Properties

Label 722.2.a.k.1.1
Level $722$
Weight $2$
Character 722.1
Self dual yes
Analytic conductor $5.765$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 722 = 2 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 722.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.76519902594\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Defining polynomial: \(x^{3} - 3 x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.53209\) of defining polynomial
Character \(\chi\) \(=\) 722.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.53209 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.53209 q^{6} +2.69459 q^{7} -1.00000 q^{8} -0.652704 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.53209 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.53209 q^{6} +2.69459 q^{7} -1.00000 q^{8} -0.652704 q^{9} -2.00000 q^{10} +3.18479 q^{11} -1.53209 q^{12} +5.75877 q^{13} -2.69459 q^{14} -3.06418 q^{15} +1.00000 q^{16} -6.51754 q^{17} +0.652704 q^{18} +2.00000 q^{20} -4.12836 q^{21} -3.18479 q^{22} -0.694593 q^{23} +1.53209 q^{24} -1.00000 q^{25} -5.75877 q^{26} +5.59627 q^{27} +2.69459 q^{28} +2.82295 q^{29} +3.06418 q^{30} -2.45336 q^{31} -1.00000 q^{32} -4.87939 q^{33} +6.51754 q^{34} +5.38919 q^{35} -0.652704 q^{36} +4.36959 q^{37} -8.82295 q^{39} -2.00000 q^{40} +0.347296 q^{41} +4.12836 q^{42} +6.06418 q^{43} +3.18479 q^{44} -1.30541 q^{45} +0.694593 q^{46} +7.88713 q^{47} -1.53209 q^{48} +0.260830 q^{49} +1.00000 q^{50} +9.98545 q^{51} +5.75877 q^{52} -8.21213 q^{53} -5.59627 q^{54} +6.36959 q^{55} -2.69459 q^{56} -2.82295 q^{58} -0.573978 q^{59} -3.06418 q^{60} +2.93582 q^{61} +2.45336 q^{62} -1.75877 q^{63} +1.00000 q^{64} +11.5175 q^{65} +4.87939 q^{66} +4.95811 q^{67} -6.51754 q^{68} +1.06418 q^{69} -5.38919 q^{70} +8.45336 q^{71} +0.652704 q^{72} +15.7665 q^{73} -4.36959 q^{74} +1.53209 q^{75} +8.58172 q^{77} +8.82295 q^{78} +9.06418 q^{79} +2.00000 q^{80} -6.61587 q^{81} -0.347296 q^{82} +8.47565 q^{83} -4.12836 q^{84} -13.0351 q^{85} -6.06418 q^{86} -4.32501 q^{87} -3.18479 q^{88} -7.73917 q^{89} +1.30541 q^{90} +15.5175 q^{91} -0.694593 q^{92} +3.75877 q^{93} -7.88713 q^{94} +1.53209 q^{96} +0.347296 q^{97} -0.260830 q^{98} -2.07873 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} + 6 q^{5} + 6 q^{7} - 3 q^{8} - 3 q^{9} + O(q^{10}) \) \( 3 q - 3 q^{2} + 3 q^{4} + 6 q^{5} + 6 q^{7} - 3 q^{8} - 3 q^{9} - 6 q^{10} + 6 q^{11} + 6 q^{13} - 6 q^{14} + 3 q^{16} + 3 q^{17} + 3 q^{18} + 6 q^{20} + 6 q^{21} - 6 q^{22} - 3 q^{25} - 6 q^{26} + 3 q^{27} + 6 q^{28} - 12 q^{29} + 6 q^{31} - 3 q^{32} - 9 q^{33} - 3 q^{34} + 12 q^{35} - 3 q^{36} + 6 q^{37} - 6 q^{39} - 6 q^{40} - 6 q^{42} + 9 q^{43} + 6 q^{44} - 6 q^{45} - 6 q^{47} + 15 q^{49} + 3 q^{50} + 12 q^{51} + 6 q^{52} - 3 q^{54} + 12 q^{55} - 6 q^{56} + 12 q^{58} + 6 q^{59} + 18 q^{61} - 6 q^{62} + 6 q^{63} + 3 q^{64} + 12 q^{65} + 9 q^{66} + 18 q^{67} + 3 q^{68} - 6 q^{69} - 12 q^{70} + 12 q^{71} + 3 q^{72} + 12 q^{73} - 6 q^{74} - 6 q^{77} + 6 q^{78} + 18 q^{79} + 6 q^{80} - 9 q^{81} + 6 q^{83} + 6 q^{84} + 6 q^{85} - 9 q^{86} - 18 q^{87} - 6 q^{88} - 9 q^{89} + 6 q^{90} + 24 q^{91} + 6 q^{94} - 15 q^{98} - 15 q^{99} + O(q^{100}) \)

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
\(3\) −1.53209 −0.884552 −0.442276 0.896879i \(-0.645829\pi\)
−0.442276 + 0.896879i \(0.645829\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 1.53209 0.625473
\(7\) 2.69459 1.01846 0.509230 0.860630i \(-0.329930\pi\)
0.509230 + 0.860630i \(0.329930\pi\)
\(8\) −1.00000 −0.353553
\(9\) −0.652704 −0.217568
\(10\) −2.00000 −0.632456
\(11\) 3.18479 0.960251 0.480126 0.877200i \(-0.340591\pi\)
0.480126 + 0.877200i \(0.340591\pi\)
\(12\) −1.53209 −0.442276
\(13\) 5.75877 1.59720 0.798598 0.601865i \(-0.205576\pi\)
0.798598 + 0.601865i \(0.205576\pi\)
\(14\) −2.69459 −0.720160
\(15\) −3.06418 −0.791167
\(16\) 1.00000 0.250000
\(17\) −6.51754 −1.58074 −0.790368 0.612632i \(-0.790111\pi\)
−0.790368 + 0.612632i \(0.790111\pi\)
\(18\) 0.652704 0.153844
\(19\) 0 0
\(20\) 2.00000 0.447214
\(21\) −4.12836 −0.900881
\(22\) −3.18479 −0.679000
\(23\) −0.694593 −0.144833 −0.0724163 0.997374i \(-0.523071\pi\)
−0.0724163 + 0.997374i \(0.523071\pi\)
\(24\) 1.53209 0.312736
\(25\) −1.00000 −0.200000
\(26\) −5.75877 −1.12939
\(27\) 5.59627 1.07700
\(28\) 2.69459 0.509230
\(29\) 2.82295 0.524208 0.262104 0.965040i \(-0.415584\pi\)
0.262104 + 0.965040i \(0.415584\pi\)
\(30\) 3.06418 0.559440
\(31\) −2.45336 −0.440637 −0.220319 0.975428i \(-0.570710\pi\)
−0.220319 + 0.975428i \(0.570710\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.87939 −0.849392
\(34\) 6.51754 1.11775
\(35\) 5.38919 0.910939
\(36\) −0.652704 −0.108784
\(37\) 4.36959 0.718355 0.359178 0.933269i \(-0.383057\pi\)
0.359178 + 0.933269i \(0.383057\pi\)
\(38\) 0 0
\(39\) −8.82295 −1.41280
\(40\) −2.00000 −0.316228
\(41\) 0.347296 0.0542386 0.0271193 0.999632i \(-0.491367\pi\)
0.0271193 + 0.999632i \(0.491367\pi\)
\(42\) 4.12836 0.637019
\(43\) 6.06418 0.924778 0.462389 0.886677i \(-0.346992\pi\)
0.462389 + 0.886677i \(0.346992\pi\)
\(44\) 3.18479 0.480126
\(45\) −1.30541 −0.194599
\(46\) 0.694593 0.102412
\(47\) 7.88713 1.15046 0.575228 0.817993i \(-0.304913\pi\)
0.575228 + 0.817993i \(0.304913\pi\)
\(48\) −1.53209 −0.221138
\(49\) 0.260830 0.0372614
\(50\) 1.00000 0.141421
\(51\) 9.98545 1.39824
\(52\) 5.75877 0.798598
\(53\) −8.21213 −1.12802 −0.564012 0.825767i \(-0.690743\pi\)
−0.564012 + 0.825767i \(0.690743\pi\)
\(54\) −5.59627 −0.761555
\(55\) 6.36959 0.858875
\(56\) −2.69459 −0.360080
\(57\) 0 0
\(58\) −2.82295 −0.370671
\(59\) −0.573978 −0.0747256 −0.0373628 0.999302i \(-0.511896\pi\)
−0.0373628 + 0.999302i \(0.511896\pi\)
\(60\) −3.06418 −0.395584
\(61\) 2.93582 0.375894 0.187947 0.982179i \(-0.439817\pi\)
0.187947 + 0.982179i \(0.439817\pi\)
\(62\) 2.45336 0.311577
\(63\) −1.75877 −0.221584
\(64\) 1.00000 0.125000
\(65\) 11.5175 1.42858
\(66\) 4.87939 0.600611
\(67\) 4.95811 0.605730 0.302865 0.953034i \(-0.402057\pi\)
0.302865 + 0.953034i \(0.402057\pi\)
\(68\) −6.51754 −0.790368
\(69\) 1.06418 0.128112
\(70\) −5.38919 −0.644131
\(71\) 8.45336 1.00323 0.501615 0.865091i \(-0.332739\pi\)
0.501615 + 0.865091i \(0.332739\pi\)
\(72\) 0.652704 0.0769219
\(73\) 15.7665 1.84533 0.922665 0.385602i \(-0.126006\pi\)
0.922665 + 0.385602i \(0.126006\pi\)
\(74\) −4.36959 −0.507954
\(75\) 1.53209 0.176910
\(76\) 0 0
\(77\) 8.58172 0.977978
\(78\) 8.82295 0.999002
\(79\) 9.06418 1.01980 0.509900 0.860234i \(-0.329682\pi\)
0.509900 + 0.860234i \(0.329682\pi\)
\(80\) 2.00000 0.223607
\(81\) −6.61587 −0.735096
\(82\) −0.347296 −0.0383525
\(83\) 8.47565 0.930324 0.465162 0.885226i \(-0.345996\pi\)
0.465162 + 0.885226i \(0.345996\pi\)
\(84\) −4.12836 −0.450441
\(85\) −13.0351 −1.41385
\(86\) −6.06418 −0.653917
\(87\) −4.32501 −0.463689
\(88\) −3.18479 −0.339500
\(89\) −7.73917 −0.820350 −0.410175 0.912007i \(-0.634532\pi\)
−0.410175 + 0.912007i \(0.634532\pi\)
\(90\) 1.30541 0.137602
\(91\) 15.5175 1.62668
\(92\) −0.694593 −0.0724163
\(93\) 3.75877 0.389766
\(94\) −7.88713 −0.813495
\(95\) 0 0
\(96\) 1.53209 0.156368
\(97\) 0.347296 0.0352626 0.0176313 0.999845i \(-0.494387\pi\)
0.0176313 + 0.999845i \(0.494387\pi\)
\(98\) −0.260830 −0.0263478
\(99\) −2.07873 −0.208920
\(100\) −1.00000 −0.100000
\(101\) −0.369585 −0.0367751 −0.0183875 0.999831i \(-0.505853\pi\)
−0.0183875 + 0.999831i \(0.505853\pi\)
\(102\) −9.98545 −0.988707
\(103\) −8.58172 −0.845582 −0.422791 0.906227i \(-0.638950\pi\)
−0.422791 + 0.906227i \(0.638950\pi\)
\(104\) −5.75877 −0.564694
\(105\) −8.25671 −0.805772
\(106\) 8.21213 0.797633
\(107\) 11.4534 1.10724 0.553619 0.832770i \(-0.313246\pi\)
0.553619 + 0.832770i \(0.313246\pi\)
\(108\) 5.59627 0.538501
\(109\) −8.69459 −0.832791 −0.416395 0.909184i \(-0.636707\pi\)
−0.416395 + 0.909184i \(0.636707\pi\)
\(110\) −6.36959 −0.607316
\(111\) −6.69459 −0.635423
\(112\) 2.69459 0.254615
\(113\) −2.85978 −0.269026 −0.134513 0.990912i \(-0.542947\pi\)
−0.134513 + 0.990912i \(0.542947\pi\)
\(114\) 0 0
\(115\) −1.38919 −0.129542
\(116\) 2.82295 0.262104
\(117\) −3.75877 −0.347498
\(118\) 0.573978 0.0528390
\(119\) −17.5621 −1.60992
\(120\) 3.06418 0.279720
\(121\) −0.857097 −0.0779179
\(122\) −2.93582 −0.265797
\(123\) −0.532089 −0.0479768
\(124\) −2.45336 −0.220319
\(125\) −12.0000 −1.07331
\(126\) 1.75877 0.156684
\(127\) −9.72967 −0.863369 −0.431685 0.902025i \(-0.642081\pi\)
−0.431685 + 0.902025i \(0.642081\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −9.29086 −0.818015
\(130\) −11.5175 −1.01016
\(131\) −6.46791 −0.565104 −0.282552 0.959252i \(-0.591181\pi\)
−0.282552 + 0.959252i \(0.591181\pi\)
\(132\) −4.87939 −0.424696
\(133\) 0 0
\(134\) −4.95811 −0.428316
\(135\) 11.1925 0.963300
\(136\) 6.51754 0.558875
\(137\) −11.6604 −0.996219 −0.498109 0.867114i \(-0.665972\pi\)
−0.498109 + 0.867114i \(0.665972\pi\)
\(138\) −1.06418 −0.0905888
\(139\) −8.26352 −0.700902 −0.350451 0.936581i \(-0.613972\pi\)
−0.350451 + 0.936581i \(0.613972\pi\)
\(140\) 5.38919 0.455469
\(141\) −12.0838 −1.01764
\(142\) −8.45336 −0.709390
\(143\) 18.3405 1.53371
\(144\) −0.652704 −0.0543920
\(145\) 5.64590 0.468866
\(146\) −15.7665 −1.30485
\(147\) −0.399615 −0.0329597
\(148\) 4.36959 0.359178
\(149\) −16.4534 −1.34791 −0.673956 0.738771i \(-0.735406\pi\)
−0.673956 + 0.738771i \(0.735406\pi\)
\(150\) −1.53209 −0.125095
\(151\) −4.65539 −0.378850 −0.189425 0.981895i \(-0.560662\pi\)
−0.189425 + 0.981895i \(0.560662\pi\)
\(152\) 0 0
\(153\) 4.25402 0.343917
\(154\) −8.58172 −0.691535
\(155\) −4.90673 −0.394118
\(156\) −8.82295 −0.706401
\(157\) 8.45336 0.674652 0.337326 0.941388i \(-0.390478\pi\)
0.337326 + 0.941388i \(0.390478\pi\)
\(158\) −9.06418 −0.721107
\(159\) 12.5817 0.997795
\(160\) −2.00000 −0.158114
\(161\) −1.87164 −0.147506
\(162\) 6.61587 0.519792
\(163\) 17.0496 1.33543 0.667715 0.744417i \(-0.267272\pi\)
0.667715 + 0.744417i \(0.267272\pi\)
\(164\) 0.347296 0.0271193
\(165\) −9.75877 −0.759719
\(166\) −8.47565 −0.657838
\(167\) 3.19253 0.247046 0.123523 0.992342i \(-0.460581\pi\)
0.123523 + 0.992342i \(0.460581\pi\)
\(168\) 4.12836 0.318510
\(169\) 20.1634 1.55103
\(170\) 13.0351 0.999745
\(171\) 0 0
\(172\) 6.06418 0.462389
\(173\) −9.63041 −0.732187 −0.366093 0.930578i \(-0.619305\pi\)
−0.366093 + 0.930578i \(0.619305\pi\)
\(174\) 4.32501 0.327878
\(175\) −2.69459 −0.203692
\(176\) 3.18479 0.240063
\(177\) 0.879385 0.0660986
\(178\) 7.73917 0.580075
\(179\) −18.8161 −1.40638 −0.703192 0.711000i \(-0.748243\pi\)
−0.703192 + 0.711000i \(0.748243\pi\)
\(180\) −1.30541 −0.0972993
\(181\) −2.77837 −0.206515 −0.103257 0.994655i \(-0.532927\pi\)
−0.103257 + 0.994655i \(0.532927\pi\)
\(182\) −15.5175 −1.15024
\(183\) −4.49794 −0.332497
\(184\) 0.694593 0.0512061
\(185\) 8.73917 0.642517
\(186\) −3.75877 −0.275606
\(187\) −20.7570 −1.51790
\(188\) 7.88713 0.575228
\(189\) 15.0797 1.09688
\(190\) 0 0
\(191\) 9.56212 0.691891 0.345945 0.938255i \(-0.387558\pi\)
0.345945 + 0.938255i \(0.387558\pi\)
\(192\) −1.53209 −0.110569
\(193\) 23.6810 1.70459 0.852297 0.523058i \(-0.175209\pi\)
0.852297 + 0.523058i \(0.175209\pi\)
\(194\) −0.347296 −0.0249344
\(195\) −17.6459 −1.26365
\(196\) 0.260830 0.0186307
\(197\) −22.9222 −1.63314 −0.816570 0.577247i \(-0.804127\pi\)
−0.816570 + 0.577247i \(0.804127\pi\)
\(198\) 2.07873 0.147729
\(199\) 10.0838 0.714820 0.357410 0.933948i \(-0.383660\pi\)
0.357410 + 0.933948i \(0.383660\pi\)
\(200\) 1.00000 0.0707107
\(201\) −7.59627 −0.535799
\(202\) 0.369585 0.0260039
\(203\) 7.60670 0.533885
\(204\) 9.98545 0.699121
\(205\) 0.694593 0.0485125
\(206\) 8.58172 0.597917
\(207\) 0.453363 0.0315109
\(208\) 5.75877 0.399299
\(209\) 0 0
\(210\) 8.25671 0.569767
\(211\) 22.3601 1.53933 0.769666 0.638447i \(-0.220423\pi\)
0.769666 + 0.638447i \(0.220423\pi\)
\(212\) −8.21213 −0.564012
\(213\) −12.9513 −0.887409
\(214\) −11.4534 −0.782936
\(215\) 12.1284 0.827147
\(216\) −5.59627 −0.380778
\(217\) −6.61081 −0.448771
\(218\) 8.69459 0.588872
\(219\) −24.1557 −1.63229
\(220\) 6.36959 0.429437
\(221\) −37.5330 −2.52474
\(222\) 6.69459 0.449312
\(223\) 9.27631 0.621188 0.310594 0.950543i \(-0.399472\pi\)
0.310594 + 0.950543i \(0.399472\pi\)
\(224\) −2.69459 −0.180040
\(225\) 0.652704 0.0435136
\(226\) 2.85978 0.190230
\(227\) −7.73648 −0.513488 −0.256744 0.966479i \(-0.582650\pi\)
−0.256744 + 0.966479i \(0.582650\pi\)
\(228\) 0 0
\(229\) −23.0351 −1.52220 −0.761101 0.648634i \(-0.775341\pi\)
−0.761101 + 0.648634i \(0.775341\pi\)
\(230\) 1.38919 0.0916002
\(231\) −13.1480 −0.865072
\(232\) −2.82295 −0.185336
\(233\) 8.39961 0.550277 0.275139 0.961405i \(-0.411276\pi\)
0.275139 + 0.961405i \(0.411276\pi\)
\(234\) 3.75877 0.245719
\(235\) 15.7743 1.02900
\(236\) −0.573978 −0.0373628
\(237\) −13.8871 −0.902066
\(238\) 17.5621 1.13838
\(239\) −15.7297 −1.01747 −0.508734 0.860924i \(-0.669886\pi\)
−0.508734 + 0.860924i \(0.669886\pi\)
\(240\) −3.06418 −0.197792
\(241\) −17.5030 −1.12747 −0.563733 0.825957i \(-0.690635\pi\)
−0.563733 + 0.825957i \(0.690635\pi\)
\(242\) 0.857097 0.0550963
\(243\) −6.65270 −0.426771
\(244\) 2.93582 0.187947
\(245\) 0.521660 0.0333276
\(246\) 0.532089 0.0339247
\(247\) 0 0
\(248\) 2.45336 0.155789
\(249\) −12.9855 −0.822920
\(250\) 12.0000 0.758947
\(251\) 8.56717 0.540755 0.270378 0.962754i \(-0.412851\pi\)
0.270378 + 0.962754i \(0.412851\pi\)
\(252\) −1.75877 −0.110792
\(253\) −2.21213 −0.139076
\(254\) 9.72967 0.610494
\(255\) 19.9709 1.25063
\(256\) 1.00000 0.0625000
\(257\) 8.02465 0.500564 0.250282 0.968173i \(-0.419477\pi\)
0.250282 + 0.968173i \(0.419477\pi\)
\(258\) 9.29086 0.578424
\(259\) 11.7743 0.731616
\(260\) 11.5175 0.714288
\(261\) −1.84255 −0.114051
\(262\) 6.46791 0.399589
\(263\) −5.96080 −0.367559 −0.183779 0.982968i \(-0.558833\pi\)
−0.183779 + 0.982968i \(0.558833\pi\)
\(264\) 4.87939 0.300305
\(265\) −16.4243 −1.00893
\(266\) 0 0
\(267\) 11.8571 0.725643
\(268\) 4.95811 0.302865
\(269\) 1.14796 0.0699921 0.0349961 0.999387i \(-0.488858\pi\)
0.0349961 + 0.999387i \(0.488858\pi\)
\(270\) −11.1925 −0.681156
\(271\) 20.0547 1.21824 0.609118 0.793080i \(-0.291524\pi\)
0.609118 + 0.793080i \(0.291524\pi\)
\(272\) −6.51754 −0.395184
\(273\) −23.7743 −1.43888
\(274\) 11.6604 0.704433
\(275\) −3.18479 −0.192050
\(276\) 1.06418 0.0640560
\(277\) 17.3601 1.04307 0.521533 0.853231i \(-0.325360\pi\)
0.521533 + 0.853231i \(0.325360\pi\)
\(278\) 8.26352 0.495613
\(279\) 1.60132 0.0958685
\(280\) −5.38919 −0.322065
\(281\) −2.92127 −0.174269 −0.0871343 0.996197i \(-0.527771\pi\)
−0.0871343 + 0.996197i \(0.527771\pi\)
\(282\) 12.0838 0.719579
\(283\) −9.48070 −0.563569 −0.281785 0.959478i \(-0.590926\pi\)
−0.281785 + 0.959478i \(0.590926\pi\)
\(284\) 8.45336 0.501615
\(285\) 0 0
\(286\) −18.3405 −1.08450
\(287\) 0.935822 0.0552398
\(288\) 0.652704 0.0384609
\(289\) 25.4783 1.49873
\(290\) −5.64590 −0.331538
\(291\) −0.532089 −0.0311916
\(292\) 15.7665 0.922665
\(293\) 27.2918 1.59440 0.797202 0.603713i \(-0.206313\pi\)
0.797202 + 0.603713i \(0.206313\pi\)
\(294\) 0.399615 0.0233060
\(295\) −1.14796 −0.0668366
\(296\) −4.36959 −0.253977
\(297\) 17.8229 1.03419
\(298\) 16.4534 0.953118
\(299\) −4.00000 −0.231326
\(300\) 1.53209 0.0884552
\(301\) 16.3405 0.941850
\(302\) 4.65539 0.267888
\(303\) 0.566237 0.0325295
\(304\) 0 0
\(305\) 5.87164 0.336209
\(306\) −4.25402 −0.243186
\(307\) −21.3286 −1.21729 −0.608645 0.793443i \(-0.708286\pi\)
−0.608645 + 0.793443i \(0.708286\pi\)
\(308\) 8.58172 0.488989
\(309\) 13.1480 0.747961
\(310\) 4.90673 0.278683
\(311\) −29.2918 −1.66099 −0.830493 0.557030i \(-0.811941\pi\)
−0.830493 + 0.557030i \(0.811941\pi\)
\(312\) 8.82295 0.499501
\(313\) 5.56118 0.314337 0.157168 0.987572i \(-0.449763\pi\)
0.157168 + 0.987572i \(0.449763\pi\)
\(314\) −8.45336 −0.477051
\(315\) −3.51754 −0.198191
\(316\) 9.06418 0.509900
\(317\) −3.80335 −0.213617 −0.106809 0.994280i \(-0.534063\pi\)
−0.106809 + 0.994280i \(0.534063\pi\)
\(318\) −12.5817 −0.705548
\(319\) 8.99050 0.503372
\(320\) 2.00000 0.111803
\(321\) −17.5476 −0.979410
\(322\) 1.87164 0.104303
\(323\) 0 0
\(324\) −6.61587 −0.367548
\(325\) −5.75877 −0.319439
\(326\) −17.0496 −0.944292
\(327\) 13.3209 0.736647
\(328\) −0.347296 −0.0191762
\(329\) 21.2526 1.17169
\(330\) 9.75877 0.537203
\(331\) −20.4219 −1.12249 −0.561245 0.827650i \(-0.689677\pi\)
−0.561245 + 0.827650i \(0.689677\pi\)
\(332\) 8.47565 0.465162
\(333\) −2.85204 −0.156291
\(334\) −3.19253 −0.174688
\(335\) 9.91622 0.541781
\(336\) −4.12836 −0.225220
\(337\) −20.3105 −1.10638 −0.553191 0.833055i \(-0.686590\pi\)
−0.553191 + 0.833055i \(0.686590\pi\)
\(338\) −20.1634 −1.09675
\(339\) 4.38144 0.237967
\(340\) −13.0351 −0.706927
\(341\) −7.81345 −0.423122
\(342\) 0 0
\(343\) −18.1593 −0.980511
\(344\) −6.06418 −0.326959
\(345\) 2.12836 0.114587
\(346\) 9.63041 0.517734
\(347\) 5.21482 0.279946 0.139973 0.990155i \(-0.455298\pi\)
0.139973 + 0.990155i \(0.455298\pi\)
\(348\) −4.32501 −0.231845
\(349\) 14.3405 0.767629 0.383814 0.923410i \(-0.374610\pi\)
0.383814 + 0.923410i \(0.374610\pi\)
\(350\) 2.69459 0.144032
\(351\) 32.2276 1.72018
\(352\) −3.18479 −0.169750
\(353\) 26.2499 1.39714 0.698571 0.715541i \(-0.253820\pi\)
0.698571 + 0.715541i \(0.253820\pi\)
\(354\) −0.879385 −0.0467388
\(355\) 16.9067 0.897316
\(356\) −7.73917 −0.410175
\(357\) 26.9067 1.42405
\(358\) 18.8161 0.994464
\(359\) −33.6905 −1.77812 −0.889058 0.457795i \(-0.848639\pi\)
−0.889058 + 0.457795i \(0.848639\pi\)
\(360\) 1.30541 0.0688010
\(361\) 0 0
\(362\) 2.77837 0.146028
\(363\) 1.31315 0.0689224
\(364\) 15.5175 0.813340
\(365\) 31.5330 1.65051
\(366\) 4.49794 0.235111
\(367\) −10.4688 −0.546469 −0.273235 0.961947i \(-0.588094\pi\)
−0.273235 + 0.961947i \(0.588094\pi\)
\(368\) −0.694593 −0.0362081
\(369\) −0.226682 −0.0118006
\(370\) −8.73917 −0.454328
\(371\) −22.1284 −1.14885
\(372\) 3.75877 0.194883
\(373\) −23.9026 −1.23763 −0.618815 0.785537i \(-0.712387\pi\)
−0.618815 + 0.785537i \(0.712387\pi\)
\(374\) 20.7570 1.07332
\(375\) 18.3851 0.949401
\(376\) −7.88713 −0.406747
\(377\) 16.2567 0.837263
\(378\) −15.0797 −0.775614
\(379\) −17.8135 −0.915016 −0.457508 0.889206i \(-0.651258\pi\)
−0.457508 + 0.889206i \(0.651258\pi\)
\(380\) 0 0
\(381\) 14.9067 0.763695
\(382\) −9.56212 −0.489241
\(383\) −25.0797 −1.28151 −0.640755 0.767745i \(-0.721379\pi\)
−0.640755 + 0.767745i \(0.721379\pi\)
\(384\) 1.53209 0.0781841
\(385\) 17.1634 0.874730
\(386\) −23.6810 −1.20533
\(387\) −3.95811 −0.201202
\(388\) 0.347296 0.0176313
\(389\) −9.41828 −0.477526 −0.238763 0.971078i \(-0.576742\pi\)
−0.238763 + 0.971078i \(0.576742\pi\)
\(390\) 17.6459 0.893535
\(391\) 4.52704 0.228942
\(392\) −0.260830 −0.0131739
\(393\) 9.90941 0.499864
\(394\) 22.9222 1.15480
\(395\) 18.1284 0.912137
\(396\) −2.07873 −0.104460
\(397\) 6.85204 0.343894 0.171947 0.985106i \(-0.444994\pi\)
0.171947 + 0.985106i \(0.444994\pi\)
\(398\) −10.0838 −0.505454
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −4.74691 −0.237049 −0.118525 0.992951i \(-0.537816\pi\)
−0.118525 + 0.992951i \(0.537816\pi\)
\(402\) 7.59627 0.378867
\(403\) −14.1284 −0.703784
\(404\) −0.369585 −0.0183875
\(405\) −13.2317 −0.657490
\(406\) −7.60670 −0.377514
\(407\) 13.9162 0.689802
\(408\) −9.98545 −0.494354
\(409\) −31.6928 −1.56711 −0.783555 0.621322i \(-0.786596\pi\)
−0.783555 + 0.621322i \(0.786596\pi\)
\(410\) −0.694593 −0.0343035
\(411\) 17.8648 0.881207
\(412\) −8.58172 −0.422791
\(413\) −1.54664 −0.0761050
\(414\) −0.453363 −0.0222816
\(415\) 16.9513 0.832107
\(416\) −5.75877 −0.282347
\(417\) 12.6604 0.619985
\(418\) 0 0
\(419\) −11.0101 −0.537879 −0.268939 0.963157i \(-0.586673\pi\)
−0.268939 + 0.963157i \(0.586673\pi\)
\(420\) −8.25671 −0.402886
\(421\) 8.66550 0.422330 0.211165 0.977450i \(-0.432274\pi\)
0.211165 + 0.977450i \(0.432274\pi\)
\(422\) −22.3601 −1.08847
\(423\) −5.14796 −0.250302
\(424\) 8.21213 0.398816
\(425\) 6.51754 0.316147
\(426\) 12.9513 0.627493
\(427\) 7.91085 0.382833
\(428\) 11.4534 0.553619
\(429\) −28.0993 −1.35665
\(430\) −12.1284 −0.584881
\(431\) 29.8871 1.43961 0.719806 0.694175i \(-0.244231\pi\)
0.719806 + 0.694175i \(0.244231\pi\)
\(432\) 5.59627 0.269251
\(433\) −9.26083 −0.445047 −0.222524 0.974927i \(-0.571429\pi\)
−0.222524 + 0.974927i \(0.571429\pi\)
\(434\) 6.61081 0.317329
\(435\) −8.65002 −0.414736
\(436\) −8.69459 −0.416395
\(437\) 0 0
\(438\) 24.1557 1.15420
\(439\) −20.8621 −0.995696 −0.497848 0.867264i \(-0.665876\pi\)
−0.497848 + 0.867264i \(0.665876\pi\)
\(440\) −6.36959 −0.303658
\(441\) −0.170245 −0.00810689
\(442\) 37.5330 1.78526
\(443\) 23.8280 1.13210 0.566051 0.824370i \(-0.308470\pi\)
0.566051 + 0.824370i \(0.308470\pi\)
\(444\) −6.69459 −0.317711
\(445\) −15.4783 −0.733744
\(446\) −9.27631 −0.439246
\(447\) 25.2080 1.19230
\(448\) 2.69459 0.127308
\(449\) 2.18210 0.102980 0.0514899 0.998674i \(-0.483603\pi\)
0.0514899 + 0.998674i \(0.483603\pi\)
\(450\) −0.652704 −0.0307687
\(451\) 1.10607 0.0520827
\(452\) −2.85978 −0.134513
\(453\) 7.13247 0.335113
\(454\) 7.73648 0.363091
\(455\) 31.0351 1.45495
\(456\) 0 0
\(457\) 1.78106 0.0833144 0.0416572 0.999132i \(-0.486736\pi\)
0.0416572 + 0.999132i \(0.486736\pi\)
\(458\) 23.0351 1.07636
\(459\) −36.4739 −1.70246
\(460\) −1.38919 −0.0647711
\(461\) −15.4884 −0.721369 −0.360684 0.932688i \(-0.617457\pi\)
−0.360684 + 0.932688i \(0.617457\pi\)
\(462\) 13.1480 0.611698
\(463\) −2.71007 −0.125948 −0.0629739 0.998015i \(-0.520058\pi\)
−0.0629739 + 0.998015i \(0.520058\pi\)
\(464\) 2.82295 0.131052
\(465\) 7.51754 0.348618
\(466\) −8.39961 −0.389105
\(467\) 12.9135 0.597567 0.298784 0.954321i \(-0.403419\pi\)
0.298784 + 0.954321i \(0.403419\pi\)
\(468\) −3.75877 −0.173749
\(469\) 13.3601 0.616912
\(470\) −15.7743 −0.727612
\(471\) −12.9513 −0.596765
\(472\) 0.573978 0.0264195
\(473\) 19.3131 0.888019
\(474\) 13.8871 0.637857
\(475\) 0 0
\(476\) −17.5621 −0.804958
\(477\) 5.36009 0.245422
\(478\) 15.7297 0.719459
\(479\) −18.5526 −0.847691 −0.423845 0.905735i \(-0.639320\pi\)
−0.423845 + 0.905735i \(0.639320\pi\)
\(480\) 3.06418 0.139860
\(481\) 25.1634 1.14735
\(482\) 17.5030 0.797239
\(483\) 2.86753 0.130477
\(484\) −0.857097 −0.0389589
\(485\) 0.694593 0.0315398
\(486\) 6.65270 0.301773
\(487\) −41.1735 −1.86575 −0.932876 0.360199i \(-0.882709\pi\)
−0.932876 + 0.360199i \(0.882709\pi\)
\(488\) −2.93582 −0.132898
\(489\) −26.1215 −1.18126
\(490\) −0.521660 −0.0235662
\(491\) −22.5776 −1.01891 −0.509456 0.860496i \(-0.670153\pi\)
−0.509456 + 0.860496i \(0.670153\pi\)
\(492\) −0.532089 −0.0239884
\(493\) −18.3987 −0.828635
\(494\) 0 0
\(495\) −4.15745 −0.186864
\(496\) −2.45336 −0.110159
\(497\) 22.7784 1.02175
\(498\) 12.9855 0.581892
\(499\) −29.3286 −1.31293 −0.656465 0.754357i \(-0.727949\pi\)
−0.656465 + 0.754357i \(0.727949\pi\)
\(500\) −12.0000 −0.536656
\(501\) −4.89124 −0.218525
\(502\) −8.56717 −0.382372
\(503\) 33.6168 1.49890 0.749450 0.662061i \(-0.230318\pi\)
0.749450 + 0.662061i \(0.230318\pi\)
\(504\) 1.75877 0.0783419
\(505\) −0.739170 −0.0328926
\(506\) 2.21213 0.0983413
\(507\) −30.8922 −1.37197
\(508\) −9.72967 −0.431685
\(509\) −4.02910 −0.178587 −0.0892933 0.996005i \(-0.528461\pi\)
−0.0892933 + 0.996005i \(0.528461\pi\)
\(510\) −19.9709 −0.884327
\(511\) 42.4843 1.87940
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −8.02465 −0.353952
\(515\) −17.1634 −0.756311
\(516\) −9.29086 −0.409007
\(517\) 25.1189 1.10473
\(518\) −11.7743 −0.517331
\(519\) 14.7547 0.647657
\(520\) −11.5175 −0.505078
\(521\) −4.98957 −0.218597 −0.109299 0.994009i \(-0.534860\pi\)
−0.109299 + 0.994009i \(0.534860\pi\)
\(522\) 1.84255 0.0806462
\(523\) 26.5526 1.16107 0.580533 0.814237i \(-0.302844\pi\)
0.580533 + 0.814237i \(0.302844\pi\)
\(524\) −6.46791 −0.282552
\(525\) 4.12836 0.180176
\(526\) 5.96080 0.259903
\(527\) 15.9899 0.696531
\(528\) −4.87939 −0.212348
\(529\) −22.5175 −0.979024
\(530\) 16.4243 0.713425
\(531\) 0.374638 0.0162579
\(532\) 0 0
\(533\) 2.00000 0.0866296
\(534\) −11.8571 −0.513107
\(535\) 22.9067 0.990344
\(536\) −4.95811 −0.214158
\(537\) 28.8280 1.24402
\(538\) −1.14796 −0.0494919
\(539\) 0.830689 0.0357803
\(540\) 11.1925 0.481650
\(541\) −11.9709 −0.514669 −0.257335 0.966322i \(-0.582844\pi\)
−0.257335 + 0.966322i \(0.582844\pi\)
\(542\) −20.0547 −0.861422
\(543\) 4.25671 0.182673
\(544\) 6.51754 0.279437
\(545\) −17.3892 −0.744871
\(546\) 23.7743 1.01744
\(547\) −3.19665 −0.136679 −0.0683395 0.997662i \(-0.521770\pi\)
−0.0683395 + 0.997662i \(0.521770\pi\)
\(548\) −11.6604 −0.498109
\(549\) −1.91622 −0.0817824
\(550\) 3.18479 0.135800
\(551\) 0 0
\(552\) −1.06418 −0.0452944
\(553\) 24.4243 1.03863
\(554\) −17.3601 −0.737560
\(555\) −13.3892 −0.568339
\(556\) −8.26352 −0.350451
\(557\) −23.9763 −1.01591 −0.507954 0.861384i \(-0.669598\pi\)
−0.507954 + 0.861384i \(0.669598\pi\)
\(558\) −1.60132 −0.0677892
\(559\) 34.9222 1.47705
\(560\) 5.38919 0.227735
\(561\) 31.8016 1.34266
\(562\) 2.92127 0.123227
\(563\) 8.75702 0.369064 0.184532 0.982826i \(-0.440923\pi\)
0.184532 + 0.982826i \(0.440923\pi\)
\(564\) −12.0838 −0.508819
\(565\) −5.71957 −0.240624
\(566\) 9.48070 0.398504
\(567\) −17.8271 −0.748666
\(568\) −8.45336 −0.354695
\(569\) 36.4201 1.52681 0.763406 0.645919i \(-0.223526\pi\)
0.763406 + 0.645919i \(0.223526\pi\)
\(570\) 0 0
\(571\) 34.2131 1.43177 0.715886 0.698217i \(-0.246023\pi\)
0.715886 + 0.698217i \(0.246023\pi\)
\(572\) 18.3405 0.766854
\(573\) −14.6500 −0.612013
\(574\) −0.935822 −0.0390605
\(575\) 0.694593 0.0289665
\(576\) −0.652704 −0.0271960
\(577\) 15.5098 0.645681 0.322841 0.946453i \(-0.395362\pi\)
0.322841 + 0.946453i \(0.395362\pi\)
\(578\) −25.4783 −1.05976
\(579\) −36.2814 −1.50780
\(580\) 5.64590 0.234433
\(581\) 22.8384 0.947498
\(582\) 0.532089 0.0220558
\(583\) −26.1539 −1.08319
\(584\) −15.7665 −0.652423
\(585\) −7.51754 −0.310812
\(586\) −27.2918 −1.12741
\(587\) −11.5817 −0.478029 −0.239014 0.971016i \(-0.576824\pi\)
−0.239014 + 0.971016i \(0.576824\pi\)
\(588\) −0.399615 −0.0164798
\(589\) 0 0
\(590\) 1.14796 0.0472606
\(591\) 35.1189 1.44460
\(592\) 4.36959 0.179589
\(593\) −46.3928 −1.90512 −0.952562 0.304344i \(-0.901563\pi\)
−0.952562 + 0.304344i \(0.901563\pi\)
\(594\) −17.8229 −0.731284
\(595\) −35.1242 −1.43995
\(596\) −16.4534 −0.673956
\(597\) −15.4492 −0.632295
\(598\) 4.00000 0.163572
\(599\) −25.6323 −1.04731 −0.523653 0.851931i \(-0.675431\pi\)
−0.523653 + 0.851931i \(0.675431\pi\)
\(600\) −1.53209 −0.0625473
\(601\) −7.99226 −0.326011 −0.163006 0.986625i \(-0.552119\pi\)
−0.163006 + 0.986625i \(0.552119\pi\)
\(602\) −16.3405 −0.665989
\(603\) −3.23618 −0.131787
\(604\) −4.65539 −0.189425
\(605\) −1.71419 −0.0696919
\(606\) −0.566237 −0.0230018
\(607\) 26.9905 1.09551 0.547755 0.836639i \(-0.315482\pi\)
0.547755 + 0.836639i \(0.315482\pi\)
\(608\) 0 0
\(609\) −11.6541 −0.472249
\(610\) −5.87164 −0.237736
\(611\) 45.4201 1.83750
\(612\) 4.25402 0.171959
\(613\) −14.2513 −0.575606 −0.287803 0.957690i \(-0.592925\pi\)
−0.287803 + 0.957690i \(0.592925\pi\)
\(614\) 21.3286 0.860753
\(615\) −1.06418 −0.0429118
\(616\) −8.58172 −0.345767
\(617\) −30.5604 −1.23031 −0.615157 0.788405i \(-0.710907\pi\)
−0.615157 + 0.788405i \(0.710907\pi\)
\(618\) −13.1480 −0.528888
\(619\) −28.6750 −1.15255 −0.576273 0.817258i \(-0.695493\pi\)
−0.576273 + 0.817258i \(0.695493\pi\)
\(620\) −4.90673 −0.197059
\(621\) −3.88713 −0.155985
\(622\) 29.2918 1.17449
\(623\) −20.8539 −0.835494
\(624\) −8.82295 −0.353201
\(625\) −19.0000 −0.760000
\(626\) −5.56118 −0.222270
\(627\) 0 0
\(628\) 8.45336 0.337326
\(629\) −28.4789 −1.13553
\(630\) 3.51754 0.140142
\(631\) −4.49794 −0.179060 −0.0895301 0.995984i \(-0.528537\pi\)
−0.0895301 + 0.995984i \(0.528537\pi\)
\(632\) −9.06418 −0.360554
\(633\) −34.2576 −1.36162
\(634\) 3.80335 0.151050
\(635\) −19.4593 −0.772221
\(636\) 12.5817 0.498898
\(637\) 1.50206 0.0595138
\(638\) −8.99050 −0.355937
\(639\) −5.51754 −0.218271
\(640\) −2.00000 −0.0790569
\(641\) 11.6081 0.458493 0.229247 0.973368i \(-0.426374\pi\)
0.229247 + 0.973368i \(0.426374\pi\)
\(642\) 17.5476 0.692547
\(643\) 26.0455 1.02713 0.513567 0.858049i \(-0.328324\pi\)
0.513567 + 0.858049i \(0.328324\pi\)
\(644\) −1.87164 −0.0737531
\(645\) −18.5817 −0.731654
\(646\) 0 0
\(647\) −2.31490 −0.0910082 −0.0455041 0.998964i \(-0.514489\pi\)
−0.0455041 + 0.998964i \(0.514489\pi\)
\(648\) 6.61587 0.259896
\(649\) −1.82800 −0.0717553
\(650\) 5.75877 0.225878
\(651\) 10.1284 0.396962
\(652\) 17.0496 0.667715
\(653\) 3.30541 0.129351 0.0646753 0.997906i \(-0.479399\pi\)
0.0646753 + 0.997906i \(0.479399\pi\)
\(654\) −13.3209 −0.520888
\(655\) −12.9358 −0.505444
\(656\) 0.347296 0.0135596
\(657\) −10.2909 −0.401485
\(658\) −21.2526 −0.828512
\(659\) 13.7493 0.535596 0.267798 0.963475i \(-0.413704\pi\)
0.267798 + 0.963475i \(0.413704\pi\)
\(660\) −9.75877 −0.379860
\(661\) 3.33450 0.129697 0.0648486 0.997895i \(-0.479344\pi\)
0.0648486 + 0.997895i \(0.479344\pi\)
\(662\) 20.4219 0.793720
\(663\) 57.5039 2.23327
\(664\) −8.47565 −0.328919
\(665\) 0 0
\(666\) 2.85204 0.110514
\(667\) −1.96080 −0.0759225
\(668\) 3.19253 0.123523
\(669\) −14.2121 −0.549473
\(670\) −9.91622 −0.383097
\(671\) 9.34998 0.360952
\(672\) 4.12836 0.159255
\(673\) −38.9810 −1.50261 −0.751304 0.659957i \(-0.770575\pi\)
−0.751304 + 0.659957i \(0.770575\pi\)
\(674\) 20.3105 0.782330
\(675\) −5.59627 −0.215400
\(676\) 20.1634 0.775517
\(677\) 43.5877 1.67521 0.837606 0.546275i \(-0.183955\pi\)
0.837606 + 0.546275i \(0.183955\pi\)
\(678\) −4.38144 −0.168268
\(679\) 0.935822 0.0359136
\(680\) 13.0351 0.499873
\(681\) 11.8530 0.454207
\(682\) 7.81345 0.299193
\(683\) −32.9317 −1.26010 −0.630048 0.776556i \(-0.716965\pi\)
−0.630048 + 0.776556i \(0.716965\pi\)
\(684\) 0 0
\(685\) −23.3209 −0.891045
\(686\) 18.1593 0.693326
\(687\) 35.2918 1.34647
\(688\) 6.06418 0.231195
\(689\) −47.2918 −1.80167
\(690\) −2.12836 −0.0810251
\(691\) 34.3209 1.30563 0.652814 0.757518i \(-0.273588\pi\)
0.652814 + 0.757518i \(0.273588\pi\)
\(692\) −9.63041 −0.366093
\(693\) −5.60132 −0.212777
\(694\) −5.21482 −0.197952
\(695\) −16.5270 −0.626906
\(696\) 4.32501 0.163939
\(697\) −2.26352 −0.0857369
\(698\) −14.3405 −0.542796
\(699\) −12.8690 −0.486749
\(700\) −2.69459 −0.101846
\(701\) 6.45336 0.243740 0.121870 0.992546i \(-0.461111\pi\)
0.121870 + 0.992546i \(0.461111\pi\)
\(702\) −32.2276 −1.21635
\(703\) 0 0
\(704\) 3.18479 0.120031
\(705\) −24.1676 −0.910203
\(706\) −26.2499 −0.987928
\(707\) −0.995881 −0.0374540
\(708\) 0.879385 0.0330493
\(709\) −3.37908 −0.126904 −0.0634520 0.997985i \(-0.520211\pi\)
−0.0634520 + 0.997985i \(0.520211\pi\)
\(710\) −16.9067 −0.634498
\(711\) −5.91622 −0.221876
\(712\) 7.73917 0.290038
\(713\) 1.70409 0.0638186
\(714\) −26.9067 −1.00696
\(715\) 36.6810 1.37179
\(716\) −18.8161 −0.703192
\(717\) 24.0993 0.900003
\(718\) 33.6905 1.25732
\(719\) −31.2026 −1.16366 −0.581831 0.813310i \(-0.697664\pi\)
−0.581831 + 0.813310i \(0.697664\pi\)
\(720\) −1.30541 −0.0486497
\(721\) −23.1242 −0.861192
\(722\) 0 0
\(723\) 26.8161 0.997303
\(724\) −2.77837 −0.103257
\(725\) −2.82295 −0.104842
\(726\) −1.31315 −0.0487355
\(727\) −25.6614 −0.951728 −0.475864 0.879519i \(-0.657865\pi\)
−0.475864 + 0.879519i \(0.657865\pi\)
\(728\) −15.5175 −0.575118
\(729\) 30.0401 1.11260
\(730\) −31.5330 −1.16709
\(731\) −39.5235 −1.46183
\(732\) −4.49794 −0.166249
\(733\) −23.8735 −0.881788 −0.440894 0.897559i \(-0.645339\pi\)
−0.440894 + 0.897559i \(0.645339\pi\)
\(734\) 10.4688 0.386412
\(735\) −0.799229 −0.0294800
\(736\) 0.694593 0.0256030
\(737\) 15.7906 0.581653
\(738\) 0.226682 0.00834426
\(739\) −45.9404 −1.68994 −0.844972 0.534810i \(-0.820383\pi\)
−0.844972 + 0.534810i \(0.820383\pi\)
\(740\) 8.73917 0.321258
\(741\) 0 0
\(742\) 22.1284 0.812357
\(743\) 51.0114 1.87143 0.935713 0.352763i \(-0.114758\pi\)
0.935713 + 0.352763i \(0.114758\pi\)
\(744\) −3.75877 −0.137803
\(745\) −32.9067 −1.20561
\(746\) 23.9026 0.875137
\(747\) −5.53209 −0.202409
\(748\) −20.7570 −0.758952
\(749\) 30.8621 1.12768
\(750\) −18.3851 −0.671328
\(751\) 36.3696 1.32715 0.663573 0.748112i \(-0.269039\pi\)
0.663573 + 0.748112i \(0.269039\pi\)
\(752\) 7.88713 0.287614
\(753\) −13.1257 −0.478326
\(754\) −16.2567 −0.592034
\(755\) −9.31078 −0.338854
\(756\) 15.0797 0.548442
\(757\) −5.80335 −0.210926 −0.105463 0.994423i \(-0.533633\pi\)
−0.105463 + 0.994423i \(0.533633\pi\)
\(758\) 17.8135 0.647014
\(759\) 3.38919 0.123020
\(760\) 0 0
\(761\) 22.6355 0.820535 0.410268 0.911965i \(-0.365435\pi\)
0.410268 + 0.911965i \(0.365435\pi\)
\(762\) −14.9067 −0.540014
\(763\) −23.4284 −0.848165
\(764\) 9.56212 0.345945
\(765\) 8.50805 0.307609
\(766\) 25.0797 0.906165
\(767\) −3.30541 −0.119351
\(768\) −1.53209 −0.0552845
\(769\) 10.9026 0.393158 0.196579 0.980488i \(-0.437017\pi\)
0.196579 + 0.980488i \(0.437017\pi\)
\(770\) −17.1634 −0.618527
\(771\) −12.2945 −0.442775
\(772\) 23.6810 0.852297
\(773\) 5.25072 0.188855 0.0944277 0.995532i \(-0.469898\pi\)
0.0944277 + 0.995532i \(0.469898\pi\)
\(774\) 3.95811 0.142271
\(775\) 2.45336 0.0881274
\(776\) −0.347296 −0.0124672
\(777\) −18.0392 −0.647153
\(778\) 9.41828 0.337662
\(779\) 0 0
\(780\) −17.6459 −0.631824
\(781\) 26.9222 0.963352
\(782\) −4.52704 −0.161886
\(783\) 15.7980 0.564573
\(784\) 0.260830 0.00931535
\(785\) 16.9067 0.603427
\(786\) −9.90941 −0.353457
\(787\) 2.38743 0.0851027 0.0425514 0.999094i \(-0.486451\pi\)
0.0425514 + 0.999094i \(0.486451\pi\)
\(788\) −22.9222 −0.816570
\(789\) 9.13247 0.325125
\(790\) −18.1284 −0.644978
\(791\) −7.70596 −0.273992
\(792\) 2.07873 0.0738643
\(793\) 16.9067 0.600375
\(794\) −6.85204 −0.243170
\(795\) 25.1634 0.892455
\(796\) 10.0838 0.357410
\(797\) −31.0951 −1.10145 −0.550723 0.834688i \(-0.685648\pi\)
−0.550723 + 0.834688i \(0.685648\pi\)
\(798\) 0 0
\(799\) −51.4047 −1.81857
\(800\) 1.00000 0.0353553
\(801\) 5.05138 0.178482
\(802\) 4.74691 0.167619
\(803\) 50.2131 1.77198
\(804\) −7.59627 −0.267900
\(805\) −3.74329 −0.131934
\(806\) 14.1284 0.497650
\(807\) −1.75877 −0.0619117
\(808\) 0.369585 0.0130020
\(809\) 22.3037 0.784155 0.392077 0.919932i \(-0.371757\pi\)
0.392077 + 0.919932i \(0.371757\pi\)
\(810\) 13.2317 0.464916
\(811\) −3.31902 −0.116547 −0.0582733 0.998301i \(-0.518559\pi\)
−0.0582733 + 0.998301i \(0.518559\pi\)
\(812\) 7.60670 0.266943
\(813\) −30.7256 −1.07759
\(814\) −13.9162 −0.487763
\(815\) 34.0993 1.19444
\(816\) 9.98545 0.349561
\(817\) 0 0
\(818\) 31.6928 1.10811
\(819\) −10.1284 −0.353913
\(820\) 0.694593 0.0242562
\(821\) −1.83244 −0.0639527 −0.0319764 0.999489i \(-0.510180\pi\)
−0.0319764 + 0.999489i \(0.510180\pi\)
\(822\) −17.8648 −0.623108
\(823\) −34.2377 −1.19345 −0.596726 0.802445i \(-0.703532\pi\)
−0.596726 + 0.802445i \(0.703532\pi\)
\(824\) 8.58172 0.298958
\(825\) 4.87939 0.169878
\(826\) 1.54664 0.0538144
\(827\) 19.8658 0.690801 0.345400 0.938455i \(-0.387743\pi\)
0.345400 + 0.938455i \(0.387743\pi\)
\(828\) 0.453363 0.0157555
\(829\) 35.7351 1.24113 0.620565 0.784155i \(-0.286903\pi\)
0.620565 + 0.784155i \(0.286903\pi\)
\(830\) −16.9513 −0.588388
\(831\) −26.5972 −0.922647
\(832\) 5.75877 0.199649
\(833\) −1.69997 −0.0589004
\(834\) −12.6604 −0.438395
\(835\) 6.38507 0.220964
\(836\) 0 0
\(837\) −13.7297 −0.474567
\(838\) 11.0101 0.380338
\(839\) 54.3952 1.87793 0.938965 0.344013i \(-0.111786\pi\)
0.938965 + 0.344013i \(0.111786\pi\)
\(840\) 8.25671 0.284884
\(841\) −21.0310 −0.725206
\(842\) −8.66550 −0.298633
\(843\) 4.47565 0.154150
\(844\) 22.3601 0.769666
\(845\) 40.3269 1.38729
\(846\) 5.14796 0.176990
\(847\) −2.30953 −0.0793563
\(848\) −8.21213 −0.282006
\(849\) 14.5253 0.498506
\(850\) −6.51754 −0.223550
\(851\) −3.03508 −0.104041
\(852\) −12.9513 −0.443704
\(853\) 15.9763 0.547017 0.273509 0.961870i \(-0.411816\pi\)
0.273509 + 0.961870i \(0.411816\pi\)
\(854\) −7.91085 −0.270704
\(855\) 0 0
\(856\) −11.4534 −0.391468
\(857\) 26.0145 0.888640 0.444320 0.895868i \(-0.353445\pi\)
0.444320 + 0.895868i \(0.353445\pi\)
\(858\) 28.0993 0.959293
\(859\) 53.2704 1.81756 0.908782 0.417271i \(-0.137014\pi\)
0.908782 + 0.417271i \(0.137014\pi\)
\(860\) 12.1284 0.413573
\(861\) −1.43376 −0.0488625
\(862\) −29.8871 −1.01796
\(863\) −3.23173 −0.110010 −0.0550048 0.998486i \(-0.517517\pi\)
−0.0550048 + 0.998486i \(0.517517\pi\)
\(864\) −5.59627 −0.190389
\(865\) −19.2608 −0.654888
\(866\) 9.26083 0.314696
\(867\) −39.0351 −1.32570
\(868\) −6.61081 −0.224386
\(869\) 28.8675 0.979264
\(870\) 8.65002 0.293263
\(871\) 28.5526 0.967469
\(872\) 8.69459 0.294436
\(873\) −0.226682 −0.00767201
\(874\) 0 0
\(875\) −32.3351 −1.09313
\(876\) −24.1557 −0.816145
\(877\) −11.7243 −0.395901 −0.197951 0.980212i \(-0.563429\pi\)
−0.197951 + 0.980212i \(0.563429\pi\)
\(878\) 20.8621 0.704063
\(879\) −41.8135 −1.41033
\(880\) 6.36959 0.214719
\(881\) 27.0473 0.911246 0.455623 0.890173i \(-0.349417\pi\)
0.455623 + 0.890173i \(0.349417\pi\)
\(882\) 0.170245 0.00573243
\(883\) 14.4810 0.487325 0.243663 0.969860i \(-0.421651\pi\)
0.243663 + 0.969860i \(0.421651\pi\)
\(884\) −37.5330 −1.26237
\(885\) 1.75877 0.0591204
\(886\) −23.8280 −0.800517
\(887\) 11.5912 0.389195 0.194597 0.980883i \(-0.437660\pi\)
0.194597 + 0.980883i \(0.437660\pi\)
\(888\) 6.69459 0.224656
\(889\) −26.2175 −0.879307
\(890\) 15.4783 0.518835
\(891\) −21.0702 −0.705877
\(892\) 9.27631 0.310594
\(893\) 0 0
\(894\) −25.2080 −0.843082
\(895\) −37.6323 −1.25791
\(896\) −2.69459 −0.0900200
\(897\) 6.12836 0.204620
\(898\) −2.18210 −0.0728178
\(899\) −6.92572 −0.230986
\(900\) 0.652704 0.0217568
\(901\) 53.5229 1.78311
\(902\) −1.10607 −0.0368280
\(903\) −25.0351 −0.833115
\(904\) 2.85978 0.0951150
\(905\) −5.55674 −0.184712
\(906\) −7.13247 −0.236961
\(907\) −41.9786 −1.39388 −0.696939 0.717130i \(-0.745455\pi\)
−0.696939 + 0.717130i \(0.745455\pi\)
\(908\) −7.73648 −0.256744
\(909\) 0.241230 0.00800108
\(910\) −31.0351 −1.02880
\(911\) 44.8675 1.48653 0.743264 0.668999i \(-0.233277\pi\)
0.743264 + 0.668999i \(0.233277\pi\)
\(912\) 0 0
\(913\) 26.9932 0.893344
\(914\) −1.78106 −0.0589122
\(915\) −8.99588 −0.297395
\(916\) −23.0351 −0.761101
\(917\) −17.4284 −0.575536
\(918\) 36.4739 1.20382
\(919\) −32.5270 −1.07297 −0.536484 0.843911i \(-0.680248\pi\)
−0.536484 + 0.843911i \(0.680248\pi\)
\(920\) 1.38919 0.0458001
\(921\) 32.6774 1.07676
\(922\) 15.4884 0.510085
\(923\) 48.6810 1.60235
\(924\) −13.1480 −0.432536
\(925\) −4.36959 −0.143671
\(926\) 2.71007 0.0890586
\(927\) 5.60132 0.183971
\(928\) −2.82295 −0.0926678
\(929\) 11.8402 0.388464 0.194232 0.980956i \(-0.437779\pi\)
0.194232 + 0.980956i \(0.437779\pi\)
\(930\) −7.51754 −0.246510
\(931\) 0 0
\(932\) 8.39961 0.275139
\(933\) 44.8776 1.46923
\(934\) −12.9135 −0.422544
\(935\) −41.5140 −1.35765
\(936\) 3.75877 0.122859
\(937\) −0.763823 −0.0249530 −0.0124765 0.999922i \(-0.503971\pi\)
−0.0124765 + 0.999922i \(0.503971\pi\)
\(938\) −13.3601 −0.436222
\(939\) −8.52023 −0.278047
\(940\) 15.7743 0.514499
\(941\) 21.4938 0.700679 0.350339 0.936623i \(-0.386066\pi\)
0.350339 + 0.936623i \(0.386066\pi\)
\(942\) 12.9513 0.421976
\(943\) −0.241230 −0.00785551
\(944\) −0.573978 −0.0186814
\(945\) 30.1593 0.981083
\(946\) −19.3131 −0.627925
\(947\) 31.6851 1.02963 0.514814 0.857302i \(-0.327861\pi\)
0.514814 + 0.857302i \(0.327861\pi\)
\(948\) −13.8871 −0.451033
\(949\) 90.7957 2.94735
\(950\) 0 0
\(951\) 5.82707 0.188956
\(952\) 17.5621 0.569192
\(953\) −55.8711 −1.80984 −0.904922 0.425577i \(-0.860071\pi\)
−0.904922 + 0.425577i \(0.860071\pi\)
\(954\) −5.36009 −0.173539
\(955\) 19.1242 0.618846
\(956\) −15.7297 −0.508734
\(957\) −13.7743 −0.445258
\(958\) 18.5526 0.599408
\(959\) −31.4201 −1.01461
\(960\) −3.06418 −0.0988959
\(961\) −24.9810 −0.805839
\(962\) −25.1634 −0.811302
\(963\) −7.47565 −0.240900
\(964\) −17.5030 −0.563733
\(965\) 47.3620 1.52464
\(966\) −2.86753 −0.0922611
\(967\) 21.5567 0.693218 0.346609 0.938010i \(-0.387333\pi\)
0.346609 + 0.938010i \(0.387333\pi\)
\(968\) 0.857097 0.0275481
\(969\) 0 0
\(970\) −0.694593 −0.0223020
\(971\) −48.6563 −1.56146 −0.780728 0.624871i \(-0.785151\pi\)
−0.780728 + 0.624871i \(0.785151\pi\)
\(972\) −6.65270 −0.213386
\(973\) −22.2668 −0.713841
\(974\) 41.1735 1.31929
\(975\) 8.82295 0.282560
\(976\) 2.93582 0.0939734
\(977\) 50.5482 1.61718 0.808590 0.588373i \(-0.200231\pi\)
0.808590 + 0.588373i \(0.200231\pi\)
\(978\) 26.1215 0.835275
\(979\) −24.6477 −0.787742
\(980\) 0.521660 0.0166638
\(981\) 5.67499 0.181189
\(982\) 22.5776 0.720480
\(983\) 30.0601 0.958767 0.479383 0.877606i \(-0.340860\pi\)
0.479383 + 0.877606i \(0.340860\pi\)
\(984\) 0.532089 0.0169624
\(985\) −45.8444 −1.46072
\(986\) 18.3987 0.585933
\(987\) −32.5609 −1.03642
\(988\) 0 0
\(989\) −4.21213 −0.133938
\(990\) 4.15745 0.132132
\(991\) −2.75278 −0.0874451 −0.0437225 0.999044i \(-0.513922\pi\)
−0.0437225 + 0.999044i \(0.513922\pi\)
\(992\) 2.45336 0.0778944
\(993\) 31.2882 0.992900
\(994\) −22.7784 −0.722486
\(995\) 20.1676 0.639355
\(996\) −12.9855 −0.411460
\(997\) 8.70470 0.275681 0.137840 0.990454i \(-0.455984\pi\)
0.137840 + 0.990454i \(0.455984\pi\)
\(998\) 29.3286 0.928382
\(999\) 24.4534 0.773670
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 722.2.a.k.1.1 3
3.2 odd 2 6498.2.a.bq.1.2 3
4.3 odd 2 5776.2.a.bo.1.3 3
19.2 odd 18 722.2.e.b.99.1 6
19.3 odd 18 722.2.e.m.389.1 6
19.4 even 9 722.2.e.k.415.1 6
19.5 even 9 722.2.e.k.595.1 6
19.6 even 9 722.2.e.a.245.1 6
19.7 even 3 722.2.c.l.429.3 6
19.8 odd 6 722.2.c.k.653.1 6
19.9 even 9 722.2.e.l.423.1 6
19.10 odd 18 722.2.e.b.423.1 6
19.11 even 3 722.2.c.l.653.3 6
19.12 odd 6 722.2.c.k.429.1 6
19.13 odd 18 722.2.e.m.245.1 6
19.14 odd 18 38.2.e.a.25.1 6
19.15 odd 18 38.2.e.a.35.1 yes 6
19.16 even 9 722.2.e.a.389.1 6
19.17 even 9 722.2.e.l.99.1 6
19.18 odd 2 722.2.a.l.1.3 3
57.14 even 18 342.2.u.c.253.1 6
57.53 even 18 342.2.u.c.73.1 6
57.56 even 2 6498.2.a.bl.1.2 3
76.15 even 18 304.2.u.c.225.1 6
76.71 even 18 304.2.u.c.177.1 6
76.75 even 2 5776.2.a.bn.1.1 3
95.14 odd 18 950.2.l.d.101.1 6
95.33 even 36 950.2.u.b.899.1 12
95.34 odd 18 950.2.l.d.301.1 6
95.52 even 36 950.2.u.b.899.2 12
95.53 even 36 950.2.u.b.149.2 12
95.72 even 36 950.2.u.b.149.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.2.e.a.25.1 6 19.14 odd 18
38.2.e.a.35.1 yes 6 19.15 odd 18
304.2.u.c.177.1 6 76.71 even 18
304.2.u.c.225.1 6 76.15 even 18
342.2.u.c.73.1 6 57.53 even 18
342.2.u.c.253.1 6 57.14 even 18
722.2.a.k.1.1 3 1.1 even 1 trivial
722.2.a.l.1.3 3 19.18 odd 2
722.2.c.k.429.1 6 19.12 odd 6
722.2.c.k.653.1 6 19.8 odd 6
722.2.c.l.429.3 6 19.7 even 3
722.2.c.l.653.3 6 19.11 even 3
722.2.e.a.245.1 6 19.6 even 9
722.2.e.a.389.1 6 19.16 even 9
722.2.e.b.99.1 6 19.2 odd 18
722.2.e.b.423.1 6 19.10 odd 18
722.2.e.k.415.1 6 19.4 even 9
722.2.e.k.595.1 6 19.5 even 9
722.2.e.l.99.1 6 19.17 even 9
722.2.e.l.423.1 6 19.9 even 9
722.2.e.m.245.1 6 19.13 odd 18
722.2.e.m.389.1 6 19.3 odd 18
950.2.l.d.101.1 6 95.14 odd 18
950.2.l.d.301.1 6 95.34 odd 18
950.2.u.b.149.1 12 95.72 even 36
950.2.u.b.149.2 12 95.53 even 36
950.2.u.b.899.1 12 95.33 even 36
950.2.u.b.899.2 12 95.52 even 36
5776.2.a.bn.1.1 3 76.75 even 2
5776.2.a.bo.1.3 3 4.3 odd 2
6498.2.a.bl.1.2 3 57.56 even 2
6498.2.a.bq.1.2 3 3.2 odd 2