 0.329541
\(610\) −0.194157 −0.00786120
\(611\) −6.40277 −0.259028
\(612\) 21.8276 0.882328
\(613\) −32.8020 −1.32486 −0.662430 0.749123i \(-0.730475\pi\)
−0.662430 + 0.749123i \(0.730475\pi\)
\(614\) 9.33714 0.376816
\(615\) 0.102289 0.00412467
\(616\) −1.96119 −0.0790187
\(617\) −5.77642 −0.232550 −0.116275 0.993217i \(-0.537095\pi\)
−0.116275 + 0.993217i \(0.537095\pi\)
\(618\) −33.0784 −1.33061
\(619\) 20.4060 0.820186 0.410093 0.912044i \(-0.365496\pi\)
0.410093 + 0.912044i \(0.365496\pi\)
\(620\) −0.000666036 0 −2.67486e−5 0
\(621\) −33.9158 −1.36099
\(622\) −13.9792 −0.560516
\(623\) −11.9087 −0.477112
\(624\) −4.22296 −0.169054
\(625\) 24.9949 0.999795
\(626\) 5.19525 0.207644
\(627\) 8.72767 0.348549
\(628\) 14.4011 0.574666
\(629\) −7.13520 −0.284499
\(630\) 0.0659855 0.00262893
\(631\) 1.05713 0.0420836 0.0210418 0.999779i \(-0.493302\pi\)
0.0210418 + 0.999779i \(0.493302\pi\)
\(632\) 2.65372 0.105559
\(633\) −71.2730 −2.83285
\(634\) −7.35207 −0.291988
\(635\) −0.224762 −0.00891941
\(636\) −31.0199 −1.23002
\(637\) −9.81743 −0.388981
\(638\) 9.35266 0.370275
\(639\) −36.2063 −1.43230
\(640\) 0.0184852 0.000730693 0
\(641\) 12.5999 0.497668 0.248834 0.968546i \(-0.419953\pi\)
0.248834 + 0.968546i \(0.419953\pi\)
\(642\) −24.3820 −0.962281
\(643\) −49.2429 −1.94195 −0.970975 0.239182i \(-0.923121\pi\)
−0.970975 + 0.239182i \(0.923121\pi\)
\(644\) 6.37866 0.251354
\(645\) −0.320395 −0.0126155
\(646\) 6.16445 0.242537
\(647\) −46.3985 −1.82411 −0.912056 0.410065i \(-0.865506\pi\)
−0.912056 + 0.410065i \(0.865506\pi\)
\(648\) −2.08230 −0.0818005
\(649\) −7.39727 −0.290368
\(650\) −7.69525 −0.301832
\(651\) 0.0779367 0.00305458
\(652\) 12.9831 0.508456
\(653\) −15.7993 −0.618275 −0.309137 0.951017i \(-0.600040\pi\)
−0.309137 + 0.951017i \(0.600040\pi\)
\(654\) −46.3074 −1.81076
\(655\) 0.0152468 0.000595742 0
\(656\) −2.01682 −0.0787437
\(657\) 35.2628 1.37573
\(658\) −3.27960 −0.127852
\(659\) 24.6396 0.959822 0.479911 0.877317i \(-0.340669\pi\)
0.479911 + 0.877317i \(0.340669\pi\)
\(660\) 0.126167 0.00491104
\(661\) −16.1127 −0.626710 −0.313355 0.949636i \(-0.601453\pi\)
−0.313355 + 0.949636i \(0.601453\pi\)
\(662\) −2.03960 −0.0792713
\(663\) −20.3579 −0.790636
\(664\) 1.27427 0.0494514
\(665\) 0.0186353 0.000722647 0
\(666\) −6.70161 −0.259682
\(667\) −30.4190 −1.17783
\(668\) 6.79378 0.262859
\(669\) −37.4407 −1.44754
\(670\) 0.0802594 0.00310069
\(671\) 26.1285 1.00868
\(672\) −2.16306 −0.0834420
\(673\) 33.7196 1.29979 0.649897 0.760022i \(-0.274812\pi\)
0.649897 + 0.760022i \(0.274812\pi\)
\(674\) 15.4529 0.595223
\(675\) 20.9579 0.806669
\(676\) −10.6310 −0.408885
\(677\) −2.36642 −0.0909491 −0.0454745 0.998965i \(-0.514480\pi\)
−0.0454745 + 0.998965i \(0.514480\pi\)
\(678\) −13.5346 −0.519791
\(679\) 1.99200 0.0764458
\(680\) 0.0891131 0.00341733
\(681\) 56.4323 2.16249
\(682\) 0.0896310 0.00343215
\(683\) −25.3441 −0.969765 −0.484883 0.874579i \(-0.661138\pi\)
−0.484883 + 0.874579i \(0.661138\pi\)
\(684\) 5.78985 0.221380
\(685\) −0.425324 −0.0162508
\(686\) −10.5473 −0.402697
\(687\) −8.54579 −0.326042
\(688\) 6.31722 0.240842
\(689\) 17.4016 0.662947
\(690\) −0.410350 −0.0156218
\(691\) −43.1346 −1.64092 −0.820458 0.571707i \(-0.806281\pi\)
−0.820458 + 0.571707i \(0.806281\pi\)
\(692\) −14.6890 −0.558394
\(693\) −8.87993 −0.337321
\(694\) 6.31059 0.239547
\(695\) 0.409749 0.0155427
\(696\) 10.3154 0.391002
\(697\) −9.72264 −0.368271
\(698\) 14.4514 0.546995
\(699\) 10.5811 0.400216
\(700\) −3.94162 −0.148979
\(701\) −33.6047 −1.26923 −0.634615 0.772828i \(-0.718841\pi\)
−0.634615 + 0.772828i \(0.718841\pi\)
\(702\) −6.45192 −0.243512
\(703\) −1.89264 −0.0713822
\(704\) −2.48763 −0.0937560
\(705\) 0.210982 0.00794604
\(706\) −11.9084 −0.448180
\(707\) 5.81361 0.218643
\(708\) −8.15869 −0.306622
\(709\) −44.9350 −1.68757 −0.843784 0.536682i \(-0.819677\pi\)
−0.843784 + 0.536682i \(0.819677\pi\)
\(710\) −0.147816 −0.00554742
\(711\) 12.0156 0.450618
\(712\) −15.1053 −0.566095
\(713\) −0.291519 −0.0109175
\(714\) −10.4276 −0.390244
\(715\) −0.0707771 −0.00264691
\(716\) 10.5999 0.396137
\(717\) −47.3895 −1.76979
\(718\) −1.32615 −0.0494916
\(719\) −7.55784 −0.281860 −0.140930 0.990020i \(-0.545009\pi\)
−0.140930 + 0.990020i \(0.545009\pi\)
\(720\) 0.0836978 0.00311923
\(721\) 9.50482 0.353978
\(722\) −17.3649 −0.646253
\(723\) −70.0432 −2.60493
\(724\) −11.7055 −0.435030
\(725\) 18.7971 0.698105
\(726\) 13.2018 0.489965
\(727\) 19.0693 0.707241 0.353621 0.935389i \(-0.384950\pi\)
0.353621 + 0.935389i \(0.384950\pi\)
\(728\) 1.21344 0.0449729
\(729\) −43.9318 −1.62710
\(730\) 0.143964 0.00532833
\(731\) 30.4539 1.12638
\(732\) 28.8180 1.06514
\(733\) −39.2998 −1.45157 −0.725786 0.687921i \(-0.758524\pi\)
−0.725786 + 0.687921i \(0.758524\pi\)
\(734\) −6.64307 −0.245200
\(735\) 0.323501 0.0119325
\(736\) 8.09086 0.298233
\(737\) −10.8008 −0.397853
\(738\) −9.13181 −0.336147
\(739\) 16.1536 0.594219 0.297110 0.954843i \(-0.403977\pi\)
0.297110 + 0.954843i \(0.403977\pi\)
\(740\) −0.0273599 −0.00100577
\(741\) −5.40001 −0.198374
\(742\) 8.91334 0.327219
\(743\) −7.02866 −0.257856 −0.128928 0.991654i \(-0.541154\pi\)
−0.128928 + 0.991654i \(0.541154\pi\)
\(744\) 0.0988570 0.00362427
\(745\) −0.0783673 −0.00287116
\(746\) 21.9878 0.805029
\(747\) 5.76969 0.211102
\(748\) −11.9923 −0.438482
\(749\) 7.00599 0.255993
\(750\) 0.507160 0.0185189
\(751\) 22.0628 0.805083 0.402541 0.915402i \(-0.368127\pi\)
0.402541 + 0.915402i \(0.368127\pi\)
\(752\) −4.15993 −0.151697
\(753\) −32.7912 −1.19498
\(754\) −5.78671 −0.210739
\(755\) −0.229129 −0.00833886
\(756\) −3.30477 −0.120193
\(757\) 26.7813 0.973384 0.486692 0.873574i \(-0.338203\pi\)
0.486692 + 0.873574i \(0.338203\pi\)
\(758\) −12.0688 −0.438359
\(759\) 55.2224 2.00445
\(760\) 0.0236376 0.000857425 0
\(761\) 40.3205 1.46162 0.730809 0.682582i \(-0.239143\pi\)
0.730809 + 0.682582i \(0.239143\pi\)
\(762\) 33.3605 1.20852
\(763\) 13.3061 0.481713
\(764\) 22.1834 0.802566
\(765\) 0.403488 0.0145881
\(766\) −28.9615 −1.04642
\(767\) 4.57686 0.165261
\(768\) −2.74369 −0.0990043
\(769\) −44.8943 −1.61893 −0.809465 0.587168i \(-0.800243\pi\)
−0.809465 + 0.587168i \(0.800243\pi\)
\(770\) −0.0362531 −0.00130647
\(771\) −38.6681 −1.39260
\(772\) 4.63212 0.166714
\(773\) −31.4851 −1.13244 −0.566221 0.824254i \(-0.691595\pi\)
−0.566221 + 0.824254i \(0.691595\pi\)
\(774\) 28.6032 1.02812
\(775\) 0.180141 0.00647086
\(776\) 2.52670 0.0907033
\(777\) 3.20154 0.114855
\(778\) −3.08570 −0.110628
\(779\) −2.57897 −0.0924010
\(780\) −0.0780624 −0.00279508
\(781\) 19.8921 0.711796
\(782\) 39.0042 1.39479
\(783\) 15.7600 0.563217
\(784\) −6.37846 −0.227802
\(785\) 0.266207 0.00950135
\(786\) −2.26302 −0.0807193
\(787\) −34.6110 −1.23375 −0.616874 0.787062i \(-0.711601\pi\)
−0.616874 + 0.787062i \(0.711601\pi\)
\(788\) 3.87619 0.138084
\(789\) −50.5693 −1.80031
\(790\) 0.0490546 0.00174528
\(791\) 3.88905 0.138279
\(792\) −11.2635 −0.400232
\(793\) −16.1663 −0.574083
\(794\) 7.76995 0.275745
\(795\) −0.573410 −0.0203368
\(796\) −2.00985 −0.0712374
\(797\) 34.8024 1.23276 0.616382 0.787447i \(-0.288598\pi\)
0.616382 + 0.787447i \(0.288598\pi\)
\(798\) −2.76597 −0.0979142
\(799\) −20.0541 −0.709462
\(800\) −4.99966 −0.176765
\(801\) −68.3941 −2.41659
\(802\) 34.9117 1.23278
\(803\) −19.3737 −0.683685
\(804\) −11.9126 −0.420124
\(805\) 0.117911 0.00415582
\(806\) −0.0554568 −0.00195338
\(807\) −2.22190 −0.0782145
\(808\) 7.37414 0.259421
\(809\) 40.0745 1.40894 0.704471 0.709732i \(-0.251184\pi\)
0.704471 + 0.709732i \(0.251184\pi\)
\(810\) −0.0384918 −0.00135246
\(811\) −47.9594 −1.68408 −0.842041 0.539414i \(-0.818646\pi\)
−0.842041 + 0.539414i \(0.818646\pi\)
\(812\) −2.96404 −0.104017
\(813\) 23.6599 0.829788
\(814\) 3.68193 0.129052
\(815\) 0.239995 0.00840666
\(816\) −13.2267 −0.463027
\(817\) 8.07800 0.282613
\(818\) −34.9325 −1.22139
\(819\) 5.49422 0.191984
\(820\) −0.0372814 −0.00130192
\(821\) 23.1030 0.806299 0.403149 0.915134i \(-0.367916\pi\)
0.403149 + 0.915134i \(0.367916\pi\)
\(822\) 63.1290 2.20188
\(823\) −22.1407 −0.771776 −0.385888 0.922546i \(-0.626105\pi\)
−0.385888 + 0.922546i \(0.626105\pi\)
\(824\) 12.0562 0.419996
\(825\) −34.1240 −1.18805
\(826\) 2.34434 0.0815700
\(827\) 31.4763 1.09454 0.547270 0.836956i \(-0.315667\pi\)
0.547270 + 0.836956i \(0.315667\pi\)
\(828\) 36.6340 1.27312
\(829\) −20.9745 −0.728474 −0.364237 0.931306i \(-0.618670\pi\)
−0.364237 + 0.931306i \(0.618670\pi\)
\(830\) 0.0235553 0.000817615 0
\(831\) −88.1185 −3.05680
\(832\) 1.53915 0.0533606
\(833\) −30.7491 −1.06539
\(834\) −60.8173 −2.10593
\(835\) 0.125585 0.00434604
\(836\) −3.18100 −0.110017
\(837\) 0.151036 0.00522056
\(838\) 30.3730 1.04922
\(839\) 20.0722 0.692970 0.346485 0.938055i \(-0.387375\pi\)
0.346485 + 0.938055i \(0.387375\pi\)
\(840\) −0.0399847 −0.00137960
\(841\) −14.8649 −0.512583
\(842\) 23.0913 0.795778
\(843\) −20.1858 −0.695235
\(844\) 25.9771 0.894168
\(845\) −0.196517 −0.00676038
\(846\) −18.8354 −0.647575
\(847\) −3.79344 −0.130344
\(848\) 11.3059 0.388247
\(849\) 41.5934 1.42748
\(850\) −24.1022 −0.826699
\(851\) −11.9753 −0.410506
\(852\) 21.9397 0.751640
\(853\) 24.7291 0.846708 0.423354 0.905964i \(-0.360853\pi\)
0.423354 + 0.905964i \(0.360853\pi\)
\(854\) −8.28064 −0.283358
\(855\) 0.107027 0.00366023
\(856\) 8.88659 0.303737
\(857\) −48.0887 −1.64268 −0.821339 0.570440i \(-0.806773\pi\)
−0.821339 + 0.570440i \(0.806773\pi\)
\(858\) 10.5052 0.358640
\(859\) −13.3622 −0.455912 −0.227956 0.973671i \(-0.573204\pi\)
−0.227956 + 0.973671i \(0.573204\pi\)
\(860\) 0.116775 0.00398200
\(861\) 4.36251 0.148674
\(862\) 10.3071 0.351063
\(863\) −13.8921 −0.472893 −0.236447 0.971644i \(-0.575983\pi\)
−0.236447 + 0.971644i \(0.575983\pi\)
\(864\) −4.19186 −0.142610
\(865\) −0.271530 −0.00923231
\(866\) −39.1152 −1.32919
\(867\) −17.1202 −0.581431
\(868\) −0.0284058 −0.000964156 0
\(869\) −6.60146 −0.223939
\(870\) 0.190682 0.00646471
\(871\) 6.68272 0.226435
\(872\) 16.8778 0.571554
\(873\) 11.4405 0.387201
\(874\) 10.3460 0.349959
\(875\) −0.145729 −0.00492652
\(876\) −21.3679 −0.721956
\(877\) 43.3498 1.46382 0.731910 0.681401i \(-0.238629\pi\)
0.731910 + 0.681401i \(0.238629\pi\)
\(878\) 6.05495 0.204345
\(879\) −79.6717 −2.68726
\(880\) −0.0459844 −0.00155013
\(881\) −23.1256 −0.779122 −0.389561 0.921001i \(-0.627373\pi\)
−0.389561 + 0.921001i \(0.627373\pi\)
\(882\) −28.8805 −0.972458
\(883\) −40.1517 −1.35121 −0.675605 0.737263i \(-0.736118\pi\)
−0.675605 + 0.737263i \(0.736118\pi\)
\(884\) 7.41991 0.249559
\(885\) −0.150815 −0.00506960
\(886\) 31.4493 1.05656
\(887\) 2.84685 0.0955880 0.0477940 0.998857i \(-0.484781\pi\)
0.0477940 + 0.998857i \(0.484781\pi\)
\(888\) 4.06092 0.136276
\(889\) −9.58589 −0.321500
\(890\) −0.279225 −0.00935964
\(891\) 5.17999 0.173536
\(892\) 13.6461 0.456906
\(893\) −5.31941 −0.178007
\(894\) 11.6317 0.389023
\(895\) 0.195942 0.00654960
\(896\) 0.788378 0.0263379
\(897\) −34.1674 −1.14082
\(898\) 14.3148 0.477691
\(899\) 0.135463 0.00451796
\(900\) −22.6376 −0.754585
\(901\) 54.5033 1.81577
\(902\) 5.01711 0.167051
\(903\) −13.6645 −0.454727
\(904\) 4.93298 0.164068
\(905\) −0.216378 −0.00719265
\(906\) 34.0087 1.12986
\(907\) 33.0557 1.09760 0.548798 0.835955i \(-0.315086\pi\)
0.548798 + 0.835955i \(0.315086\pi\)
\(908\) −20.5680 −0.682575
\(909\) 33.3888 1.10744
\(910\) 0.0224306 0.000743568 0
\(911\) −40.3586 −1.33714 −0.668570 0.743649i \(-0.733093\pi\)
−0.668570 + 0.743649i \(0.733093\pi\)
\(912\) −3.50843 −0.116176
\(913\) −3.16992 −0.104909
\(914\) −12.6993 −0.420057
\(915\) 0.532707 0.0176108
\(916\) 3.11471 0.102913
\(917\) 0.650262 0.0214735
\(918\) −20.2080 −0.666964
\(919\) 38.4416 1.26807 0.634035 0.773304i \(-0.281398\pi\)
0.634035 + 0.773304i \(0.281398\pi\)
\(920\) 0.149561 0.00493090
\(921\) −25.6182 −0.844148
\(922\) 14.6325 0.481895
\(923\) −12.3077 −0.405113
\(924\) 5.38090 0.177019
\(925\) 7.39997 0.243310
\(926\) −3.91048 −0.128506
\(927\) 54.5882 1.79291
\(928\) −3.75967 −0.123417
\(929\) 34.6265 1.13606 0.568030 0.823008i \(-0.307706\pi\)
0.568030 + 0.823008i \(0.307706\pi\)
\(930\) 0.00182739 5.99226e−5 0
\(931\) −8.15631 −0.267312
\(932\) −3.85654 −0.126325
\(933\) 38.3546 1.25567
\(934\) 37.7693 1.23585
\(935\) −0.221680 −0.00724972
\(936\) 6.96901 0.227789
\(937\) 23.4427 0.765838 0.382919 0.923782i \(-0.374919\pi\)
0.382919 + 0.923782i \(0.374919\pi\)
\(938\) 3.42299 0.111765
\(939\) −14.2541 −0.465166
\(940\) −0.0768972 −0.00250811
\(941\) 16.0696 0.523852 0.261926 0.965088i \(-0.415642\pi\)
0.261926 + 0.965088i \(0.415642\pi\)
\(942\) −39.5121 −1.28737
\(943\) −16.3178 −0.531382
\(944\) 2.97362 0.0967831
\(945\) −0.0610895 −0.00198724
\(946\) −15.7149 −0.510935
\(947\) 57.3229 1.86274 0.931371 0.364071i \(-0.118613\pi\)
0.931371 + 0.364071i \(0.118613\pi\)
\(948\) −7.28097 −0.236475
\(949\) 11.9870 0.389114
\(950\) −6.39320 −0.207423
\(951\) 20.1718 0.654115
\(952\) 3.80059 0.123178
\(953\) −15.8131 −0.512237 −0.256118 0.966645i \(-0.582444\pi\)
−0.256118 + 0.966645i \(0.582444\pi\)
\(954\) 51.1912 1.65738
\(955\) 0.410065 0.0132694
\(956\) 17.2722 0.558622
\(957\) −25.6608 −0.829495
\(958\) 2.74746 0.0887663
\(959\) −18.1397 −0.585760
\(960\) −0.0507177 −0.00163691
\(961\) −30.9987 −0.999958
\(962\) −2.27810 −0.0734488
\(963\) 40.2369 1.29662
\(964\) 25.5288 0.822229
\(965\) 0.0856257 0.00275639
\(966\) −17.5010 −0.563087
\(967\) −19.7814 −0.636126 −0.318063 0.948070i \(-0.603032\pi\)
−0.318063 + 0.948070i \(0.603032\pi\)
\(968\) −4.81170 −0.154654
\(969\) −16.9133 −0.543334
\(970\) 0.0467067 0.00149966
\(971\) −48.6621 −1.56164 −0.780821 0.624755i \(-0.785199\pi\)
−0.780821 + 0.624755i \(0.785199\pi\)
\(972\) 18.2888 0.586613
\(973\) 17.4754 0.560236
\(974\) −2.34055 −0.0749962
\(975\) 21.1133 0.676168
\(976\) −10.5034 −0.336205
\(977\) −42.1880 −1.34971 −0.674856 0.737949i \(-0.735794\pi\)
−0.674856 + 0.737949i \(0.735794\pi\)
\(978\) −35.6215 −1.13905
\(979\) 37.5764 1.20095
\(980\) −0.117907 −0.00376641
\(981\) 76.4197 2.43989
\(982\) 0.808608 0.0258037
\(983\) −34.4412 −1.09850 −0.549251 0.835657i \(-0.685087\pi\)
−0.549251 + 0.835657i \(0.685087\pi\)
\(984\) 5.53353 0.176403
\(985\) 0.0716523 0.00228303
\(986\) −18.1245 −0.577202
\(987\) 8.99819 0.286415
\(988\) 1.96816 0.0626155
\(989\) 51.1117 1.62526
\(990\) −0.208209 −0.00661732
\(991\) −40.3170 −1.28071 −0.640355 0.768079i \(-0.721213\pi\)
−0.640355 + 0.768079i \(0.721213\pi\)
\(992\) −0.0360307 −0.00114398
\(993\) 5.59602 0.177584
\(994\) −6.30420 −0.199957
\(995\) −0.0371526 −0.00117782
\(996\) −3.49621 −0.110782
\(997\) −6.90638 −0.218727 −0.109364 0.994002i \(-0.534881\pi\)
−0.109364 + 0.994002i \(0.534881\pi\)
\(998\) −4.23519 −0.134063
\(999\) 6.20436 0.196297
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 6046.2.a.f.1.6 67
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6046.2.a.f.1.6 67 1.1 even 1 trivial