Properties

Label 2001.2.a.i.1.5
Level $2001$
Weight $2$
Character 2001.1
Self dual yes
Analytic conductor $15.978$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2001 = 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2001.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.9780654445\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 3 x^{6} - 5 x^{5} + 18 x^{4} + 4 x^{3} - 26 x^{2} + x + 8\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-0.586186\) of defining polynomial
Character \(\chi\) \(=\) 2001.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.586186 q^{2} +1.00000 q^{3} -1.65639 q^{4} -0.842866 q^{5} +0.586186 q^{6} +1.23837 q^{7} -2.14332 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.586186 q^{2} +1.00000 q^{3} -1.65639 q^{4} -0.842866 q^{5} +0.586186 q^{6} +1.23837 q^{7} -2.14332 q^{8} +1.00000 q^{9} -0.494076 q^{10} +1.47020 q^{11} -1.65639 q^{12} -3.26028 q^{13} +0.725912 q^{14} -0.842866 q^{15} +2.05639 q^{16} -4.93493 q^{17} +0.586186 q^{18} +0.260886 q^{19} +1.39611 q^{20} +1.23837 q^{21} +0.861812 q^{22} +1.00000 q^{23} -2.14332 q^{24} -4.28958 q^{25} -1.91113 q^{26} +1.00000 q^{27} -2.05121 q^{28} -1.00000 q^{29} -0.494076 q^{30} -0.101716 q^{31} +5.49207 q^{32} +1.47020 q^{33} -2.89279 q^{34} -1.04378 q^{35} -1.65639 q^{36} -3.33423 q^{37} +0.152928 q^{38} -3.26028 q^{39} +1.80653 q^{40} -0.732484 q^{41} +0.725912 q^{42} -5.01063 q^{43} -2.43522 q^{44} -0.842866 q^{45} +0.586186 q^{46} -8.71377 q^{47} +2.05639 q^{48} -5.46645 q^{49} -2.51449 q^{50} -4.93493 q^{51} +5.40027 q^{52} -4.24097 q^{53} +0.586186 q^{54} -1.23918 q^{55} -2.65422 q^{56} +0.260886 q^{57} -0.586186 q^{58} +8.76778 q^{59} +1.39611 q^{60} -1.88602 q^{61} -0.0596245 q^{62} +1.23837 q^{63} -0.893403 q^{64} +2.74797 q^{65} +0.861812 q^{66} -3.21685 q^{67} +8.17415 q^{68} +1.00000 q^{69} -0.611847 q^{70} -1.72395 q^{71} -2.14332 q^{72} -2.64612 q^{73} -1.95448 q^{74} -4.28958 q^{75} -0.432129 q^{76} +1.82065 q^{77} -1.91113 q^{78} +0.465342 q^{79} -1.73326 q^{80} +1.00000 q^{81} -0.429372 q^{82} -8.34128 q^{83} -2.05121 q^{84} +4.15949 q^{85} -2.93716 q^{86} -1.00000 q^{87} -3.15112 q^{88} -5.13593 q^{89} -0.494076 q^{90} -4.03741 q^{91} -1.65639 q^{92} -0.101716 q^{93} -5.10789 q^{94} -0.219892 q^{95} +5.49207 q^{96} -6.87730 q^{97} -3.20436 q^{98} +1.47020 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q - 3q^{2} + 7q^{3} + 5q^{4} - 3q^{5} - 3q^{6} - 5q^{7} - 6q^{8} + 7q^{9} + O(q^{10}) \) \( 7q - 3q^{2} + 7q^{3} + 5q^{4} - 3q^{5} - 3q^{6} - 5q^{7} - 6q^{8} + 7q^{9} + 3q^{10} - 4q^{11} + 5q^{12} - 18q^{13} - 2q^{14} - 3q^{15} - 7q^{16} - 3q^{17} - 3q^{18} - 4q^{19} - 2q^{20} - 5q^{21} - 26q^{22} + 7q^{23} - 6q^{24} - 8q^{25} - 7q^{26} + 7q^{27} - 6q^{28} - 7q^{29} + 3q^{30} - 22q^{31} + 5q^{32} - 4q^{33} + 9q^{34} + 3q^{35} + 5q^{36} - 25q^{37} + 14q^{38} - 18q^{39} - 10q^{40} - 13q^{41} - 2q^{42} - 2q^{43} + 4q^{44} - 3q^{45} - 3q^{46} - 25q^{47} - 7q^{48} - 8q^{49} + 19q^{50} - 3q^{51} - 12q^{52} - 5q^{53} - 3q^{54} - 15q^{55} + 18q^{56} - 4q^{57} + 3q^{58} + 11q^{59} - 2q^{60} - 33q^{61} + 28q^{62} - 5q^{63} - 14q^{64} - 2q^{65} - 26q^{66} + 8q^{67} + 12q^{68} + 7q^{69} - 22q^{70} - 6q^{71} - 6q^{72} + 15q^{73} + 34q^{74} - 8q^{75} - 28q^{76} - q^{77} - 7q^{78} - 15q^{79} - 12q^{80} + 7q^{81} - 14q^{82} + 21q^{83} - 6q^{84} - 28q^{85} - 12q^{86} - 7q^{87} - 13q^{88} + 8q^{89} + 3q^{90} + 6q^{91} + 5q^{92} - 22q^{93} - 35q^{94} - 25q^{95} + 5q^{96} + 13q^{97} + q^{98} - 4q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.586186 0.414496 0.207248 0.978288i \(-0.433549\pi\)
0.207248 + 0.978288i \(0.433549\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.65639 −0.828193
\(5\) −0.842866 −0.376941 −0.188471 0.982079i \(-0.560353\pi\)
−0.188471 + 0.982079i \(0.560353\pi\)
\(6\) 0.586186 0.239309
\(7\) 1.23837 0.468058 0.234029 0.972230i \(-0.424809\pi\)
0.234029 + 0.972230i \(0.424809\pi\)
\(8\) −2.14332 −0.757779
\(9\) 1.00000 0.333333
\(10\) −0.494076 −0.156241
\(11\) 1.47020 0.443283 0.221641 0.975128i \(-0.428859\pi\)
0.221641 + 0.975128i \(0.428859\pi\)
\(12\) −1.65639 −0.478158
\(13\) −3.26028 −0.904238 −0.452119 0.891958i \(-0.649332\pi\)
−0.452119 + 0.891958i \(0.649332\pi\)
\(14\) 0.725912 0.194008
\(15\) −0.842866 −0.217627
\(16\) 2.05639 0.514097
\(17\) −4.93493 −1.19690 −0.598448 0.801161i \(-0.704216\pi\)
−0.598448 + 0.801161i \(0.704216\pi\)
\(18\) 0.586186 0.138165
\(19\) 0.260886 0.0598514 0.0299257 0.999552i \(-0.490473\pi\)
0.0299257 + 0.999552i \(0.490473\pi\)
\(20\) 1.39611 0.312180
\(21\) 1.23837 0.270234
\(22\) 0.861812 0.183739
\(23\) 1.00000 0.208514
\(24\) −2.14332 −0.437504
\(25\) −4.28958 −0.857915
\(26\) −1.91113 −0.374803
\(27\) 1.00000 0.192450
\(28\) −2.05121 −0.387643
\(29\) −1.00000 −0.185695
\(30\) −0.494076 −0.0902055
\(31\) −0.101716 −0.0182688 −0.00913438 0.999958i \(-0.502908\pi\)
−0.00913438 + 0.999958i \(0.502908\pi\)
\(32\) 5.49207 0.970870
\(33\) 1.47020 0.255929
\(34\) −2.89279 −0.496109
\(35\) −1.04378 −0.176430
\(36\) −1.65639 −0.276064
\(37\) −3.33423 −0.548144 −0.274072 0.961709i \(-0.588371\pi\)
−0.274072 + 0.961709i \(0.588371\pi\)
\(38\) 0.152928 0.0248082
\(39\) −3.26028 −0.522062
\(40\) 1.80653 0.285638
\(41\) −0.732484 −0.114395 −0.0571974 0.998363i \(-0.518216\pi\)
−0.0571974 + 0.998363i \(0.518216\pi\)
\(42\) 0.725912 0.112011
\(43\) −5.01063 −0.764114 −0.382057 0.924139i \(-0.624784\pi\)
−0.382057 + 0.924139i \(0.624784\pi\)
\(44\) −2.43522 −0.367124
\(45\) −0.842866 −0.125647
\(46\) 0.586186 0.0864284
\(47\) −8.71377 −1.27103 −0.635517 0.772087i \(-0.719213\pi\)
−0.635517 + 0.772087i \(0.719213\pi\)
\(48\) 2.05639 0.296814
\(49\) −5.46645 −0.780922
\(50\) −2.51449 −0.355602
\(51\) −4.93493 −0.691029
\(52\) 5.40027 0.748883
\(53\) −4.24097 −0.582541 −0.291271 0.956641i \(-0.594078\pi\)
−0.291271 + 0.956641i \(0.594078\pi\)
\(54\) 0.586186 0.0797698
\(55\) −1.23918 −0.167091
\(56\) −2.65422 −0.354685
\(57\) 0.260886 0.0345552
\(58\) −0.586186 −0.0769700
\(59\) 8.76778 1.14147 0.570734 0.821135i \(-0.306659\pi\)
0.570734 + 0.821135i \(0.306659\pi\)
\(60\) 1.39611 0.180237
\(61\) −1.88602 −0.241481 −0.120740 0.992684i \(-0.538527\pi\)
−0.120740 + 0.992684i \(0.538527\pi\)
\(62\) −0.0596245 −0.00757232
\(63\) 1.23837 0.156019
\(64\) −0.893403 −0.111675
\(65\) 2.74797 0.340844
\(66\) 0.861812 0.106082
\(67\) −3.21685 −0.393001 −0.196500 0.980504i \(-0.562958\pi\)
−0.196500 + 0.980504i \(0.562958\pi\)
\(68\) 8.17415 0.991262
\(69\) 1.00000 0.120386
\(70\) −0.611847 −0.0731297
\(71\) −1.72395 −0.204595 −0.102297 0.994754i \(-0.532619\pi\)
−0.102297 + 0.994754i \(0.532619\pi\)
\(72\) −2.14332 −0.252593
\(73\) −2.64612 −0.309705 −0.154852 0.987938i \(-0.549490\pi\)
−0.154852 + 0.987938i \(0.549490\pi\)
\(74\) −1.95448 −0.227203
\(75\) −4.28958 −0.495318
\(76\) −0.432129 −0.0495685
\(77\) 1.82065 0.207482
\(78\) −1.91113 −0.216393
\(79\) 0.465342 0.0523551 0.0261775 0.999657i \(-0.491666\pi\)
0.0261775 + 0.999657i \(0.491666\pi\)
\(80\) −1.73326 −0.193784
\(81\) 1.00000 0.111111
\(82\) −0.429372 −0.0474162
\(83\) −8.34128 −0.915575 −0.457787 0.889062i \(-0.651358\pi\)
−0.457787 + 0.889062i \(0.651358\pi\)
\(84\) −2.05121 −0.223806
\(85\) 4.15949 0.451160
\(86\) −2.93716 −0.316722
\(87\) −1.00000 −0.107211
\(88\) −3.15112 −0.335910
\(89\) −5.13593 −0.544408 −0.272204 0.962240i \(-0.587753\pi\)
−0.272204 + 0.962240i \(0.587753\pi\)
\(90\) −0.494076 −0.0520802
\(91\) −4.03741 −0.423236
\(92\) −1.65639 −0.172690
\(93\) −0.101716 −0.0105475
\(94\) −5.10789 −0.526838
\(95\) −0.219892 −0.0225605
\(96\) 5.49207 0.560532
\(97\) −6.87730 −0.698284 −0.349142 0.937070i \(-0.613527\pi\)
−0.349142 + 0.937070i \(0.613527\pi\)
\(98\) −3.20436 −0.323689
\(99\) 1.47020 0.147761
\(100\) 7.10520 0.710520
\(101\) −15.4552 −1.53785 −0.768923 0.639342i \(-0.779207\pi\)
−0.768923 + 0.639342i \(0.779207\pi\)
\(102\) −2.89279 −0.286429
\(103\) 9.59885 0.945803 0.472901 0.881115i \(-0.343207\pi\)
0.472901 + 0.881115i \(0.343207\pi\)
\(104\) 6.98782 0.685212
\(105\) −1.04378 −0.101862
\(106\) −2.48599 −0.241461
\(107\) 18.4638 1.78496 0.892480 0.451088i \(-0.148964\pi\)
0.892480 + 0.451088i \(0.148964\pi\)
\(108\) −1.65639 −0.159386
\(109\) −13.6145 −1.30403 −0.652017 0.758204i \(-0.726077\pi\)
−0.652017 + 0.758204i \(0.726077\pi\)
\(110\) −0.726391 −0.0692587
\(111\) −3.33423 −0.316471
\(112\) 2.54656 0.240627
\(113\) −9.49007 −0.892750 −0.446375 0.894846i \(-0.647285\pi\)
−0.446375 + 0.894846i \(0.647285\pi\)
\(114\) 0.152928 0.0143230
\(115\) −0.842866 −0.0785976
\(116\) 1.65639 0.153792
\(117\) −3.26028 −0.301413
\(118\) 5.13955 0.473134
\(119\) −6.11125 −0.560217
\(120\) 1.80653 0.164913
\(121\) −8.83851 −0.803501
\(122\) −1.10556 −0.100093
\(123\) −0.732484 −0.0660459
\(124\) 0.168481 0.0151301
\(125\) 7.82987 0.700325
\(126\) 0.725912 0.0646694
\(127\) 19.1350 1.69796 0.848980 0.528426i \(-0.177217\pi\)
0.848980 + 0.528426i \(0.177217\pi\)
\(128\) −11.5078 −1.01716
\(129\) −5.01063 −0.441161
\(130\) 1.61082 0.141279
\(131\) 14.4038 1.25846 0.629232 0.777218i \(-0.283370\pi\)
0.629232 + 0.777218i \(0.283370\pi\)
\(132\) −2.43522 −0.211959
\(133\) 0.323073 0.0280140
\(134\) −1.88567 −0.162897
\(135\) −0.842866 −0.0725423
\(136\) 10.5771 0.906983
\(137\) 10.9664 0.936922 0.468461 0.883484i \(-0.344809\pi\)
0.468461 + 0.883484i \(0.344809\pi\)
\(138\) 0.586186 0.0498994
\(139\) −10.7820 −0.914519 −0.457259 0.889333i \(-0.651169\pi\)
−0.457259 + 0.889333i \(0.651169\pi\)
\(140\) 1.72890 0.146118
\(141\) −8.71377 −0.733832
\(142\) −1.01055 −0.0848038
\(143\) −4.79326 −0.400833
\(144\) 2.05639 0.171366
\(145\) 0.842866 0.0699962
\(146\) −1.55112 −0.128371
\(147\) −5.46645 −0.450865
\(148\) 5.52277 0.453969
\(149\) −14.9598 −1.22555 −0.612776 0.790257i \(-0.709947\pi\)
−0.612776 + 0.790257i \(0.709947\pi\)
\(150\) −2.51449 −0.205307
\(151\) −4.51476 −0.367406 −0.183703 0.982982i \(-0.558808\pi\)
−0.183703 + 0.982982i \(0.558808\pi\)
\(152\) −0.559163 −0.0453541
\(153\) −4.93493 −0.398966
\(154\) 1.06724 0.0860005
\(155\) 0.0857330 0.00688624
\(156\) 5.40027 0.432368
\(157\) −16.2289 −1.29521 −0.647604 0.761977i \(-0.724229\pi\)
−0.647604 + 0.761977i \(0.724229\pi\)
\(158\) 0.272777 0.0217010
\(159\) −4.24097 −0.336330
\(160\) −4.62908 −0.365961
\(161\) 1.23837 0.0975969
\(162\) 0.586186 0.0460551
\(163\) −2.54577 −0.199400 −0.0997001 0.995018i \(-0.531788\pi\)
−0.0997001 + 0.995018i \(0.531788\pi\)
\(164\) 1.21328 0.0947410
\(165\) −1.23918 −0.0964703
\(166\) −4.88954 −0.379502
\(167\) 19.0688 1.47559 0.737794 0.675026i \(-0.235868\pi\)
0.737794 + 0.675026i \(0.235868\pi\)
\(168\) −2.65422 −0.204777
\(169\) −2.37061 −0.182354
\(170\) 2.43823 0.187004
\(171\) 0.260886 0.0199505
\(172\) 8.29954 0.632834
\(173\) −9.37891 −0.713065 −0.356533 0.934283i \(-0.616041\pi\)
−0.356533 + 0.934283i \(0.616041\pi\)
\(174\) −0.586186 −0.0444386
\(175\) −5.31207 −0.401554
\(176\) 3.02331 0.227890
\(177\) 8.76778 0.659026
\(178\) −3.01061 −0.225655
\(179\) 4.33101 0.323715 0.161857 0.986814i \(-0.448252\pi\)
0.161857 + 0.986814i \(0.448252\pi\)
\(180\) 1.39611 0.104060
\(181\) 20.4714 1.52163 0.760813 0.648971i \(-0.224801\pi\)
0.760813 + 0.648971i \(0.224801\pi\)
\(182\) −2.36667 −0.175430
\(183\) −1.88602 −0.139419
\(184\) −2.14332 −0.158008
\(185\) 2.81031 0.206618
\(186\) −0.0596245 −0.00437188
\(187\) −7.25535 −0.530564
\(188\) 14.4334 1.05266
\(189\) 1.23837 0.0900778
\(190\) −0.128898 −0.00935122
\(191\) 7.10652 0.514210 0.257105 0.966383i \(-0.417231\pi\)
0.257105 + 0.966383i \(0.417231\pi\)
\(192\) −0.893403 −0.0644758
\(193\) 2.37896 0.171242 0.0856208 0.996328i \(-0.472713\pi\)
0.0856208 + 0.996328i \(0.472713\pi\)
\(194\) −4.03138 −0.289436
\(195\) 2.74797 0.196787
\(196\) 9.05455 0.646754
\(197\) 4.61033 0.328472 0.164236 0.986421i \(-0.447484\pi\)
0.164236 + 0.986421i \(0.447484\pi\)
\(198\) 0.861812 0.0612463
\(199\) −12.7702 −0.905254 −0.452627 0.891700i \(-0.649513\pi\)
−0.452627 + 0.891700i \(0.649513\pi\)
\(200\) 9.19394 0.650110
\(201\) −3.21685 −0.226899
\(202\) −9.05959 −0.637431
\(203\) −1.23837 −0.0869162
\(204\) 8.17415 0.572305
\(205\) 0.617386 0.0431201
\(206\) 5.62671 0.392031
\(207\) 1.00000 0.0695048
\(208\) −6.70439 −0.464866
\(209\) 0.383556 0.0265311
\(210\) −0.611847 −0.0422214
\(211\) −8.75135 −0.602468 −0.301234 0.953550i \(-0.597398\pi\)
−0.301234 + 0.953550i \(0.597398\pi\)
\(212\) 7.02468 0.482457
\(213\) −1.72395 −0.118123
\(214\) 10.8232 0.739858
\(215\) 4.22329 0.288026
\(216\) −2.14332 −0.145835
\(217\) −0.125962 −0.00855084
\(218\) −7.98064 −0.540517
\(219\) −2.64612 −0.178808
\(220\) 2.05257 0.138384
\(221\) 16.0892 1.08228
\(222\) −1.95448 −0.131176
\(223\) 29.3753 1.96712 0.983559 0.180586i \(-0.0577993\pi\)
0.983559 + 0.180586i \(0.0577993\pi\)
\(224\) 6.80119 0.454424
\(225\) −4.28958 −0.285972
\(226\) −5.56294 −0.370041
\(227\) 21.4702 1.42503 0.712514 0.701658i \(-0.247557\pi\)
0.712514 + 0.701658i \(0.247557\pi\)
\(228\) −0.432129 −0.0286184
\(229\) 6.14700 0.406205 0.203103 0.979157i \(-0.434898\pi\)
0.203103 + 0.979157i \(0.434898\pi\)
\(230\) −0.494076 −0.0325784
\(231\) 1.82065 0.119790
\(232\) 2.14332 0.140716
\(233\) 21.0310 1.37778 0.688892 0.724864i \(-0.258097\pi\)
0.688892 + 0.724864i \(0.258097\pi\)
\(234\) −1.91113 −0.124934
\(235\) 7.34454 0.479105
\(236\) −14.5228 −0.945355
\(237\) 0.465342 0.0302272
\(238\) −3.58233 −0.232208
\(239\) −22.6235 −1.46339 −0.731697 0.681630i \(-0.761271\pi\)
−0.731697 + 0.681630i \(0.761271\pi\)
\(240\) −1.73326 −0.111881
\(241\) 16.1684 1.04150 0.520749 0.853710i \(-0.325653\pi\)
0.520749 + 0.853710i \(0.325653\pi\)
\(242\) −5.18101 −0.333048
\(243\) 1.00000 0.0641500
\(244\) 3.12398 0.199993
\(245\) 4.60748 0.294361
\(246\) −0.429372 −0.0273758
\(247\) −0.850561 −0.0541199
\(248\) 0.218010 0.0138437
\(249\) −8.34128 −0.528607
\(250\) 4.58976 0.290282
\(251\) 4.88555 0.308373 0.154186 0.988042i \(-0.450724\pi\)
0.154186 + 0.988042i \(0.450724\pi\)
\(252\) −2.05121 −0.129214
\(253\) 1.47020 0.0924308
\(254\) 11.2167 0.703797
\(255\) 4.15949 0.260477
\(256\) −4.95892 −0.309933
\(257\) 14.0680 0.877539 0.438770 0.898600i \(-0.355414\pi\)
0.438770 + 0.898600i \(0.355414\pi\)
\(258\) −2.93716 −0.182860
\(259\) −4.12899 −0.256563
\(260\) −4.55171 −0.282285
\(261\) −1.00000 −0.0618984
\(262\) 8.44329 0.521628
\(263\) −15.7419 −0.970687 −0.485344 0.874323i \(-0.661306\pi\)
−0.485344 + 0.874323i \(0.661306\pi\)
\(264\) −3.15112 −0.193938
\(265\) 3.57456 0.219584
\(266\) 0.189381 0.0116117
\(267\) −5.13593 −0.314314
\(268\) 5.32834 0.325480
\(269\) 5.80954 0.354214 0.177107 0.984192i \(-0.443326\pi\)
0.177107 + 0.984192i \(0.443326\pi\)
\(270\) −0.494076 −0.0300685
\(271\) 13.5144 0.820944 0.410472 0.911873i \(-0.365364\pi\)
0.410472 + 0.911873i \(0.365364\pi\)
\(272\) −10.1481 −0.615321
\(273\) −4.03741 −0.244355
\(274\) 6.42834 0.388350
\(275\) −6.30655 −0.380299
\(276\) −1.65639 −0.0997027
\(277\) −14.1794 −0.851956 −0.425978 0.904733i \(-0.640070\pi\)
−0.425978 + 0.904733i \(0.640070\pi\)
\(278\) −6.32027 −0.379064
\(279\) −0.101716 −0.00608958
\(280\) 2.23715 0.133695
\(281\) −0.303631 −0.0181131 −0.00905656 0.999959i \(-0.502883\pi\)
−0.00905656 + 0.999959i \(0.502883\pi\)
\(282\) −5.10789 −0.304170
\(283\) 17.4684 1.03839 0.519195 0.854656i \(-0.326232\pi\)
0.519195 + 0.854656i \(0.326232\pi\)
\(284\) 2.85552 0.169444
\(285\) −0.219892 −0.0130253
\(286\) −2.80974 −0.166144
\(287\) −0.907084 −0.0535434
\(288\) 5.49207 0.323623
\(289\) 7.35356 0.432562
\(290\) 0.494076 0.0290131
\(291\) −6.87730 −0.403155
\(292\) 4.38300 0.256495
\(293\) 18.0182 1.05263 0.526317 0.850289i \(-0.323573\pi\)
0.526317 + 0.850289i \(0.323573\pi\)
\(294\) −3.20436 −0.186882
\(295\) −7.39006 −0.430266
\(296\) 7.14632 0.415371
\(297\) 1.47020 0.0853098
\(298\) −8.76920 −0.507986
\(299\) −3.26028 −0.188547
\(300\) 7.10520 0.410219
\(301\) −6.20499 −0.357650
\(302\) −2.64649 −0.152288
\(303\) −15.4552 −0.887875
\(304\) 0.536484 0.0307694
\(305\) 1.58966 0.0910239
\(306\) −2.89279 −0.165370
\(307\) 14.8992 0.850340 0.425170 0.905113i \(-0.360214\pi\)
0.425170 + 0.905113i \(0.360214\pi\)
\(308\) −3.01570 −0.171835
\(309\) 9.59885 0.546060
\(310\) 0.0502555 0.00285432
\(311\) −1.83728 −0.104183 −0.0520914 0.998642i \(-0.516589\pi\)
−0.0520914 + 0.998642i \(0.516589\pi\)
\(312\) 6.98782 0.395607
\(313\) 4.46857 0.252579 0.126289 0.991993i \(-0.459693\pi\)
0.126289 + 0.991993i \(0.459693\pi\)
\(314\) −9.51315 −0.536858
\(315\) −1.04378 −0.0588101
\(316\) −0.770786 −0.0433601
\(317\) 15.9563 0.896194 0.448097 0.893985i \(-0.352102\pi\)
0.448097 + 0.893985i \(0.352102\pi\)
\(318\) −2.48599 −0.139408
\(319\) −1.47020 −0.0823155
\(320\) 0.753019 0.0420950
\(321\) 18.4638 1.03055
\(322\) 0.725912 0.0404535
\(323\) −1.28746 −0.0716360
\(324\) −1.65639 −0.0920215
\(325\) 13.9852 0.775759
\(326\) −1.49230 −0.0826506
\(327\) −13.6145 −0.752885
\(328\) 1.56995 0.0866860
\(329\) −10.7908 −0.594918
\(330\) −0.726391 −0.0399865
\(331\) −35.5195 −1.95233 −0.976164 0.217035i \(-0.930362\pi\)
−0.976164 + 0.217035i \(0.930362\pi\)
\(332\) 13.8164 0.758273
\(333\) −3.33423 −0.182715
\(334\) 11.1779 0.611625
\(335\) 2.71137 0.148138
\(336\) 2.54656 0.138926
\(337\) −20.6824 −1.12664 −0.563322 0.826237i \(-0.690477\pi\)
−0.563322 + 0.826237i \(0.690477\pi\)
\(338\) −1.38962 −0.0755851
\(339\) −9.49007 −0.515430
\(340\) −6.88971 −0.373647
\(341\) −0.149543 −0.00809822
\(342\) 0.152928 0.00826939
\(343\) −15.4380 −0.833575
\(344\) 10.7394 0.579029
\(345\) −0.842866 −0.0453784
\(346\) −5.49778 −0.295563
\(347\) 13.4784 0.723557 0.361779 0.932264i \(-0.382170\pi\)
0.361779 + 0.932264i \(0.382170\pi\)
\(348\) 1.65639 0.0887916
\(349\) −32.9095 −1.76160 −0.880801 0.473486i \(-0.842995\pi\)
−0.880801 + 0.473486i \(0.842995\pi\)
\(350\) −3.11386 −0.166443
\(351\) −3.26028 −0.174021
\(352\) 8.07445 0.430370
\(353\) 23.8249 1.26807 0.634035 0.773305i \(-0.281398\pi\)
0.634035 + 0.773305i \(0.281398\pi\)
\(354\) 5.13955 0.273164
\(355\) 1.45306 0.0771202
\(356\) 8.50709 0.450875
\(357\) −6.11125 −0.323442
\(358\) 2.53877 0.134178
\(359\) 26.2175 1.38370 0.691852 0.722039i \(-0.256795\pi\)
0.691852 + 0.722039i \(0.256795\pi\)
\(360\) 1.80653 0.0952126
\(361\) −18.9319 −0.996418
\(362\) 12.0000 0.630708
\(363\) −8.83851 −0.463901
\(364\) 6.68751 0.350521
\(365\) 2.23032 0.116740
\(366\) −1.10556 −0.0577886
\(367\) 9.01887 0.470781 0.235391 0.971901i \(-0.424363\pi\)
0.235391 + 0.971901i \(0.424363\pi\)
\(368\) 2.05639 0.107197
\(369\) −0.732484 −0.0381316
\(370\) 1.64736 0.0856422
\(371\) −5.25187 −0.272663
\(372\) 0.168481 0.00873534
\(373\) −8.59391 −0.444976 −0.222488 0.974935i \(-0.571418\pi\)
−0.222488 + 0.974935i \(0.571418\pi\)
\(374\) −4.25298 −0.219916
\(375\) 7.82987 0.404333
\(376\) 18.6764 0.963162
\(377\) 3.26028 0.167913
\(378\) 0.725912 0.0373369
\(379\) 8.48972 0.436088 0.218044 0.975939i \(-0.430032\pi\)
0.218044 + 0.975939i \(0.430032\pi\)
\(380\) 0.364226 0.0186844
\(381\) 19.1350 0.980317
\(382\) 4.16574 0.213138
\(383\) 10.6436 0.543865 0.271933 0.962316i \(-0.412337\pi\)
0.271933 + 0.962316i \(0.412337\pi\)
\(384\) −11.5078 −0.587257
\(385\) −1.53456 −0.0782085
\(386\) 1.39451 0.0709789
\(387\) −5.01063 −0.254705
\(388\) 11.3915 0.578314
\(389\) −14.6896 −0.744794 −0.372397 0.928074i \(-0.621464\pi\)
−0.372397 + 0.928074i \(0.621464\pi\)
\(390\) 1.61082 0.0815672
\(391\) −4.93493 −0.249570
\(392\) 11.7164 0.591766
\(393\) 14.4038 0.726574
\(394\) 2.70251 0.136150
\(395\) −0.392221 −0.0197348
\(396\) −2.43522 −0.122375
\(397\) 7.75333 0.389128 0.194564 0.980890i \(-0.437671\pi\)
0.194564 + 0.980890i \(0.437671\pi\)
\(398\) −7.48570 −0.375224
\(399\) 0.323073 0.0161739
\(400\) −8.82103 −0.441052
\(401\) −1.17706 −0.0587797 −0.0293899 0.999568i \(-0.509356\pi\)
−0.0293899 + 0.999568i \(0.509356\pi\)
\(402\) −1.88567 −0.0940487
\(403\) 0.331623 0.0165193
\(404\) 25.5997 1.27363
\(405\) −0.842866 −0.0418823
\(406\) −0.725912 −0.0360264
\(407\) −4.90199 −0.242982
\(408\) 10.5771 0.523647
\(409\) −23.6328 −1.16857 −0.584283 0.811550i \(-0.698624\pi\)
−0.584283 + 0.811550i \(0.698624\pi\)
\(410\) 0.361903 0.0178731
\(411\) 10.9664 0.540932
\(412\) −15.8994 −0.783307
\(413\) 10.8577 0.534273
\(414\) 0.586186 0.0288095
\(415\) 7.03058 0.345118
\(416\) −17.9057 −0.877897
\(417\) −10.7820 −0.527998
\(418\) 0.224835 0.0109970
\(419\) 18.6252 0.909901 0.454951 0.890517i \(-0.349657\pi\)
0.454951 + 0.890517i \(0.349657\pi\)
\(420\) 1.72890 0.0843615
\(421\) −31.0986 −1.51565 −0.757827 0.652456i \(-0.773739\pi\)
−0.757827 + 0.652456i \(0.773739\pi\)
\(422\) −5.12992 −0.249720
\(423\) −8.71377 −0.423678
\(424\) 9.08975 0.441437
\(425\) 21.1688 1.02684
\(426\) −1.01055 −0.0489615
\(427\) −2.33559 −0.113027
\(428\) −30.5831 −1.47829
\(429\) −4.79326 −0.231421
\(430\) 2.47563 0.119386
\(431\) 9.88560 0.476173 0.238086 0.971244i \(-0.423480\pi\)
0.238086 + 0.971244i \(0.423480\pi\)
\(432\) 2.05639 0.0989380
\(433\) −8.91141 −0.428255 −0.214127 0.976806i \(-0.568691\pi\)
−0.214127 + 0.976806i \(0.568691\pi\)
\(434\) −0.0738370 −0.00354429
\(435\) 0.842866 0.0404123
\(436\) 22.5509 1.07999
\(437\) 0.260886 0.0124799
\(438\) −1.55112 −0.0741153
\(439\) −31.9814 −1.52639 −0.763193 0.646170i \(-0.776370\pi\)
−0.763193 + 0.646170i \(0.776370\pi\)
\(440\) 2.65597 0.126618
\(441\) −5.46645 −0.260307
\(442\) 9.43128 0.448600
\(443\) 20.8775 0.991921 0.495960 0.868345i \(-0.334816\pi\)
0.495960 + 0.868345i \(0.334816\pi\)
\(444\) 5.52277 0.262099
\(445\) 4.32890 0.205210
\(446\) 17.2194 0.815363
\(447\) −14.9598 −0.707573
\(448\) −1.10636 −0.0522706
\(449\) −13.7130 −0.647158 −0.323579 0.946201i \(-0.604886\pi\)
−0.323579 + 0.946201i \(0.604886\pi\)
\(450\) −2.51449 −0.118534
\(451\) −1.07690 −0.0507092
\(452\) 15.7192 0.739370
\(453\) −4.51476 −0.212122
\(454\) 12.5855 0.590668
\(455\) 3.40300 0.159535
\(456\) −0.559163 −0.0261852
\(457\) 12.8613 0.601627 0.300813 0.953683i \(-0.402742\pi\)
0.300813 + 0.953683i \(0.402742\pi\)
\(458\) 3.60328 0.168370
\(459\) −4.93493 −0.230343
\(460\) 1.39611 0.0650940
\(461\) −24.1332 −1.12400 −0.561998 0.827139i \(-0.689967\pi\)
−0.561998 + 0.827139i \(0.689967\pi\)
\(462\) 1.06724 0.0496524
\(463\) −40.5906 −1.88640 −0.943202 0.332219i \(-0.892203\pi\)
−0.943202 + 0.332219i \(0.892203\pi\)
\(464\) −2.05639 −0.0954654
\(465\) 0.0857330 0.00397577
\(466\) 12.3281 0.571086
\(467\) −16.4110 −0.759413 −0.379706 0.925107i \(-0.623975\pi\)
−0.379706 + 0.925107i \(0.623975\pi\)
\(468\) 5.40027 0.249628
\(469\) −3.98363 −0.183947
\(470\) 4.30526 0.198587
\(471\) −16.2289 −0.747788
\(472\) −18.7922 −0.864979
\(473\) −7.36664 −0.338718
\(474\) 0.272777 0.0125291
\(475\) −1.11909 −0.0513475
\(476\) 10.1226 0.463968
\(477\) −4.24097 −0.194180
\(478\) −13.2616 −0.606571
\(479\) −11.5781 −0.529017 −0.264508 0.964383i \(-0.585210\pi\)
−0.264508 + 0.964383i \(0.585210\pi\)
\(480\) −4.62908 −0.211287
\(481\) 10.8705 0.495652
\(482\) 9.47768 0.431697
\(483\) 1.23837 0.0563476
\(484\) 14.6400 0.665454
\(485\) 5.79664 0.263212
\(486\) 0.586186 0.0265899
\(487\) −5.57197 −0.252490 −0.126245 0.991999i \(-0.540293\pi\)
−0.126245 + 0.991999i \(0.540293\pi\)
\(488\) 4.04236 0.182989
\(489\) −2.54577 −0.115124
\(490\) 2.70084 0.122012
\(491\) 20.4086 0.921028 0.460514 0.887653i \(-0.347665\pi\)
0.460514 + 0.887653i \(0.347665\pi\)
\(492\) 1.21328 0.0546988
\(493\) 4.93493 0.222258
\(494\) −0.498587 −0.0224325
\(495\) −1.23918 −0.0556971
\(496\) −0.209168 −0.00939191
\(497\) −2.13488 −0.0957624
\(498\) −4.88954 −0.219106
\(499\) −32.6022 −1.45947 −0.729737 0.683728i \(-0.760357\pi\)
−0.729737 + 0.683728i \(0.760357\pi\)
\(500\) −12.9693 −0.580004
\(501\) 19.0688 0.851931
\(502\) 2.86384 0.127819
\(503\) 6.33199 0.282330 0.141165 0.989986i \(-0.454915\pi\)
0.141165 + 0.989986i \(0.454915\pi\)
\(504\) −2.65422 −0.118228
\(505\) 13.0266 0.579677
\(506\) 0.861812 0.0383122
\(507\) −2.37061 −0.105282
\(508\) −31.6950 −1.40624
\(509\) 27.0573 1.19929 0.599646 0.800265i \(-0.295308\pi\)
0.599646 + 0.800265i \(0.295308\pi\)
\(510\) 2.43823 0.107967
\(511\) −3.27686 −0.144960
\(512\) 20.1088 0.888693
\(513\) 0.260886 0.0115184
\(514\) 8.24647 0.363736
\(515\) −8.09054 −0.356512
\(516\) 8.29954 0.365367
\(517\) −12.8110 −0.563427
\(518\) −2.42036 −0.106344
\(519\) −9.37891 −0.411688
\(520\) −5.88979 −0.258284
\(521\) −5.85670 −0.256587 −0.128293 0.991736i \(-0.540950\pi\)
−0.128293 + 0.991736i \(0.540950\pi\)
\(522\) −0.586186 −0.0256567
\(523\) 8.29727 0.362814 0.181407 0.983408i \(-0.441935\pi\)
0.181407 + 0.983408i \(0.441935\pi\)
\(524\) −23.8582 −1.04225
\(525\) −5.31207 −0.231838
\(526\) −9.22768 −0.402346
\(527\) 0.501962 0.0218658
\(528\) 3.02331 0.131572
\(529\) 1.00000 0.0434783
\(530\) 2.09536 0.0910166
\(531\) 8.76778 0.380489
\(532\) −0.535133 −0.0232010
\(533\) 2.38810 0.103440
\(534\) −3.01061 −0.130282
\(535\) −15.5625 −0.672824
\(536\) 6.89474 0.297807
\(537\) 4.33101 0.186897
\(538\) 3.40547 0.146820
\(539\) −8.03679 −0.346169
\(540\) 1.39611 0.0600791
\(541\) 11.0825 0.476473 0.238237 0.971207i \(-0.423431\pi\)
0.238237 + 0.971207i \(0.423431\pi\)
\(542\) 7.92197 0.340278
\(543\) 20.4714 0.878511
\(544\) −27.1030 −1.16203
\(545\) 11.4752 0.491544
\(546\) −2.36667 −0.101284
\(547\) 21.4790 0.918375 0.459187 0.888339i \(-0.348141\pi\)
0.459187 + 0.888339i \(0.348141\pi\)
\(548\) −18.1646 −0.775952
\(549\) −1.88602 −0.0804935
\(550\) −3.69681 −0.157632
\(551\) −0.260886 −0.0111141
\(552\) −2.14332 −0.0912258
\(553\) 0.576264 0.0245052
\(554\) −8.31175 −0.353132
\(555\) 2.81031 0.119291
\(556\) 17.8592 0.757398
\(557\) 38.2929 1.62252 0.811262 0.584683i \(-0.198781\pi\)
0.811262 + 0.584683i \(0.198781\pi\)
\(558\) −0.0596245 −0.00252411
\(559\) 16.3360 0.690940
\(560\) −2.14641 −0.0907023
\(561\) −7.25535 −0.306321
\(562\) −0.177984 −0.00750781
\(563\) −4.52231 −0.190592 −0.0952962 0.995449i \(-0.530380\pi\)
−0.0952962 + 0.995449i \(0.530380\pi\)
\(564\) 14.4334 0.607754
\(565\) 7.99885 0.336514
\(566\) 10.2397 0.430409
\(567\) 1.23837 0.0520065
\(568\) 3.69497 0.155038
\(569\) 44.8603 1.88064 0.940321 0.340289i \(-0.110525\pi\)
0.940321 + 0.340289i \(0.110525\pi\)
\(570\) −0.128898 −0.00539893
\(571\) 0.417218 0.0174600 0.00873001 0.999962i \(-0.497221\pi\)
0.00873001 + 0.999962i \(0.497221\pi\)
\(572\) 7.93949 0.331967
\(573\) 7.10652 0.296879
\(574\) −0.531719 −0.0221935
\(575\) −4.28958 −0.178888
\(576\) −0.893403 −0.0372251
\(577\) −11.8343 −0.492670 −0.246335 0.969185i \(-0.579226\pi\)
−0.246335 + 0.969185i \(0.579226\pi\)
\(578\) 4.31055 0.179295
\(579\) 2.37896 0.0988663
\(580\) −1.39611 −0.0579704
\(581\) −10.3296 −0.428542
\(582\) −4.03138 −0.167106
\(583\) −6.23508 −0.258230
\(584\) 5.67148 0.234688
\(585\) 2.74797 0.113615
\(586\) 10.5620 0.436312
\(587\) −18.5466 −0.765501 −0.382750 0.923852i \(-0.625023\pi\)
−0.382750 + 0.923852i \(0.625023\pi\)
\(588\) 9.05455 0.373403
\(589\) −0.0265363 −0.00109341
\(590\) −4.33195 −0.178343
\(591\) 4.61033 0.189643
\(592\) −6.85646 −0.281799
\(593\) −29.0821 −1.19426 −0.597129 0.802145i \(-0.703692\pi\)
−0.597129 + 0.802145i \(0.703692\pi\)
\(594\) 0.861812 0.0353606
\(595\) 5.15096 0.211169
\(596\) 24.7792 1.01499
\(597\) −12.7702 −0.522649
\(598\) −1.91113 −0.0781518
\(599\) 29.5204 1.20617 0.603085 0.797677i \(-0.293938\pi\)
0.603085 + 0.797677i \(0.293938\pi\)
\(600\) 9.19394 0.375341
\(601\) 3.49768 0.142673 0.0713367 0.997452i \(-0.477274\pi\)
0.0713367 + 0.997452i \(0.477274\pi\)
\(602\) −3.63728 −0.148244
\(603\) −3.21685 −0.131000
\(604\) 7.47819 0.304283
\(605\) 7.44967 0.302872
\(606\) −9.05959 −0.368021
\(607\) 7.88945 0.320223 0.160111 0.987099i \(-0.448815\pi\)
0.160111 + 0.987099i \(0.448815\pi\)
\(608\) 1.43281 0.0581079
\(609\) −1.23837 −0.0501811
\(610\) 0.931839 0.0377291
\(611\) 28.4093 1.14932
\(612\) 8.17415 0.330421
\(613\) −33.4560 −1.35127 −0.675637 0.737235i \(-0.736131\pi\)
−0.675637 + 0.737235i \(0.736131\pi\)
\(614\) 8.73368 0.352463
\(615\) 0.617386 0.0248954
\(616\) −3.90223 −0.157225
\(617\) −33.2433 −1.33832 −0.669162 0.743117i \(-0.733347\pi\)
−0.669162 + 0.743117i \(0.733347\pi\)
\(618\) 5.62671 0.226339
\(619\) −23.1547 −0.930668 −0.465334 0.885135i \(-0.654066\pi\)
−0.465334 + 0.885135i \(0.654066\pi\)
\(620\) −0.142007 −0.00570314
\(621\) 1.00000 0.0401286
\(622\) −1.07699 −0.0431834
\(623\) −6.36016 −0.254815
\(624\) −6.70439 −0.268390
\(625\) 14.8484 0.593934
\(626\) 2.61941 0.104693
\(627\) 0.383556 0.0153177
\(628\) 26.8813 1.07268
\(629\) 16.4542 0.656071
\(630\) −0.611847 −0.0243766
\(631\) −17.7854 −0.708027 −0.354014 0.935240i \(-0.615183\pi\)
−0.354014 + 0.935240i \(0.615183\pi\)
\(632\) −0.997378 −0.0396735
\(633\) −8.75135 −0.347835
\(634\) 9.35334 0.371469
\(635\) −16.1283 −0.640030
\(636\) 7.02468 0.278547
\(637\) 17.8221 0.706139
\(638\) −0.861812 −0.0341194
\(639\) −1.72395 −0.0681983
\(640\) 9.69956 0.383409
\(641\) 27.7374 1.09556 0.547782 0.836621i \(-0.315472\pi\)
0.547782 + 0.836621i \(0.315472\pi\)
\(642\) 10.8232 0.427157
\(643\) 15.3293 0.604528 0.302264 0.953224i \(-0.402258\pi\)
0.302264 + 0.953224i \(0.402258\pi\)
\(644\) −2.05121 −0.0808291
\(645\) 4.22329 0.166292
\(646\) −0.754689 −0.0296928
\(647\) −41.3220 −1.62454 −0.812268 0.583284i \(-0.801767\pi\)
−0.812268 + 0.583284i \(0.801767\pi\)
\(648\) −2.14332 −0.0841976
\(649\) 12.8904 0.505993
\(650\) 8.19793 0.321549
\(651\) −0.125962 −0.00493683
\(652\) 4.21678 0.165142
\(653\) −31.9364 −1.24977 −0.624883 0.780718i \(-0.714854\pi\)
−0.624883 + 0.780718i \(0.714854\pi\)
\(654\) −7.98064 −0.312068
\(655\) −12.1405 −0.474367
\(656\) −1.50627 −0.0588100
\(657\) −2.64612 −0.103235
\(658\) −6.32543 −0.246591
\(659\) −39.4143 −1.53536 −0.767681 0.640833i \(-0.778589\pi\)
−0.767681 + 0.640833i \(0.778589\pi\)
\(660\) 2.05257 0.0798960
\(661\) 16.5791 0.644850 0.322425 0.946595i \(-0.395502\pi\)
0.322425 + 0.946595i \(0.395502\pi\)
\(662\) −20.8210 −0.809232
\(663\) 16.0892 0.624854
\(664\) 17.8781 0.693803
\(665\) −0.272307 −0.0105596
\(666\) −1.95448 −0.0757344
\(667\) −1.00000 −0.0387202
\(668\) −31.5853 −1.22207
\(669\) 29.3753 1.13572
\(670\) 1.58937 0.0614026
\(671\) −2.77284 −0.107044
\(672\) 6.80119 0.262362
\(673\) −13.0131 −0.501618 −0.250809 0.968037i \(-0.580697\pi\)
−0.250809 + 0.968037i \(0.580697\pi\)
\(674\) −12.1237 −0.466989
\(675\) −4.28958 −0.165106
\(676\) 3.92664 0.151025
\(677\) 12.2182 0.469583 0.234792 0.972046i \(-0.424559\pi\)
0.234792 + 0.972046i \(0.424559\pi\)
\(678\) −5.56294 −0.213644
\(679\) −8.51661 −0.326838
\(680\) −8.91512 −0.341879
\(681\) 21.4702 0.822740
\(682\) −0.0876601 −0.00335668
\(683\) 1.56826 0.0600076 0.0300038 0.999550i \(-0.490448\pi\)
0.0300038 + 0.999550i \(0.490448\pi\)
\(684\) −0.432129 −0.0165228
\(685\) −9.24319 −0.353164
\(686\) −9.04955 −0.345513
\(687\) 6.14700 0.234523
\(688\) −10.3038 −0.392829
\(689\) 13.8267 0.526756
\(690\) −0.494076 −0.0188091
\(691\) 6.57025 0.249944 0.124972 0.992160i \(-0.460116\pi\)
0.124972 + 0.992160i \(0.460116\pi\)
\(692\) 15.5351 0.590556
\(693\) 1.82065 0.0691607
\(694\) 7.90084 0.299912
\(695\) 9.08779 0.344720
\(696\) 2.14332 0.0812424
\(697\) 3.61476 0.136919
\(698\) −19.2911 −0.730177
\(699\) 21.0310 0.795464
\(700\) 8.79883 0.332565
\(701\) 16.6273 0.628004 0.314002 0.949422i \(-0.398330\pi\)
0.314002 + 0.949422i \(0.398330\pi\)
\(702\) −1.91113 −0.0721308
\(703\) −0.869854 −0.0328072
\(704\) −1.31348 −0.0495038
\(705\) 7.34454 0.276611
\(706\) 13.9658 0.525610
\(707\) −19.1391 −0.719801
\(708\) −14.5228 −0.545801
\(709\) −9.39341 −0.352777 −0.176389 0.984321i \(-0.556442\pi\)
−0.176389 + 0.984321i \(0.556442\pi\)
\(710\) 0.851761 0.0319660
\(711\) 0.465342 0.0174517
\(712\) 11.0080 0.412541
\(713\) −0.101716 −0.00380930
\(714\) −3.58233 −0.134065
\(715\) 4.04008 0.151090
\(716\) −7.17382 −0.268098
\(717\) −22.6235 −0.844891
\(718\) 15.3683 0.573540
\(719\) −0.289428 −0.0107939 −0.00539693 0.999985i \(-0.501718\pi\)
−0.00539693 + 0.999985i \(0.501718\pi\)
\(720\) −1.73326 −0.0645947
\(721\) 11.8869 0.442691
\(722\) −11.0976 −0.413011
\(723\) 16.1684 0.601309
\(724\) −33.9085 −1.26020
\(725\) 4.28958 0.159311
\(726\) −5.18101 −0.192285
\(727\) 12.8787 0.477644 0.238822 0.971063i \(-0.423239\pi\)
0.238822 + 0.971063i \(0.423239\pi\)
\(728\) 8.65347 0.320719
\(729\) 1.00000 0.0370370
\(730\) 1.30738 0.0483884
\(731\) 24.7271 0.914566
\(732\) 3.12398 0.115466
\(733\) −33.0973 −1.22248 −0.611239 0.791446i \(-0.709328\pi\)
−0.611239 + 0.791446i \(0.709328\pi\)
\(734\) 5.28673 0.195137
\(735\) 4.60748 0.169950
\(736\) 5.49207 0.202440
\(737\) −4.72942 −0.174210
\(738\) −0.429372 −0.0158054
\(739\) 17.1818 0.632043 0.316022 0.948752i \(-0.397653\pi\)
0.316022 + 0.948752i \(0.397653\pi\)
\(740\) −4.65495 −0.171119
\(741\) −0.850561 −0.0312461
\(742\) −3.07857 −0.113018
\(743\) −21.8990 −0.803395 −0.401697 0.915772i \(-0.631580\pi\)
−0.401697 + 0.915772i \(0.631580\pi\)
\(744\) 0.218010 0.00799265
\(745\) 12.6091 0.461961
\(746\) −5.03763 −0.184441
\(747\) −8.34128 −0.305192
\(748\) 12.0177 0.439409
\(749\) 22.8649 0.835465
\(750\) 4.58976 0.167594
\(751\) 50.1136 1.82867 0.914336 0.404956i \(-0.132713\pi\)
0.914336 + 0.404956i \(0.132713\pi\)
\(752\) −17.9189 −0.653435
\(753\) 4.88555 0.178039
\(754\) 1.91113 0.0695991
\(755\) 3.80534 0.138490
\(756\) −2.05121 −0.0746018
\(757\) 49.4866 1.79862 0.899311 0.437309i \(-0.144068\pi\)
0.899311 + 0.437309i \(0.144068\pi\)
\(758\) 4.97655 0.180757
\(759\) 1.47020 0.0533650
\(760\) 0.471300 0.0170958
\(761\) −18.5155 −0.671186 −0.335593 0.942007i \(-0.608937\pi\)
−0.335593 + 0.942007i \(0.608937\pi\)
\(762\) 11.2167 0.406338
\(763\) −16.8598 −0.610364
\(764\) −11.7711 −0.425865
\(765\) 4.15949 0.150387
\(766\) 6.23916 0.225430
\(767\) −28.5854 −1.03216
\(768\) −4.95892 −0.178940
\(769\) 9.36922 0.337863 0.168931 0.985628i \(-0.445968\pi\)
0.168931 + 0.985628i \(0.445968\pi\)
\(770\) −0.899538 −0.0324171
\(771\) 14.0680 0.506647
\(772\) −3.94048 −0.141821
\(773\) 3.24015 0.116540 0.0582700 0.998301i \(-0.481442\pi\)
0.0582700 + 0.998301i \(0.481442\pi\)
\(774\) −2.93716 −0.105574
\(775\) 0.436319 0.0156730
\(776\) 14.7403 0.529145
\(777\) −4.12899 −0.148127
\(778\) −8.61085 −0.308714
\(779\) −0.191095 −0.00684670
\(780\) −4.55171 −0.162977
\(781\) −2.53455 −0.0906934
\(782\) −2.89279 −0.103446
\(783\) −1.00000 −0.0357371
\(784\) −11.2411 −0.401469
\(785\) 13.6788 0.488217
\(786\) 8.44329 0.301162
\(787\) 34.2199 1.21981 0.609905 0.792475i \(-0.291208\pi\)
0.609905 + 0.792475i \(0.291208\pi\)
\(788\) −7.63648 −0.272038
\(789\) −15.7419 −0.560427
\(790\) −0.229914 −0.00817998
\(791\) −11.7522 −0.417859
\(792\) −3.15112 −0.111970
\(793\) 6.14896 0.218356
\(794\) 4.54489 0.161292
\(795\) 3.57456 0.126777
\(796\) 21.1524 0.749725
\(797\) −9.08039 −0.321644 −0.160822 0.986983i \(-0.551415\pi\)
−0.160822 + 0.986983i \(0.551415\pi\)
\(798\) 0.189381 0.00670400
\(799\) 43.0019 1.52130
\(800\) −23.5587 −0.832924
\(801\) −5.13593 −0.181469
\(802\) −0.689978 −0.0243640
\(803\) −3.89033 −0.137287
\(804\) 5.32834 0.187916
\(805\) −1.04378 −0.0367883
\(806\) 0.194392 0.00684718
\(807\) 5.80954 0.204505
\(808\) 33.1254 1.16535
\(809\) −2.55646 −0.0898804 −0.0449402 0.998990i \(-0.514310\pi\)
−0.0449402 + 0.998990i \(0.514310\pi\)
\(810\) −0.494076 −0.0173601
\(811\) 1.90277 0.0668153 0.0334077 0.999442i \(-0.489364\pi\)
0.0334077 + 0.999442i \(0.489364\pi\)
\(812\) 2.05121 0.0719834
\(813\) 13.5144 0.473972
\(814\) −2.87348 −0.100715
\(815\) 2.14574 0.0751621
\(816\) −10.1481 −0.355256
\(817\) −1.30720 −0.0457333
\(818\) −13.8532 −0.484366
\(819\) −4.03741 −0.141079
\(820\) −1.02263 −0.0357118
\(821\) −13.7415 −0.479581 −0.239790 0.970825i \(-0.577079\pi\)
−0.239790 + 0.970825i \(0.577079\pi\)
\(822\) 6.42834 0.224214
\(823\) −10.1853 −0.355039 −0.177519 0.984117i \(-0.556807\pi\)
−0.177519 + 0.984117i \(0.556807\pi\)
\(824\) −20.5734 −0.716709
\(825\) −6.30655 −0.219566
\(826\) 6.36464 0.221454
\(827\) 26.7391 0.929809 0.464905 0.885361i \(-0.346089\pi\)
0.464905 + 0.885361i \(0.346089\pi\)
\(828\) −1.65639 −0.0575634
\(829\) −51.5489 −1.79037 −0.895184 0.445697i \(-0.852956\pi\)
−0.895184 + 0.445697i \(0.852956\pi\)
\(830\) 4.12123 0.143050
\(831\) −14.1794 −0.491877
\(832\) 2.91274 0.100981
\(833\) 26.9766 0.934683
\(834\) −6.32027 −0.218853
\(835\) −16.0724 −0.556210
\(836\) −0.635316 −0.0219729
\(837\) −0.101716 −0.00351582
\(838\) 10.9178 0.377150
\(839\) 4.54360 0.156862 0.0784312 0.996920i \(-0.475009\pi\)
0.0784312 + 0.996920i \(0.475009\pi\)
\(840\) 2.23715 0.0771889
\(841\) 1.00000 0.0344828
\(842\) −18.2296 −0.628232
\(843\) −0.303631 −0.0104576
\(844\) 14.4956 0.498960
\(845\) 1.99810 0.0687368
\(846\) −5.10789 −0.175613
\(847\) −10.9453 −0.376085
\(848\) −8.72107 −0.299483
\(849\) 17.4684 0.599515
\(850\) 12.4088 0.425620
\(851\) −3.33423 −0.114296
\(852\) 2.85552 0.0978286
\(853\) −14.2400 −0.487568 −0.243784 0.969830i \(-0.578389\pi\)
−0.243784 + 0.969830i \(0.578389\pi\)
\(854\) −1.36909 −0.0468492
\(855\) −0.219892 −0.00752015
\(856\) −39.5738 −1.35260
\(857\) −2.90828 −0.0993451 −0.0496726 0.998766i \(-0.515818\pi\)
−0.0496726 + 0.998766i \(0.515818\pi\)
\(858\) −2.80974 −0.0959230
\(859\) 0.525329 0.0179240 0.00896200 0.999960i \(-0.497147\pi\)
0.00896200 + 0.999960i \(0.497147\pi\)
\(860\) −6.99540 −0.238541
\(861\) −0.907084 −0.0309133
\(862\) 5.79480 0.197372
\(863\) −35.2839 −1.20108 −0.600539 0.799595i \(-0.705047\pi\)
−0.600539 + 0.799595i \(0.705047\pi\)
\(864\) 5.49207 0.186844
\(865\) 7.90516 0.268784
\(866\) −5.22374 −0.177510
\(867\) 7.35356 0.249740
\(868\) 0.208641 0.00708175
\(869\) 0.684147 0.0232081
\(870\) 0.494076 0.0167507
\(871\) 10.4878 0.355366
\(872\) 29.1803 0.988170
\(873\) −6.87730 −0.232761
\(874\) 0.152928 0.00517286
\(875\) 9.69624 0.327793
\(876\) 4.38300 0.148088
\(877\) −32.9988 −1.11429 −0.557145 0.830415i \(-0.688103\pi\)
−0.557145 + 0.830415i \(0.688103\pi\)
\(878\) −18.7470 −0.632681
\(879\) 18.0182 0.607738
\(880\) −2.54824 −0.0859012
\(881\) −8.43036 −0.284026 −0.142013 0.989865i \(-0.545358\pi\)
−0.142013 + 0.989865i \(0.545358\pi\)
\(882\) −3.20436 −0.107896
\(883\) −26.7930 −0.901656 −0.450828 0.892611i \(-0.648871\pi\)
−0.450828 + 0.892611i \(0.648871\pi\)
\(884\) −26.6500 −0.896336
\(885\) −7.39006 −0.248414
\(886\) 12.2381 0.411147
\(887\) 18.0659 0.606594 0.303297 0.952896i \(-0.401913\pi\)
0.303297 + 0.952896i \(0.401913\pi\)
\(888\) 7.14632 0.239815
\(889\) 23.6962 0.794744
\(890\) 2.53754 0.0850585
\(891\) 1.47020 0.0492536
\(892\) −48.6569 −1.62915
\(893\) −2.27330 −0.0760732
\(894\) −8.76920 −0.293286
\(895\) −3.65046 −0.122021
\(896\) −14.2509 −0.476089
\(897\) −3.26028 −0.108857
\(898\) −8.03838 −0.268244
\(899\) 0.101716 0.00339242
\(900\) 7.10520 0.236840
\(901\) 20.9289 0.697242
\(902\) −0.631264 −0.0210188
\(903\) −6.20499 −0.206489
\(904\) 20.3403 0.676507
\(905\) −17.2546 −0.573563
\(906\) −2.64649 −0.0879237
\(907\) 50.2869 1.66975 0.834874 0.550441i \(-0.185540\pi\)
0.834874 + 0.550441i \(0.185540\pi\)
\(908\) −35.5630 −1.18020
\(909\) −15.4552 −0.512615
\(910\) 1.99479 0.0661266
\(911\) −25.2794 −0.837544 −0.418772 0.908091i \(-0.637539\pi\)
−0.418772 + 0.908091i \(0.637539\pi\)
\(912\) 0.536484 0.0177647
\(913\) −12.2634 −0.405858
\(914\) 7.53912 0.249372
\(915\) 1.58966 0.0525527
\(916\) −10.1818 −0.336416
\(917\) 17.8371 0.589034
\(918\) −2.89279 −0.0954762
\(919\) −48.0884 −1.58629 −0.793146 0.609032i \(-0.791558\pi\)
−0.793146 + 0.609032i \(0.791558\pi\)
\(920\) 1.80653 0.0595596
\(921\) 14.8992 0.490944
\(922\) −14.1465 −0.465891
\(923\) 5.62054 0.185002
\(924\) −3.01570 −0.0992091
\(925\) 14.3024 0.470261
\(926\) −23.7936 −0.781907
\(927\) 9.59885 0.315268
\(928\) −5.49207 −0.180286
\(929\) −32.8357 −1.07731 −0.538653 0.842528i \(-0.681066\pi\)
−0.538653 + 0.842528i \(0.681066\pi\)
\(930\) 0.0502555 0.00164794
\(931\) −1.42612 −0.0467393
\(932\) −34.8354 −1.14107
\(933\) −1.83728 −0.0601500
\(934\) −9.61992 −0.314773
\(935\) 6.11528 0.199991
\(936\) 6.98782 0.228404
\(937\) −28.5880 −0.933930 −0.466965 0.884276i \(-0.654653\pi\)
−0.466965 + 0.884276i \(0.654653\pi\)
\(938\) −2.33515 −0.0762453
\(939\) 4.46857 0.145826
\(940\) −12.1654 −0.396791
\(941\) −13.1403 −0.428361 −0.214181 0.976794i \(-0.568708\pi\)
−0.214181 + 0.976794i \(0.568708\pi\)
\(942\) −9.51315 −0.309955
\(943\) −0.732484 −0.0238530
\(944\) 18.0299 0.586825
\(945\) −1.04378 −0.0339540
\(946\) −4.31822 −0.140397
\(947\) −35.6856 −1.15963 −0.579813 0.814750i \(-0.696874\pi\)
−0.579813 + 0.814750i \(0.696874\pi\)
\(948\) −0.770786 −0.0250340
\(949\) 8.62708 0.280047
\(950\) −0.655996 −0.0212833
\(951\) 15.9563 0.517418
\(952\) 13.0984 0.424521
\(953\) −14.8964 −0.482541 −0.241270 0.970458i \(-0.577564\pi\)
−0.241270 + 0.970458i \(0.577564\pi\)
\(954\) −2.48599 −0.0804870
\(955\) −5.98985 −0.193827
\(956\) 37.4733 1.21197
\(957\) −1.47020 −0.0475249
\(958\) −6.78691 −0.219275
\(959\) 13.5804 0.438534
\(960\) 0.753019 0.0243036
\(961\) −30.9897 −0.999666
\(962\) 6.37213 0.205446
\(963\) 18.4638 0.594986
\(964\) −26.7811 −0.862561
\(965\) −2.00515 −0.0645480
\(966\) 0.725912 0.0233558
\(967\) −15.5500 −0.500053 −0.250026 0.968239i \(-0.580439\pi\)
−0.250026 + 0.968239i \(0.580439\pi\)
\(968\) 18.9438 0.608876
\(969\) −1.28746 −0.0413591
\(970\) 3.39791 0.109100
\(971\) −40.0985 −1.28682 −0.643410 0.765522i \(-0.722481\pi\)
−0.643410 + 0.765522i \(0.722481\pi\)
\(972\) −1.65639 −0.0531286
\(973\) −13.3521 −0.428048
\(974\) −3.26621 −0.104656
\(975\) 13.9852 0.447885
\(976\) −3.87840 −0.124144
\(977\) −12.6678 −0.405277 −0.202639 0.979254i \(-0.564952\pi\)
−0.202639 + 0.979254i \(0.564952\pi\)
\(978\) −1.49230 −0.0477183
\(979\) −7.55086 −0.241326
\(980\) −7.63177 −0.243788
\(981\) −13.6145 −0.434678
\(982\) 11.9632 0.381762
\(983\) −41.8116 −1.33358 −0.666792 0.745244i \(-0.732333\pi\)
−0.666792 + 0.745244i \(0.732333\pi\)
\(984\) 1.56995 0.0500482
\(985\) −3.88589 −0.123815
\(986\) 2.89279 0.0921251
\(987\) −10.7908 −0.343476
\(988\) 1.40886 0.0448217
\(989\) −5.01063 −0.159329
\(990\) −0.726391 −0.0230862
\(991\) 6.44884 0.204854 0.102427 0.994741i \(-0.467339\pi\)
0.102427 + 0.994741i \(0.467339\pi\)
\(992\) −0.558632 −0.0177366
\(993\) −35.5195 −1.12718
\(994\) −1.25143 −0.0396931
\(995\) 10.7635 0.341227
\(996\) 13.8164 0.437789
\(997\) 28.3269 0.897123 0.448562 0.893752i \(-0.351936\pi\)
0.448562 + 0.893752i \(0.351936\pi\)
\(998\) −19.1109 −0.604946
\(999\) −3.33423 −0.105490
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 2001.2.a.i.1.5 7
3.2 odd 2 6003.2.a.j.1.3 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2001.2.a.i.1.5 7 1.1 even 1 trivial
6003.2.a.j.1.3 7 3.2 odd 2