Properties

Label 6422.2.a.q.1.3
Level $6422$
Weight $2$
Character 6422.1
Self dual yes
Analytic conductor $51.280$
Analytic rank $0$
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: \(0\)
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: no (minimal twist has level 494)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.24698\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +3.24698 q^{3} +1.00000 q^{4} -1.55496 q^{5} -3.24698 q^{6} +3.60388 q^{7} -1.00000 q^{8} +7.54288 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +3.24698 q^{3} +1.00000 q^{4} -1.55496 q^{5} -3.24698 q^{6} +3.60388 q^{7} -1.00000 q^{8} +7.54288 q^{9} +1.55496 q^{10} +5.18598 q^{11} +3.24698 q^{12} -3.60388 q^{14} -5.04892 q^{15} +1.00000 q^{16} -5.96077 q^{17} -7.54288 q^{18} +1.00000 q^{19} -1.55496 q^{20} +11.7017 q^{21} -5.18598 q^{22} +4.29590 q^{23} -3.24698 q^{24} -2.58211 q^{25} +14.7506 q^{27} +3.60388 q^{28} +2.91185 q^{29} +5.04892 q^{30} -6.04892 q^{31} -1.00000 q^{32} +16.8388 q^{33} +5.96077 q^{34} -5.60388 q^{35} +7.54288 q^{36} +6.71379 q^{37} -1.00000 q^{38} +1.55496 q^{40} +9.83877 q^{41} -11.7017 q^{42} +5.16421 q^{43} +5.18598 q^{44} -11.7289 q^{45} -4.29590 q^{46} -11.0315 q^{47} +3.24698 q^{48} +5.98792 q^{49} +2.58211 q^{50} -19.3545 q^{51} +3.82908 q^{53} -14.7506 q^{54} -8.06398 q^{55} -3.60388 q^{56} +3.24698 q^{57} -2.91185 q^{58} -13.5254 q^{59} -5.04892 q^{60} -0.615957 q^{61} +6.04892 q^{62} +27.1836 q^{63} +1.00000 q^{64} -16.8388 q^{66} +0.987918 q^{67} -5.96077 q^{68} +13.9487 q^{69} +5.60388 q^{70} +4.71917 q^{71} -7.54288 q^{72} -6.59179 q^{73} -6.71379 q^{74} -8.38404 q^{75} +1.00000 q^{76} +18.6896 q^{77} +10.7681 q^{79} -1.55496 q^{80} +25.2664 q^{81} -9.83877 q^{82} -3.91723 q^{83} +11.7017 q^{84} +9.26875 q^{85} -5.16421 q^{86} +9.45473 q^{87} -5.18598 q^{88} -0.789856 q^{89} +11.7289 q^{90} +4.29590 q^{92} -19.6407 q^{93} +11.0315 q^{94} -1.55496 q^{95} -3.24698 q^{96} +15.9976 q^{97} -5.98792 q^{98} +39.1172 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 5 q^{3} + 3 q^{4} - 5 q^{5} - 5 q^{6} + 2 q^{7} - 3 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 5 q^{3} + 3 q^{4} - 5 q^{5} - 5 q^{6} + 2 q^{7} - 3 q^{8} + 4 q^{9} + 5 q^{10} + q^{11} + 5 q^{12} - 2 q^{14} - 6 q^{15} + 3 q^{16} - 5 q^{17} - 4 q^{18} + 3 q^{19} - 5 q^{20} + 8 q^{21} - q^{22} - q^{23} - 5 q^{24} - 2 q^{25} + 8 q^{27} + 2 q^{28} + 5 q^{29} + 6 q^{30} - 9 q^{31} - 3 q^{32} + 18 q^{33} + 5 q^{34} - 8 q^{35} + 4 q^{36} + 12 q^{37} - 3 q^{38} + 5 q^{40} - 3 q^{41} - 8 q^{42} + 4 q^{43} + q^{44} - 2 q^{45} + q^{46} - 8 q^{47} + 5 q^{48} - q^{49} + 2 q^{50} - 13 q^{51} + q^{53} - 8 q^{54} + 10 q^{55} - 2 q^{56} + 5 q^{57} - 5 q^{58} - 6 q^{59} - 6 q^{60} - 12 q^{61} + 9 q^{62} + 26 q^{63} + 3 q^{64} - 18 q^{66} - 16 q^{67} - 5 q^{68} + 10 q^{69} + 8 q^{70} + 3 q^{71} - 4 q^{72} + 8 q^{73} - 12 q^{74} - 15 q^{75} + 3 q^{76} + 10 q^{77} + 12 q^{79} - 5 q^{80} + 27 q^{81} + 3 q^{82} - 5 q^{83} + 8 q^{84} + 20 q^{85} - 4 q^{86} + 6 q^{87} - q^{88} + 21 q^{89} + 2 q^{90} - q^{92} - 22 q^{93} + 8 q^{94} - 5 q^{95} - 5 q^{96} + 7 q^{97} + q^{98} + 55 q^{99}+O(q^{100}) \) 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
\(3\) 3.24698 1.87464 0.937322 0.348464i \(-0.113297\pi\)
0.937322 + 0.348464i \(0.113297\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.55496 −0.695398 −0.347699 0.937606i \(-0.613037\pi\)
−0.347699 + 0.937606i \(0.613037\pi\)
\(6\) −3.24698 −1.32557
\(7\) 3.60388 1.36214 0.681068 0.732220i \(-0.261516\pi\)
0.681068 + 0.732220i \(0.261516\pi\)
\(8\) −1.00000 −0.353553
\(9\) 7.54288 2.51429
\(10\) 1.55496 0.491721
\(11\) 5.18598 1.56363 0.781816 0.623509i \(-0.214294\pi\)
0.781816 + 0.623509i \(0.214294\pi\)
\(12\) 3.24698 0.937322
\(13\) 0 0
\(14\) −3.60388 −0.963176
\(15\) −5.04892 −1.30362
\(16\) 1.00000 0.250000
\(17\) −5.96077 −1.44570 −0.722850 0.691005i \(-0.757168\pi\)
−0.722850 + 0.691005i \(0.757168\pi\)
\(18\) −7.54288 −1.77787
\(19\) 1.00000 0.229416
\(20\) −1.55496 −0.347699
\(21\) 11.7017 2.55352
\(22\) −5.18598 −1.10565
\(23\) 4.29590 0.895756 0.447878 0.894095i \(-0.352180\pi\)
0.447878 + 0.894095i \(0.352180\pi\)
\(24\) −3.24698 −0.662787
\(25\) −2.58211 −0.516421
\(26\) 0 0
\(27\) 14.7506 2.83876
\(28\) 3.60388 0.681068
\(29\) 2.91185 0.540718 0.270359 0.962760i \(-0.412858\pi\)
0.270359 + 0.962760i \(0.412858\pi\)
\(30\) 5.04892 0.921802
\(31\) −6.04892 −1.08642 −0.543209 0.839598i \(-0.682791\pi\)
−0.543209 + 0.839598i \(0.682791\pi\)
\(32\) −1.00000 −0.176777
\(33\) 16.8388 2.93125
\(34\) 5.96077 1.02226
\(35\) −5.60388 −0.947228
\(36\) 7.54288 1.25715
\(37\) 6.71379 1.10374 0.551870 0.833930i \(-0.313914\pi\)
0.551870 + 0.833930i \(0.313914\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) 1.55496 0.245860
\(41\) 9.83877 1.53656 0.768279 0.640115i \(-0.221113\pi\)
0.768279 + 0.640115i \(0.221113\pi\)
\(42\) −11.7017 −1.80561
\(43\) 5.16421 0.787535 0.393767 0.919210i \(-0.371172\pi\)
0.393767 + 0.919210i \(0.371172\pi\)
\(44\) 5.18598 0.781816
\(45\) −11.7289 −1.74843
\(46\) −4.29590 −0.633395
\(47\) −11.0315 −1.60910 −0.804552 0.593882i \(-0.797594\pi\)
−0.804552 + 0.593882i \(0.797594\pi\)
\(48\) 3.24698 0.468661
\(49\) 5.98792 0.855417
\(50\) 2.58211 0.365165
\(51\) −19.3545 −2.71017
\(52\) 0 0
\(53\) 3.82908 0.525965 0.262983 0.964801i \(-0.415294\pi\)
0.262983 + 0.964801i \(0.415294\pi\)
\(54\) −14.7506 −2.00731
\(55\) −8.06398 −1.08735
\(56\) −3.60388 −0.481588
\(57\) 3.24698 0.430073
\(58\) −2.91185 −0.382345
\(59\) −13.5254 −1.76086 −0.880430 0.474177i \(-0.842746\pi\)
−0.880430 + 0.474177i \(0.842746\pi\)
\(60\) −5.04892 −0.651812
\(61\) −0.615957 −0.0788652 −0.0394326 0.999222i \(-0.512555\pi\)
−0.0394326 + 0.999222i \(0.512555\pi\)
\(62\) 6.04892 0.768213
\(63\) 27.1836 3.42481
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −16.8388 −2.07271
\(67\) 0.987918 0.120693 0.0603467 0.998177i \(-0.480779\pi\)
0.0603467 + 0.998177i \(0.480779\pi\)
\(68\) −5.96077 −0.722850
\(69\) 13.9487 1.67922
\(70\) 5.60388 0.669791
\(71\) 4.71917 0.560062 0.280031 0.959991i \(-0.409655\pi\)
0.280031 + 0.959991i \(0.409655\pi\)
\(72\) −7.54288 −0.888937
\(73\) −6.59179 −0.771511 −0.385756 0.922601i \(-0.626059\pi\)
−0.385756 + 0.922601i \(0.626059\pi\)
\(74\) −6.71379 −0.780462
\(75\) −8.38404 −0.968106
\(76\) 1.00000 0.114708
\(77\) 18.6896 2.12988
\(78\) 0 0
\(79\) 10.7681 1.21150 0.605752 0.795653i \(-0.292872\pi\)
0.605752 + 0.795653i \(0.292872\pi\)
\(80\) −1.55496 −0.173850
\(81\) 25.2664 2.80737
\(82\) −9.83877 −1.08651
\(83\) −3.91723 −0.429972 −0.214986 0.976617i \(-0.568971\pi\)
−0.214986 + 0.976617i \(0.568971\pi\)
\(84\) 11.7017 1.27676
\(85\) 9.26875 1.00534
\(86\) −5.16421 −0.556871
\(87\) 9.45473 1.01365
\(88\) −5.18598 −0.552827
\(89\) −0.789856 −0.0837246 −0.0418623 0.999123i \(-0.513329\pi\)
−0.0418623 + 0.999123i \(0.513329\pi\)
\(90\) 11.7289 1.23633
\(91\) 0 0
\(92\) 4.29590 0.447878
\(93\) −19.6407 −2.03665
\(94\) 11.0315 1.13781
\(95\) −1.55496 −0.159535
\(96\) −3.24698 −0.331393
\(97\) 15.9976 1.62431 0.812155 0.583441i \(-0.198294\pi\)
0.812155 + 0.583441i \(0.198294\pi\)
\(98\) −5.98792 −0.604871
\(99\) 39.1172 3.93143
\(100\) −2.58211 −0.258211
\(101\) 5.26205 0.523593 0.261797 0.965123i \(-0.415685\pi\)
0.261797 + 0.965123i \(0.415685\pi\)
\(102\) 19.3545 1.91638
\(103\) 0.396125 0.0390313 0.0195157 0.999810i \(-0.493788\pi\)
0.0195157 + 0.999810i \(0.493788\pi\)
\(104\) 0 0
\(105\) −18.1957 −1.77572
\(106\) −3.82908 −0.371914
\(107\) −0.488582 −0.0472330 −0.0236165 0.999721i \(-0.507518\pi\)
−0.0236165 + 0.999721i \(0.507518\pi\)
\(108\) 14.7506 1.41938
\(109\) −2.71379 −0.259934 −0.129967 0.991518i \(-0.541487\pi\)
−0.129967 + 0.991518i \(0.541487\pi\)
\(110\) 8.06398 0.768871
\(111\) 21.7995 2.06912
\(112\) 3.60388 0.340534
\(113\) −17.8780 −1.68182 −0.840910 0.541174i \(-0.817980\pi\)
−0.840910 + 0.541174i \(0.817980\pi\)
\(114\) −3.24698 −0.304108
\(115\) −6.67994 −0.622908
\(116\) 2.91185 0.270359
\(117\) 0 0
\(118\) 13.5254 1.24512
\(119\) −21.4819 −1.96924
\(120\) 5.04892 0.460901
\(121\) 15.8944 1.44495
\(122\) 0.615957 0.0557661
\(123\) 31.9463 2.88050
\(124\) −6.04892 −0.543209
\(125\) 11.7899 1.05452
\(126\) −27.1836 −2.42171
\(127\) −1.60388 −0.142321 −0.0711605 0.997465i \(-0.522670\pi\)
−0.0711605 + 0.997465i \(0.522670\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 16.7681 1.47635
\(130\) 0 0
\(131\) −19.3599 −1.69148 −0.845740 0.533595i \(-0.820841\pi\)
−0.845740 + 0.533595i \(0.820841\pi\)
\(132\) 16.8388 1.46563
\(133\) 3.60388 0.312496
\(134\) −0.987918 −0.0853432
\(135\) −22.9366 −1.97407
\(136\) 5.96077 0.511132
\(137\) 6.61596 0.565239 0.282620 0.959232i \(-0.408797\pi\)
0.282620 + 0.959232i \(0.408797\pi\)
\(138\) −13.9487 −1.18739
\(139\) 0.933624 0.0791890 0.0395945 0.999216i \(-0.487393\pi\)
0.0395945 + 0.999216i \(0.487393\pi\)
\(140\) −5.60388 −0.473614
\(141\) −35.8189 −3.01650
\(142\) −4.71917 −0.396024
\(143\) 0 0
\(144\) 7.54288 0.628573
\(145\) −4.52781 −0.376014
\(146\) 6.59179 0.545541
\(147\) 19.4426 1.60360
\(148\) 6.71379 0.551870
\(149\) 6.13706 0.502768 0.251384 0.967887i \(-0.419114\pi\)
0.251384 + 0.967887i \(0.419114\pi\)
\(150\) 8.38404 0.684554
\(151\) −11.2838 −0.918264 −0.459132 0.888368i \(-0.651839\pi\)
−0.459132 + 0.888368i \(0.651839\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −44.9614 −3.63491
\(154\) −18.6896 −1.50605
\(155\) 9.40581 0.755493
\(156\) 0 0
\(157\) −2.41550 −0.192778 −0.0963890 0.995344i \(-0.530729\pi\)
−0.0963890 + 0.995344i \(0.530729\pi\)
\(158\) −10.7681 −0.856663
\(159\) 12.4330 0.985998
\(160\) 1.55496 0.122930
\(161\) 15.4819 1.22014
\(162\) −25.2664 −1.98511
\(163\) 4.63102 0.362730 0.181365 0.983416i \(-0.441949\pi\)
0.181365 + 0.983416i \(0.441949\pi\)
\(164\) 9.83877 0.768279
\(165\) −26.1836 −2.03839
\(166\) 3.91723 0.304036
\(167\) 18.0344 1.39555 0.697774 0.716318i \(-0.254174\pi\)
0.697774 + 0.716318i \(0.254174\pi\)
\(168\) −11.7017 −0.902807
\(169\) 0 0
\(170\) −9.26875 −0.710881
\(171\) 7.54288 0.576818
\(172\) 5.16421 0.393767
\(173\) −22.2107 −1.68865 −0.844325 0.535831i \(-0.819999\pi\)
−0.844325 + 0.535831i \(0.819999\pi\)
\(174\) −9.45473 −0.716761
\(175\) −9.30559 −0.703436
\(176\) 5.18598 0.390908
\(177\) −43.9168 −3.30099
\(178\) 0.789856 0.0592022
\(179\) −22.0248 −1.64621 −0.823104 0.567891i \(-0.807759\pi\)
−0.823104 + 0.567891i \(0.807759\pi\)
\(180\) −11.7289 −0.874217
\(181\) 12.3134 0.915244 0.457622 0.889147i \(-0.348701\pi\)
0.457622 + 0.889147i \(0.348701\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) −4.29590 −0.316698
\(185\) −10.4397 −0.767539
\(186\) 19.6407 1.44013
\(187\) −30.9124 −2.26054
\(188\) −11.0315 −0.804552
\(189\) 53.1594 3.86678
\(190\) 1.55496 0.112809
\(191\) 9.21313 0.666639 0.333319 0.942814i \(-0.391831\pi\)
0.333319 + 0.942814i \(0.391831\pi\)
\(192\) 3.24698 0.234331
\(193\) −14.2784 −1.02778 −0.513892 0.857855i \(-0.671797\pi\)
−0.513892 + 0.857855i \(0.671797\pi\)
\(194\) −15.9976 −1.14856
\(195\) 0 0
\(196\) 5.98792 0.427708
\(197\) −11.7802 −0.839302 −0.419651 0.907685i \(-0.637848\pi\)
−0.419651 + 0.907685i \(0.637848\pi\)
\(198\) −39.1172 −2.77994
\(199\) 5.88040 0.416850 0.208425 0.978038i \(-0.433166\pi\)
0.208425 + 0.978038i \(0.433166\pi\)
\(200\) 2.58211 0.182582
\(201\) 3.20775 0.226257
\(202\) −5.26205 −0.370236
\(203\) 10.4940 0.736532
\(204\) −19.3545 −1.35509
\(205\) −15.2989 −1.06852
\(206\) −0.396125 −0.0275993
\(207\) 32.4034 2.25219
\(208\) 0 0
\(209\) 5.18598 0.358722
\(210\) 18.1957 1.25562
\(211\) −10.5894 −0.729004 −0.364502 0.931203i \(-0.618761\pi\)
−0.364502 + 0.931203i \(0.618761\pi\)
\(212\) 3.82908 0.262983
\(213\) 15.3230 1.04992
\(214\) 0.488582 0.0333988
\(215\) −8.03013 −0.547650
\(216\) −14.7506 −1.00365
\(217\) −21.7995 −1.47985
\(218\) 2.71379 0.183801
\(219\) −21.4034 −1.44631
\(220\) −8.06398 −0.543674
\(221\) 0 0
\(222\) −21.7995 −1.46309
\(223\) 1.04652 0.0700805 0.0350402 0.999386i \(-0.488844\pi\)
0.0350402 + 0.999386i \(0.488844\pi\)
\(224\) −3.60388 −0.240794
\(225\) −19.4765 −1.29843
\(226\) 17.8780 1.18923
\(227\) −7.10992 −0.471902 −0.235951 0.971765i \(-0.575820\pi\)
−0.235951 + 0.971765i \(0.575820\pi\)
\(228\) 3.24698 0.215036
\(229\) 16.6679 1.10144 0.550722 0.834689i \(-0.314353\pi\)
0.550722 + 0.834689i \(0.314353\pi\)
\(230\) 6.67994 0.440462
\(231\) 60.6848 3.99277
\(232\) −2.91185 −0.191173
\(233\) −10.5187 −0.689104 −0.344552 0.938767i \(-0.611969\pi\)
−0.344552 + 0.938767i \(0.611969\pi\)
\(234\) 0 0
\(235\) 17.1535 1.11897
\(236\) −13.5254 −0.880430
\(237\) 34.9638 2.27114
\(238\) 21.4819 1.39246
\(239\) −0.548253 −0.0354636 −0.0177318 0.999843i \(-0.505644\pi\)
−0.0177318 + 0.999843i \(0.505644\pi\)
\(240\) −5.04892 −0.325906
\(241\) −17.5743 −1.13206 −0.566031 0.824384i \(-0.691522\pi\)
−0.566031 + 0.824384i \(0.691522\pi\)
\(242\) −15.8944 −1.02173
\(243\) 37.7875 2.42407
\(244\) −0.615957 −0.0394326
\(245\) −9.31096 −0.594856
\(246\) −31.9463 −2.03682
\(247\) 0 0
\(248\) 6.04892 0.384107
\(249\) −12.7192 −0.806045
\(250\) −11.7899 −0.745656
\(251\) 4.09783 0.258653 0.129327 0.991602i \(-0.458718\pi\)
0.129327 + 0.991602i \(0.458718\pi\)
\(252\) 27.1836 1.71241
\(253\) 22.2784 1.40063
\(254\) 1.60388 0.100636
\(255\) 30.0954 1.88465
\(256\) 1.00000 0.0625000
\(257\) −6.65950 −0.415408 −0.207704 0.978192i \(-0.566599\pi\)
−0.207704 + 0.978192i \(0.566599\pi\)
\(258\) −16.7681 −1.04394
\(259\) 24.1957 1.50345
\(260\) 0 0
\(261\) 21.9638 1.35952
\(262\) 19.3599 1.19606
\(263\) 20.3763 1.25645 0.628227 0.778030i \(-0.283781\pi\)
0.628227 + 0.778030i \(0.283781\pi\)
\(264\) −16.8388 −1.03635
\(265\) −5.95407 −0.365755
\(266\) −3.60388 −0.220968
\(267\) −2.56465 −0.156954
\(268\) 0.987918 0.0603467
\(269\) −20.9638 −1.27818 −0.639091 0.769131i \(-0.720689\pi\)
−0.639091 + 0.769131i \(0.720689\pi\)
\(270\) 22.9366 1.39588
\(271\) −9.40821 −0.571508 −0.285754 0.958303i \(-0.592244\pi\)
−0.285754 + 0.958303i \(0.592244\pi\)
\(272\) −5.96077 −0.361425
\(273\) 0 0
\(274\) −6.61596 −0.399685
\(275\) −13.3907 −0.807492
\(276\) 13.9487 0.839612
\(277\) 22.0194 1.32302 0.661508 0.749938i \(-0.269917\pi\)
0.661508 + 0.749938i \(0.269917\pi\)
\(278\) −0.933624 −0.0559951
\(279\) −45.6262 −2.73157
\(280\) 5.60388 0.334896
\(281\) −24.4373 −1.45781 −0.728903 0.684617i \(-0.759969\pi\)
−0.728903 + 0.684617i \(0.759969\pi\)
\(282\) 35.8189 2.13299
\(283\) 15.9323 0.947077 0.473538 0.880773i \(-0.342977\pi\)
0.473538 + 0.880773i \(0.342977\pi\)
\(284\) 4.71917 0.280031
\(285\) −5.04892 −0.299072
\(286\) 0 0
\(287\) 35.4577 2.09300
\(288\) −7.54288 −0.444468
\(289\) 18.5308 1.09005
\(290\) 4.52781 0.265882
\(291\) 51.9439 3.04501
\(292\) −6.59179 −0.385756
\(293\) −15.5013 −0.905593 −0.452796 0.891614i \(-0.649574\pi\)
−0.452796 + 0.891614i \(0.649574\pi\)
\(294\) −19.4426 −1.13392
\(295\) 21.0315 1.22450
\(296\) −6.71379 −0.390231
\(297\) 76.4965 4.43878
\(298\) −6.13706 −0.355511
\(299\) 0 0
\(300\) −8.38404 −0.484053
\(301\) 18.6112 1.07273
\(302\) 11.2838 0.649311
\(303\) 17.0858 0.981551
\(304\) 1.00000 0.0573539
\(305\) 0.957787 0.0548427
\(306\) 44.9614 2.57027
\(307\) 10.0871 0.575700 0.287850 0.957675i \(-0.407060\pi\)
0.287850 + 0.957675i \(0.407060\pi\)
\(308\) 18.6896 1.06494
\(309\) 1.28621 0.0731698
\(310\) −9.40581 −0.534214
\(311\) −6.15644 −0.349100 −0.174550 0.984648i \(-0.555847\pi\)
−0.174550 + 0.984648i \(0.555847\pi\)
\(312\) 0 0
\(313\) 14.4101 0.814508 0.407254 0.913315i \(-0.366486\pi\)
0.407254 + 0.913315i \(0.366486\pi\)
\(314\) 2.41550 0.136315
\(315\) −42.2693 −2.38161
\(316\) 10.7681 0.605752
\(317\) 20.5241 1.15275 0.576374 0.817186i \(-0.304467\pi\)
0.576374 + 0.817186i \(0.304467\pi\)
\(318\) −12.4330 −0.697206
\(319\) 15.1008 0.845484
\(320\) −1.55496 −0.0869248
\(321\) −1.58642 −0.0885452
\(322\) −15.4819 −0.862771
\(323\) −5.96077 −0.331666
\(324\) 25.2664 1.40369
\(325\) 0 0
\(326\) −4.63102 −0.256489
\(327\) −8.81163 −0.487284
\(328\) −9.83877 −0.543255
\(329\) −39.7560 −2.19182
\(330\) 26.1836 1.44136
\(331\) 22.4155 1.23207 0.616034 0.787720i \(-0.288739\pi\)
0.616034 + 0.787720i \(0.288739\pi\)
\(332\) −3.91723 −0.214986
\(333\) 50.6413 2.77513
\(334\) −18.0344 −0.986801
\(335\) −1.53617 −0.0839300
\(336\) 11.7017 0.638381
\(337\) 25.1293 1.36888 0.684440 0.729069i \(-0.260047\pi\)
0.684440 + 0.729069i \(0.260047\pi\)
\(338\) 0 0
\(339\) −58.0495 −3.15282
\(340\) 9.26875 0.502669
\(341\) −31.3696 −1.69876
\(342\) −7.54288 −0.407872
\(343\) −3.64742 −0.196942
\(344\) −5.16421 −0.278436
\(345\) −21.6896 −1.16773
\(346\) 22.2107 1.19406
\(347\) −20.3370 −1.09175 −0.545875 0.837867i \(-0.683803\pi\)
−0.545875 + 0.837867i \(0.683803\pi\)
\(348\) 9.45473 0.506827
\(349\) −7.12498 −0.381392 −0.190696 0.981649i \(-0.561074\pi\)
−0.190696 + 0.981649i \(0.561074\pi\)
\(350\) 9.30559 0.497404
\(351\) 0 0
\(352\) −5.18598 −0.276414
\(353\) −22.4155 −1.19306 −0.596528 0.802592i \(-0.703454\pi\)
−0.596528 + 0.802592i \(0.703454\pi\)
\(354\) 43.9168 2.33415
\(355\) −7.33811 −0.389466
\(356\) −0.789856 −0.0418623
\(357\) −69.7512 −3.69163
\(358\) 22.0248 1.16404
\(359\) 6.29829 0.332411 0.166206 0.986091i \(-0.446848\pi\)
0.166206 + 0.986091i \(0.446848\pi\)
\(360\) 11.7289 0.618165
\(361\) 1.00000 0.0526316
\(362\) −12.3134 −0.647176
\(363\) 51.6088 2.70876
\(364\) 0 0
\(365\) 10.2500 0.536508
\(366\) 2.00000 0.104542
\(367\) 1.90754 0.0995729 0.0497864 0.998760i \(-0.484146\pi\)
0.0497864 + 0.998760i \(0.484146\pi\)
\(368\) 4.29590 0.223939
\(369\) 74.2127 3.86336
\(370\) 10.4397 0.542732
\(371\) 13.7995 0.716437
\(372\) −19.6407 −1.01832
\(373\) 9.57540 0.495795 0.247898 0.968786i \(-0.420260\pi\)
0.247898 + 0.968786i \(0.420260\pi\)
\(374\) 30.9124 1.59844
\(375\) 38.2814 1.97684
\(376\) 11.0315 0.568904
\(377\) 0 0
\(378\) −53.1594 −2.73423
\(379\) −11.1400 −0.572226 −0.286113 0.958196i \(-0.592363\pi\)
−0.286113 + 0.958196i \(0.592363\pi\)
\(380\) −1.55496 −0.0797677
\(381\) −5.20775 −0.266801
\(382\) −9.21313 −0.471385
\(383\) 14.0489 0.717866 0.358933 0.933363i \(-0.383141\pi\)
0.358933 + 0.933363i \(0.383141\pi\)
\(384\) −3.24698 −0.165697
\(385\) −29.0616 −1.48112
\(386\) 14.2784 0.726753
\(387\) 38.9530 1.98009
\(388\) 15.9976 0.812155
\(389\) 9.95646 0.504813 0.252406 0.967621i \(-0.418778\pi\)
0.252406 + 0.967621i \(0.418778\pi\)
\(390\) 0 0
\(391\) −25.6069 −1.29499
\(392\) −5.98792 −0.302436
\(393\) −62.8611 −3.17092
\(394\) 11.7802 0.593476
\(395\) −16.7439 −0.842478
\(396\) 39.1172 1.96571
\(397\) −16.0683 −0.806445 −0.403222 0.915102i \(-0.632110\pi\)
−0.403222 + 0.915102i \(0.632110\pi\)
\(398\) −5.88040 −0.294758
\(399\) 11.7017 0.585818
\(400\) −2.58211 −0.129105
\(401\) −4.59717 −0.229572 −0.114786 0.993390i \(-0.536618\pi\)
−0.114786 + 0.993390i \(0.536618\pi\)
\(402\) −3.20775 −0.159988
\(403\) 0 0
\(404\) 5.26205 0.261797
\(405\) −39.2881 −1.95224
\(406\) −10.4940 −0.520806
\(407\) 34.8176 1.72584
\(408\) 19.3545 0.958191
\(409\) 1.56033 0.0771536 0.0385768 0.999256i \(-0.487718\pi\)
0.0385768 + 0.999256i \(0.487718\pi\)
\(410\) 15.2989 0.755558
\(411\) 21.4819 1.05962
\(412\) 0.396125 0.0195157
\(413\) −48.7439 −2.39853
\(414\) −32.4034 −1.59254
\(415\) 6.09113 0.299002
\(416\) 0 0
\(417\) 3.03146 0.148451
\(418\) −5.18598 −0.253655
\(419\) 12.4590 0.608664 0.304332 0.952566i \(-0.401567\pi\)
0.304332 + 0.952566i \(0.401567\pi\)
\(420\) −18.1957 −0.887858
\(421\) −4.71379 −0.229736 −0.114868 0.993381i \(-0.536645\pi\)
−0.114868 + 0.993381i \(0.536645\pi\)
\(422\) 10.5894 0.515484
\(423\) −83.2089 −4.04576
\(424\) −3.82908 −0.185957
\(425\) 15.3913 0.746590
\(426\) −15.3230 −0.742404
\(427\) −2.21983 −0.107425
\(428\) −0.488582 −0.0236165
\(429\) 0 0
\(430\) 8.03013 0.387247
\(431\) −15.0707 −0.725929 −0.362965 0.931803i \(-0.618235\pi\)
−0.362965 + 0.931803i \(0.618235\pi\)
\(432\) 14.7506 0.709690
\(433\) 18.4263 0.885509 0.442755 0.896643i \(-0.354001\pi\)
0.442755 + 0.896643i \(0.354001\pi\)
\(434\) 21.7995 1.04641
\(435\) −14.7017 −0.704893
\(436\) −2.71379 −0.129967
\(437\) 4.29590 0.205501
\(438\) 21.4034 1.02269
\(439\) −38.1909 −1.82275 −0.911376 0.411575i \(-0.864979\pi\)
−0.911376 + 0.411575i \(0.864979\pi\)
\(440\) 8.06398 0.384435
\(441\) 45.1661 2.15077
\(442\) 0 0
\(443\) −31.3599 −1.48995 −0.744976 0.667091i \(-0.767539\pi\)
−0.744976 + 0.667091i \(0.767539\pi\)
\(444\) 21.7995 1.03456
\(445\) 1.22819 0.0582219
\(446\) −1.04652 −0.0495544
\(447\) 19.9269 0.942511
\(448\) 3.60388 0.170267
\(449\) 15.5104 0.731979 0.365989 0.930619i \(-0.380731\pi\)
0.365989 + 0.930619i \(0.380731\pi\)
\(450\) 19.4765 0.918131
\(451\) 51.0237 2.40261
\(452\) −17.8780 −0.840910
\(453\) −36.6383 −1.72142
\(454\) 7.10992 0.333685
\(455\) 0 0
\(456\) −3.24698 −0.152054
\(457\) 33.8538 1.58362 0.791808 0.610770i \(-0.209140\pi\)
0.791808 + 0.610770i \(0.209140\pi\)
\(458\) −16.6679 −0.778838
\(459\) −87.9251 −4.10399
\(460\) −6.67994 −0.311454
\(461\) −16.9989 −0.791719 −0.395860 0.918311i \(-0.629553\pi\)
−0.395860 + 0.918311i \(0.629553\pi\)
\(462\) −60.6848 −2.82331
\(463\) 33.5254 1.55806 0.779029 0.626988i \(-0.215712\pi\)
0.779029 + 0.626988i \(0.215712\pi\)
\(464\) 2.91185 0.135179
\(465\) 30.5405 1.41628
\(466\) 10.5187 0.487270
\(467\) −24.8659 −1.15066 −0.575329 0.817922i \(-0.695126\pi\)
−0.575329 + 0.817922i \(0.695126\pi\)
\(468\) 0 0
\(469\) 3.56033 0.164401
\(470\) −17.1535 −0.791230
\(471\) −7.84309 −0.361390
\(472\) 13.5254 0.622558
\(473\) 26.7815 1.23141
\(474\) −34.9638 −1.60594
\(475\) −2.58211 −0.118475
\(476\) −21.4819 −0.984620
\(477\) 28.8823 1.32243
\(478\) 0.548253 0.0250765
\(479\) 43.5120 1.98811 0.994057 0.108859i \(-0.0347197\pi\)
0.994057 + 0.108859i \(0.0347197\pi\)
\(480\) 5.04892 0.230450
\(481\) 0 0
\(482\) 17.5743 0.800489
\(483\) 50.2693 2.28733
\(484\) 15.8944 0.722473
\(485\) −24.8756 −1.12954
\(486\) −37.7875 −1.71407
\(487\) 9.41657 0.426705 0.213353 0.976975i \(-0.431562\pi\)
0.213353 + 0.976975i \(0.431562\pi\)
\(488\) 0.615957 0.0278831
\(489\) 15.0368 0.679989
\(490\) 9.31096 0.420626
\(491\) 1.67755 0.0757066 0.0378533 0.999283i \(-0.487948\pi\)
0.0378533 + 0.999283i \(0.487948\pi\)
\(492\) 31.9463 1.44025
\(493\) −17.3569 −0.781715
\(494\) 0 0
\(495\) −60.8256 −2.73391
\(496\) −6.04892 −0.271604
\(497\) 17.0073 0.762881
\(498\) 12.7192 0.569960
\(499\) −39.4077 −1.76413 −0.882066 0.471126i \(-0.843848\pi\)
−0.882066 + 0.471126i \(0.843848\pi\)
\(500\) 11.7899 0.527258
\(501\) 58.5575 2.61615
\(502\) −4.09783 −0.182895
\(503\) −1.86725 −0.0832565 −0.0416282 0.999133i \(-0.513255\pi\)
−0.0416282 + 0.999133i \(0.513255\pi\)
\(504\) −27.1836 −1.21085
\(505\) −8.18226 −0.364106
\(506\) −22.2784 −0.990397
\(507\) 0 0
\(508\) −1.60388 −0.0711605
\(509\) 20.8853 0.925725 0.462862 0.886430i \(-0.346822\pi\)
0.462862 + 0.886430i \(0.346822\pi\)
\(510\) −30.0954 −1.33265
\(511\) −23.7560 −1.05090
\(512\) −1.00000 −0.0441942
\(513\) 14.7506 0.651256
\(514\) 6.65950 0.293738
\(515\) −0.615957 −0.0271423
\(516\) 16.7681 0.738174
\(517\) −57.2089 −2.51605
\(518\) −24.1957 −1.06310
\(519\) −72.1178 −3.16562
\(520\) 0 0
\(521\) −6.01075 −0.263336 −0.131668 0.991294i \(-0.542033\pi\)
−0.131668 + 0.991294i \(0.542033\pi\)
\(522\) −21.9638 −0.961327
\(523\) 25.2325 1.10334 0.551670 0.834062i \(-0.313991\pi\)
0.551670 + 0.834062i \(0.313991\pi\)
\(524\) −19.3599 −0.845740
\(525\) −30.2150 −1.31869
\(526\) −20.3763 −0.888448
\(527\) 36.0562 1.57063
\(528\) 16.8388 0.732814
\(529\) −4.54527 −0.197620
\(530\) 5.95407 0.258628
\(531\) −102.021 −4.42732
\(532\) 3.60388 0.156248
\(533\) 0 0
\(534\) 2.56465 0.110983
\(535\) 0.759725 0.0328458
\(536\) −0.987918 −0.0426716
\(537\) −71.5139 −3.08605
\(538\) 20.9638 0.903812
\(539\) 31.0532 1.33756
\(540\) −22.9366 −0.987034
\(541\) −15.3459 −0.659771 −0.329885 0.944021i \(-0.607010\pi\)
−0.329885 + 0.944021i \(0.607010\pi\)
\(542\) 9.40821 0.404117
\(543\) 39.9812 1.71576
\(544\) 5.96077 0.255566
\(545\) 4.21983 0.180758
\(546\) 0 0
\(547\) −1.97525 −0.0844554 −0.0422277 0.999108i \(-0.513445\pi\)
−0.0422277 + 0.999108i \(0.513445\pi\)
\(548\) 6.61596 0.282620
\(549\) −4.64609 −0.198290
\(550\) 13.3907 0.570983
\(551\) 2.91185 0.124049
\(552\) −13.9487 −0.593696
\(553\) 38.8068 1.65023
\(554\) −22.0194 −0.935514
\(555\) −33.8974 −1.43886
\(556\) 0.933624 0.0395945
\(557\) 22.6920 0.961492 0.480746 0.876860i \(-0.340366\pi\)
0.480746 + 0.876860i \(0.340366\pi\)
\(558\) 45.6262 1.93151
\(559\) 0 0
\(560\) −5.60388 −0.236807
\(561\) −100.372 −4.23771
\(562\) 24.4373 1.03082
\(563\) 22.4373 0.945618 0.472809 0.881165i \(-0.343240\pi\)
0.472809 + 0.881165i \(0.343240\pi\)
\(564\) −35.8189 −1.50825
\(565\) 27.7995 1.16954
\(566\) −15.9323 −0.669684
\(567\) 91.0568 3.82403
\(568\) −4.71917 −0.198012
\(569\) 23.9711 1.00492 0.502459 0.864601i \(-0.332429\pi\)
0.502459 + 0.864601i \(0.332429\pi\)
\(570\) 5.04892 0.211476
\(571\) −9.84787 −0.412121 −0.206060 0.978539i \(-0.566064\pi\)
−0.206060 + 0.978539i \(0.566064\pi\)
\(572\) 0 0
\(573\) 29.9148 1.24971
\(574\) −35.4577 −1.47998
\(575\) −11.0925 −0.462587
\(576\) 7.54288 0.314287
\(577\) 25.2922 1.05293 0.526464 0.850198i \(-0.323518\pi\)
0.526464 + 0.850198i \(0.323518\pi\)
\(578\) −18.5308 −0.770779
\(579\) −46.3618 −1.92673
\(580\) −4.52781 −0.188007
\(581\) −14.1172 −0.585681
\(582\) −51.9439 −2.15314
\(583\) 19.8576 0.822416
\(584\) 6.59179 0.272770
\(585\) 0 0
\(586\) 15.5013 0.640351
\(587\) −41.2922 −1.70431 −0.852155 0.523289i \(-0.824705\pi\)
−0.852155 + 0.523289i \(0.824705\pi\)
\(588\) 19.4426 0.801801
\(589\) −6.04892 −0.249241
\(590\) −21.0315 −0.865851
\(591\) −38.2500 −1.57339
\(592\) 6.71379 0.275935
\(593\) 13.8043 0.566876 0.283438 0.958991i \(-0.408525\pi\)
0.283438 + 0.958991i \(0.408525\pi\)
\(594\) −76.4965 −3.13869
\(595\) 33.4034 1.36941
\(596\) 6.13706 0.251384
\(597\) 19.0935 0.781446
\(598\) 0 0
\(599\) 9.84309 0.402178 0.201089 0.979573i \(-0.435552\pi\)
0.201089 + 0.979573i \(0.435552\pi\)
\(600\) 8.38404 0.342277
\(601\) 14.2198 0.580039 0.290020 0.957021i \(-0.406338\pi\)
0.290020 + 0.957021i \(0.406338\pi\)
\(602\) −18.6112 −0.758535
\(603\) 7.45175 0.303459
\(604\) −11.2838 −0.459132
\(605\) −24.7151 −1.00481
\(606\) −17.0858 −0.694061
\(607\) 19.3056 0.783590 0.391795 0.920053i \(-0.371854\pi\)
0.391795 + 0.920053i \(0.371854\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 34.0737 1.38073
\(610\) −0.957787 −0.0387797
\(611\) 0 0
\(612\) −44.9614 −1.81746
\(613\) −36.6058 −1.47849 −0.739247 0.673434i \(-0.764818\pi\)
−0.739247 + 0.673434i \(0.764818\pi\)
\(614\) −10.0871 −0.407081
\(615\) −49.6752 −2.00310
\(616\) −18.6896 −0.753027
\(617\) −6.14138 −0.247242 −0.123621 0.992329i \(-0.539451\pi\)
−0.123621 + 0.992329i \(0.539451\pi\)
\(618\) −1.28621 −0.0517389
\(619\) −10.5080 −0.422351 −0.211175 0.977448i \(-0.567729\pi\)
−0.211175 + 0.977448i \(0.567729\pi\)
\(620\) 9.40581 0.377747
\(621\) 63.3672 2.54284
\(622\) 6.15644 0.246851
\(623\) −2.84654 −0.114044
\(624\) 0 0
\(625\) −5.42221 −0.216888
\(626\) −14.4101 −0.575944
\(627\) 16.8388 0.672476
\(628\) −2.41550 −0.0963890
\(629\) −40.0194 −1.59568
\(630\) 42.2693 1.68405
\(631\) −38.7090 −1.54098 −0.770491 0.637451i \(-0.779989\pi\)
−0.770491 + 0.637451i \(0.779989\pi\)
\(632\) −10.7681 −0.428331
\(633\) −34.3836 −1.36662
\(634\) −20.5241 −0.815116
\(635\) 2.49396 0.0989698
\(636\) 12.4330 0.492999
\(637\) 0 0
\(638\) −15.1008 −0.597847
\(639\) 35.5961 1.40816
\(640\) 1.55496 0.0614651
\(641\) −46.7767 −1.84757 −0.923784 0.382913i \(-0.874921\pi\)
−0.923784 + 0.382913i \(0.874921\pi\)
\(642\) 1.58642 0.0626109
\(643\) 30.6853 1.21011 0.605055 0.796183i \(-0.293151\pi\)
0.605055 + 0.796183i \(0.293151\pi\)
\(644\) 15.4819 0.610071
\(645\) −26.0737 −1.02665
\(646\) 5.96077 0.234523
\(647\) −43.3846 −1.70563 −0.852813 0.522216i \(-0.825105\pi\)
−0.852813 + 0.522216i \(0.825105\pi\)
\(648\) −25.2664 −0.992556
\(649\) −70.1426 −2.75334
\(650\) 0 0
\(651\) −70.7827 −2.77419
\(652\) 4.63102 0.181365
\(653\) −24.6025 −0.962772 −0.481386 0.876509i \(-0.659866\pi\)
−0.481386 + 0.876509i \(0.659866\pi\)
\(654\) 8.81163 0.344562
\(655\) 30.1038 1.17625
\(656\) 9.83877 0.384140
\(657\) −49.7211 −1.93980
\(658\) 39.7560 1.54985
\(659\) 25.3013 0.985598 0.492799 0.870143i \(-0.335974\pi\)
0.492799 + 0.870143i \(0.335974\pi\)
\(660\) −26.1836 −1.01919
\(661\) −20.2983 −0.789512 −0.394756 0.918786i \(-0.629171\pi\)
−0.394756 + 0.918786i \(0.629171\pi\)
\(662\) −22.4155 −0.871203
\(663\) 0 0
\(664\) 3.91723 0.152018
\(665\) −5.60388 −0.217309
\(666\) −50.6413 −1.96231
\(667\) 12.5090 0.484351
\(668\) 18.0344 0.697774
\(669\) 3.39804 0.131376
\(670\) 1.53617 0.0593475
\(671\) −3.19434 −0.123316
\(672\) −11.7017 −0.451403
\(673\) 10.8552 0.418436 0.209218 0.977869i \(-0.432908\pi\)
0.209218 + 0.977869i \(0.432908\pi\)
\(674\) −25.1293 −0.967944
\(675\) −38.0877 −1.46600
\(676\) 0 0
\(677\) 27.0204 1.03848 0.519240 0.854628i \(-0.326215\pi\)
0.519240 + 0.854628i \(0.326215\pi\)
\(678\) 58.0495 2.22938
\(679\) 57.6534 2.21253
\(680\) −9.26875 −0.355440
\(681\) −23.0858 −0.884648
\(682\) 31.3696 1.20120
\(683\) −26.8659 −1.02800 −0.513998 0.857791i \(-0.671836\pi\)
−0.513998 + 0.857791i \(0.671836\pi\)
\(684\) 7.54288 0.288409
\(685\) −10.2875 −0.393067
\(686\) 3.64742 0.139259
\(687\) 54.1202 2.06481
\(688\) 5.16421 0.196884
\(689\) 0 0
\(690\) 21.6896 0.825710
\(691\) −20.9855 −0.798327 −0.399164 0.916880i \(-0.630699\pi\)
−0.399164 + 0.916880i \(0.630699\pi\)
\(692\) −22.2107 −0.844325
\(693\) 140.974 5.35514
\(694\) 20.3370 0.771984
\(695\) −1.45175 −0.0550679
\(696\) −9.45473 −0.358381
\(697\) −58.6467 −2.22140
\(698\) 7.12498 0.269685
\(699\) −34.1540 −1.29182
\(700\) −9.30559 −0.351718
\(701\) −29.2922 −1.10635 −0.553175 0.833065i \(-0.686584\pi\)
−0.553175 + 0.833065i \(0.686584\pi\)
\(702\) 0 0
\(703\) 6.71379 0.253215
\(704\) 5.18598 0.195454
\(705\) 55.6969 2.09767
\(706\) 22.4155 0.843619
\(707\) 18.9638 0.713205
\(708\) −43.9168 −1.65049
\(709\) 11.9758 0.449762 0.224881 0.974386i \(-0.427801\pi\)
0.224881 + 0.974386i \(0.427801\pi\)
\(710\) 7.33811 0.275394
\(711\) 81.2223 3.04608
\(712\) 0.789856 0.0296011
\(713\) −25.9855 −0.973166
\(714\) 69.7512 2.61037
\(715\) 0 0
\(716\) −22.0248 −0.823104
\(717\) −1.78017 −0.0664816
\(718\) −6.29829 −0.235050
\(719\) 47.0025 1.75290 0.876449 0.481495i \(-0.159906\pi\)
0.876449 + 0.481495i \(0.159906\pi\)
\(720\) −11.7289 −0.437109
\(721\) 1.42758 0.0531660
\(722\) −1.00000 −0.0372161
\(723\) −57.0635 −2.12221
\(724\) 12.3134 0.457622
\(725\) −7.51871 −0.279238
\(726\) −51.6088 −1.91538
\(727\) −10.7832 −0.399925 −0.199962 0.979804i \(-0.564082\pi\)
−0.199962 + 0.979804i \(0.564082\pi\)
\(728\) 0 0
\(729\) 46.8961 1.73689
\(730\) −10.2500 −0.379368
\(731\) −30.7827 −1.13854
\(732\) −2.00000 −0.0739221
\(733\) 16.9554 0.626262 0.313131 0.949710i \(-0.398622\pi\)
0.313131 + 0.949710i \(0.398622\pi\)
\(734\) −1.90754 −0.0704087
\(735\) −30.2325 −1.11514
\(736\) −4.29590 −0.158349
\(737\) 5.12333 0.188720
\(738\) −74.2127 −2.73181
\(739\) 25.3817 0.933679 0.466840 0.884342i \(-0.345393\pi\)
0.466840 + 0.884342i \(0.345393\pi\)
\(740\) −10.4397 −0.383770
\(741\) 0 0
\(742\) −13.7995 −0.506597
\(743\) 8.47517 0.310924 0.155462 0.987842i \(-0.450313\pi\)
0.155462 + 0.987842i \(0.450313\pi\)
\(744\) 19.6407 0.720063
\(745\) −9.54288 −0.349624
\(746\) −9.57540 −0.350580
\(747\) −29.5472 −1.08108
\(748\) −30.9124 −1.13027
\(749\) −1.76079 −0.0643379
\(750\) −38.2814 −1.39784
\(751\) 35.0315 1.27832 0.639158 0.769075i \(-0.279283\pi\)
0.639158 + 0.769075i \(0.279283\pi\)
\(752\) −11.0315 −0.402276
\(753\) 13.3056 0.484882
\(754\) 0 0
\(755\) 17.5459 0.638559
\(756\) 53.1594 1.93339
\(757\) −37.0374 −1.34615 −0.673074 0.739575i \(-0.735026\pi\)
−0.673074 + 0.739575i \(0.735026\pi\)
\(758\) 11.1400 0.404625
\(759\) 72.3376 2.62569
\(760\) 1.55496 0.0564043
\(761\) 12.4155 0.450062 0.225031 0.974352i \(-0.427752\pi\)
0.225031 + 0.974352i \(0.427752\pi\)
\(762\) 5.20775 0.188657
\(763\) −9.78017 −0.354066
\(764\) 9.21313 0.333319
\(765\) 69.9130 2.52771
\(766\) −14.0489 −0.507608
\(767\) 0 0
\(768\) 3.24698 0.117165
\(769\) −31.8103 −1.14711 −0.573554 0.819168i \(-0.694436\pi\)
−0.573554 + 0.819168i \(0.694436\pi\)
\(770\) 29.0616 1.04731
\(771\) −21.6233 −0.778742
\(772\) −14.2784 −0.513892
\(773\) 9.54480 0.343302 0.171651 0.985158i \(-0.445090\pi\)
0.171651 + 0.985158i \(0.445090\pi\)
\(774\) −38.9530 −1.40014
\(775\) 15.6189 0.561049
\(776\) −15.9976 −0.574281
\(777\) 78.5628 2.81843
\(778\) −9.95646 −0.356956
\(779\) 9.83877 0.352511
\(780\) 0 0
\(781\) 24.4735 0.875731
\(782\) 25.6069 0.915699
\(783\) 42.9517 1.53497
\(784\) 5.98792 0.213854
\(785\) 3.75600 0.134058
\(786\) 62.8611 2.24218
\(787\) 38.5763 1.37509 0.687547 0.726139i \(-0.258687\pi\)
0.687547 + 0.726139i \(0.258687\pi\)
\(788\) −11.7802 −0.419651
\(789\) 66.1613 2.35541
\(790\) 16.7439 0.595722
\(791\) −64.4301 −2.29087
\(792\) −39.1172 −1.38997
\(793\) 0 0
\(794\) 16.0683 0.570242
\(795\) −19.3327 −0.685661
\(796\) 5.88040 0.208425
\(797\) 42.2717 1.49734 0.748671 0.662942i \(-0.230692\pi\)
0.748671 + 0.662942i \(0.230692\pi\)
\(798\) −11.7017 −0.414236
\(799\) 65.7560 2.32628
\(800\) 2.58211 0.0912912
\(801\) −5.95779 −0.210508
\(802\) 4.59717 0.162332
\(803\) −34.1849 −1.20636
\(804\) 3.20775 0.113129
\(805\) −24.0737 −0.848485
\(806\) 0 0
\(807\) −68.0689 −2.39614
\(808\) −5.26205 −0.185118
\(809\) −13.2537 −0.465975 −0.232987 0.972480i \(-0.574850\pi\)
−0.232987 + 0.972480i \(0.574850\pi\)
\(810\) 39.2881 1.38044
\(811\) 8.98313 0.315440 0.157720 0.987484i \(-0.449586\pi\)
0.157720 + 0.987484i \(0.449586\pi\)
\(812\) 10.4940 0.368266
\(813\) −30.5483 −1.07137
\(814\) −34.8176 −1.22036
\(815\) −7.20105 −0.252242
\(816\) −19.3545 −0.677543
\(817\) 5.16421 0.180673
\(818\) −1.56033 −0.0545558
\(819\) 0 0
\(820\) −15.2989 −0.534260
\(821\) 21.9302 0.765368 0.382684 0.923879i \(-0.375000\pi\)
0.382684 + 0.923879i \(0.375000\pi\)
\(822\) −21.4819 −0.749267
\(823\) −12.0567 −0.420270 −0.210135 0.977672i \(-0.567390\pi\)
−0.210135 + 0.977672i \(0.567390\pi\)
\(824\) −0.396125 −0.0137997
\(825\) −43.4795 −1.51376
\(826\) 48.7439 1.69602
\(827\) 3.49529 0.121543 0.0607715 0.998152i \(-0.480644\pi\)
0.0607715 + 0.998152i \(0.480644\pi\)
\(828\) 32.4034 1.12610
\(829\) 33.8532 1.17577 0.587886 0.808944i \(-0.299960\pi\)
0.587886 + 0.808944i \(0.299960\pi\)
\(830\) −6.09113 −0.211426
\(831\) 71.4965 2.48019
\(832\) 0 0
\(833\) −35.6926 −1.23668
\(834\) −3.03146 −0.104971
\(835\) −28.0428 −0.970461
\(836\) 5.18598 0.179361
\(837\) −89.2253 −3.08408
\(838\) −12.4590 −0.430390
\(839\) −38.2959 −1.32212 −0.661061 0.750333i \(-0.729893\pi\)
−0.661061 + 0.750333i \(0.729893\pi\)
\(840\) 18.1957 0.627810
\(841\) −20.5211 −0.707624
\(842\) 4.71379 0.162448
\(843\) −79.3473 −2.73287
\(844\) −10.5894 −0.364502
\(845\) 0 0
\(846\) 83.2089 2.86078
\(847\) 57.2814 1.96821
\(848\) 3.82908 0.131491
\(849\) 51.7318 1.77543
\(850\) −15.3913 −0.527919
\(851\) 28.8418 0.988683
\(852\) 15.3230 0.524959
\(853\) 9.33619 0.319665 0.159833 0.987144i \(-0.448905\pi\)
0.159833 + 0.987144i \(0.448905\pi\)
\(854\) 2.21983 0.0759611
\(855\) −11.7289 −0.401118
\(856\) 0.488582 0.0166994
\(857\) −47.7405 −1.63078 −0.815392 0.578910i \(-0.803478\pi\)
−0.815392 + 0.578910i \(0.803478\pi\)
\(858\) 0 0
\(859\) 10.3913 0.354548 0.177274 0.984162i \(-0.443272\pi\)
0.177274 + 0.984162i \(0.443272\pi\)
\(860\) −8.03013 −0.273825
\(861\) 115.130 3.92364
\(862\) 15.0707 0.513310
\(863\) 19.0398 0.648123 0.324061 0.946036i \(-0.394952\pi\)
0.324061 + 0.946036i \(0.394952\pi\)
\(864\) −14.7506 −0.501827
\(865\) 34.5368 1.17429
\(866\) −18.4263 −0.626150
\(867\) 60.1691 2.04345
\(868\) −21.7995 −0.739925
\(869\) 55.8431 1.89435
\(870\) 14.7017 0.498435
\(871\) 0 0
\(872\) 2.71379 0.0919006
\(873\) 120.668 4.08399
\(874\) −4.29590 −0.145311
\(875\) 42.4892 1.43640
\(876\) −21.4034 −0.723155
\(877\) −35.8684 −1.21119 −0.605595 0.795773i \(-0.707065\pi\)
−0.605595 + 0.795773i \(0.707065\pi\)
\(878\) 38.1909 1.28888
\(879\) −50.3323 −1.69766
\(880\) −8.06398 −0.271837
\(881\) −37.3631 −1.25880 −0.629398 0.777083i \(-0.716698\pi\)
−0.629398 + 0.777083i \(0.716698\pi\)
\(882\) −45.1661 −1.52082
\(883\) −19.6823 −0.662363 −0.331182 0.943567i \(-0.607447\pi\)
−0.331182 + 0.943567i \(0.607447\pi\)
\(884\) 0 0
\(885\) 68.2887 2.29550
\(886\) 31.3599 1.05356
\(887\) 35.6534 1.19712 0.598562 0.801077i \(-0.295739\pi\)
0.598562 + 0.801077i \(0.295739\pi\)
\(888\) −21.7995 −0.731545
\(889\) −5.78017 −0.193861
\(890\) −1.22819 −0.0411691
\(891\) 131.031 4.38970
\(892\) 1.04652 0.0350402
\(893\) −11.0315 −0.369154
\(894\) −19.9269 −0.666456
\(895\) 34.2476 1.14477
\(896\) −3.60388 −0.120397
\(897\) 0 0
\(898\) −15.5104 −0.517587
\(899\) −17.6136 −0.587445
\(900\) −19.4765 −0.649217
\(901\) −22.8243 −0.760388
\(902\) −51.0237 −1.69890
\(903\) 60.4301 2.01099
\(904\) 17.8780 0.594614
\(905\) −19.1468 −0.636460
\(906\) 36.6383 1.21723
\(907\) −16.5757 −0.550386 −0.275193 0.961389i \(-0.588742\pi\)
−0.275193 + 0.961389i \(0.588742\pi\)
\(908\) −7.10992 −0.235951
\(909\) 39.6910 1.31647
\(910\) 0 0
\(911\) −28.4999 −0.944245 −0.472122 0.881533i \(-0.656512\pi\)
−0.472122 + 0.881533i \(0.656512\pi\)
\(912\) 3.24698 0.107518
\(913\) −20.3147 −0.672318
\(914\) −33.8538 −1.11979
\(915\) 3.10992 0.102811
\(916\) 16.6679 0.550722
\(917\) −69.7706 −2.30403
\(918\) 87.9251 2.90196
\(919\) −53.5297 −1.76578 −0.882891 0.469577i \(-0.844406\pi\)
−0.882891 + 0.469577i \(0.844406\pi\)
\(920\) 6.67994 0.220231
\(921\) 32.7525 1.07923
\(922\) 16.9989 0.559830
\(923\) 0 0
\(924\) 60.6848 1.99638
\(925\) −17.3357 −0.569995
\(926\) −33.5254 −1.10171
\(927\) 2.98792 0.0981361
\(928\) −2.91185 −0.0955863
\(929\) −12.5133 −0.410549 −0.205275 0.978704i \(-0.565809\pi\)
−0.205275 + 0.978704i \(0.565809\pi\)
\(930\) −30.5405 −1.00146
\(931\) 5.98792 0.196246
\(932\) −10.5187 −0.344552
\(933\) −19.9898 −0.654438
\(934\) 24.8659 0.813638
\(935\) 48.0676 1.57198
\(936\) 0 0
\(937\) 44.3096 1.44753 0.723766 0.690045i \(-0.242409\pi\)
0.723766 + 0.690045i \(0.242409\pi\)
\(938\) −3.56033 −0.116249
\(939\) 46.7894 1.52691
\(940\) 17.1535 0.559484
\(941\) −43.4094 −1.41511 −0.707553 0.706660i \(-0.750201\pi\)
−0.707553 + 0.706660i \(0.750201\pi\)
\(942\) 7.84309 0.255542
\(943\) 42.2664 1.37638
\(944\) −13.5254 −0.440215
\(945\) −82.6607 −2.68895
\(946\) −26.7815 −0.870742
\(947\) −18.3338 −0.595768 −0.297884 0.954602i \(-0.596281\pi\)
−0.297884 + 0.954602i \(0.596281\pi\)
\(948\) 34.9638 1.13557
\(949\) 0 0
\(950\) 2.58211 0.0837746
\(951\) 66.6413 2.16099
\(952\) 21.4819 0.696232
\(953\) −1.43237 −0.0463990 −0.0231995 0.999731i \(-0.507385\pi\)
−0.0231995 + 0.999731i \(0.507385\pi\)
\(954\) −28.8823 −0.935099
\(955\) −14.3260 −0.463579
\(956\) −0.548253 −0.0177318
\(957\) 49.0320 1.58498
\(958\) −43.5120 −1.40581
\(959\) 23.8431 0.769933
\(960\) −5.04892 −0.162953
\(961\) 5.58940 0.180303
\(962\) 0 0
\(963\) −3.68532 −0.118758
\(964\) −17.5743 −0.566031
\(965\) 22.2024 0.714720
\(966\) −50.2693 −1.61739
\(967\) 43.6233 1.40283 0.701415 0.712753i \(-0.252552\pi\)
0.701415 + 0.712753i \(0.252552\pi\)
\(968\) −15.8944 −0.510865
\(969\) −19.3545 −0.621756
\(970\) 24.8756 0.798708
\(971\) 33.2325 1.06648 0.533241 0.845963i \(-0.320974\pi\)
0.533241 + 0.845963i \(0.320974\pi\)
\(972\) 37.7875 1.21203
\(973\) 3.36467 0.107866
\(974\) −9.41657 −0.301726
\(975\) 0 0
\(976\) −0.615957 −0.0197163
\(977\) 39.5579 1.26557 0.632785 0.774327i \(-0.281912\pi\)
0.632785 + 0.774327i \(0.281912\pi\)
\(978\) −15.0368 −0.480825
\(979\) −4.09618 −0.130914
\(980\) −9.31096 −0.297428
\(981\) −20.4698 −0.653550
\(982\) −1.67755 −0.0535327
\(983\) 6.38703 0.203715 0.101857 0.994799i \(-0.467522\pi\)
0.101857 + 0.994799i \(0.467522\pi\)
\(984\) −31.9463 −1.01841
\(985\) 18.3177 0.583649
\(986\) 17.3569 0.552756
\(987\) −129.087 −4.10888
\(988\) 0 0
\(989\) 22.1849 0.705439
\(990\) 60.8256 1.93317
\(991\) 1.76079 0.0559333 0.0279667 0.999609i \(-0.491097\pi\)
0.0279667 + 0.999609i \(0.491097\pi\)
\(992\) 6.04892 0.192053
\(993\) 72.7827 2.30969
\(994\) −17.0073 −0.539439
\(995\) −9.14377 −0.289877
\(996\) −12.7192 −0.403022
\(997\) −39.5362 −1.25212 −0.626062 0.779774i \(-0.715334\pi\)
−0.626062 + 0.779774i \(0.715334\pi\)
\(998\) 39.4077 1.24743
\(999\) 99.0326 3.13325
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.q.1.3 3
13.12 even 2 494.2.a.g.1.3 3
39.38 odd 2 4446.2.a.bd.1.2 3
52.51 odd 2 3952.2.a.m.1.1 3
247.246 odd 2 9386.2.a.u.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
494.2.a.g.1.3 3 13.12 even 2
3952.2.a.m.1.1 3 52.51 odd 2
4446.2.a.bd.1.2 3 39.38 odd 2
6422.2.a.q.1.3 3 1.1 even 1 trivial
9386.2.a.u.1.1 3 247.246 odd 2