Properties

Label 503.2.a.e.1.5
Level $503$
Weight $2$
Character 503.1
Self dual yes
Analytic conductor $4.016$
Analytic rank $1$
Dimension $10$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 503.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.01647522167\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 2 x^{9} - 9 x^{8} + 14 x^{7} + 27 x^{6} - 27 x^{5} - 34 x^{4} + 14 x^{3} + 17 x^{2} + x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-2.07227\) of defining polynomial
Character \(\chi\) \(=\) 503.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.756417 q^{2} -3.07227 q^{3} -1.42783 q^{4} +0.386144 q^{5} +2.32392 q^{6} +0.194914 q^{7} +2.59287 q^{8} +6.43884 q^{9} +O(q^{10})\) \(q-0.756417 q^{2} -3.07227 q^{3} -1.42783 q^{4} +0.386144 q^{5} +2.32392 q^{6} +0.194914 q^{7} +2.59287 q^{8} +6.43884 q^{9} -0.292086 q^{10} +2.36326 q^{11} +4.38669 q^{12} -1.22636 q^{13} -0.147436 q^{14} -1.18634 q^{15} +0.894372 q^{16} -5.04830 q^{17} -4.87045 q^{18} +4.24460 q^{19} -0.551349 q^{20} -0.598827 q^{21} -1.78761 q^{22} +1.53457 q^{23} -7.96600 q^{24} -4.85089 q^{25} +0.927639 q^{26} -10.5650 q^{27} -0.278304 q^{28} -7.31602 q^{29} +0.897367 q^{30} +3.33893 q^{31} -5.86226 q^{32} -7.26057 q^{33} +3.81863 q^{34} +0.0752647 q^{35} -9.19358 q^{36} -2.17138 q^{37} -3.21069 q^{38} +3.76771 q^{39} +1.00122 q^{40} -1.04840 q^{41} +0.452963 q^{42} -1.53067 q^{43} -3.37434 q^{44} +2.48632 q^{45} -1.16077 q^{46} -1.08422 q^{47} -2.74775 q^{48} -6.96201 q^{49} +3.66930 q^{50} +15.5098 q^{51} +1.75104 q^{52} +1.55308 q^{53} +7.99158 q^{54} +0.912558 q^{55} +0.505386 q^{56} -13.0406 q^{57} +5.53396 q^{58} +14.8451 q^{59} +1.69389 q^{60} -5.45056 q^{61} -2.52563 q^{62} +1.25502 q^{63} +2.64557 q^{64} -0.473551 q^{65} +5.49202 q^{66} -11.3202 q^{67} +7.20814 q^{68} -4.71461 q^{69} -0.0569315 q^{70} -11.4236 q^{71} +16.6951 q^{72} -0.902229 q^{73} +1.64247 q^{74} +14.9032 q^{75} -6.06059 q^{76} +0.460631 q^{77} -2.84996 q^{78} -12.7431 q^{79} +0.345356 q^{80} +13.1421 q^{81} +0.793029 q^{82} +6.10241 q^{83} +0.855025 q^{84} -1.94937 q^{85} +1.15782 q^{86} +22.4768 q^{87} +6.12763 q^{88} +6.44432 q^{89} -1.88070 q^{90} -0.239034 q^{91} -2.19111 q^{92} -10.2581 q^{93} +0.820122 q^{94} +1.63903 q^{95} +18.0105 q^{96} -12.9886 q^{97} +5.26618 q^{98} +15.2166 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 4q^{2} - 8q^{3} + 4q^{4} - q^{5} - 2q^{6} - 5q^{7} - 3q^{8} - 2q^{9} + O(q^{10}) \) \( 10q - 4q^{2} - 8q^{3} + 4q^{4} - q^{5} - 2q^{6} - 5q^{7} - 3q^{8} - 2q^{9} - 4q^{10} - 3q^{11} - 7q^{12} - 18q^{13} + q^{14} - 2q^{15} - 4q^{16} - 11q^{17} - q^{18} - 3q^{20} + q^{21} - 18q^{22} - 2q^{23} + 10q^{24} - 27q^{25} + 11q^{26} - 2q^{27} - 22q^{28} - 9q^{29} + 12q^{30} - 22q^{31} - 10q^{32} - 10q^{33} - 10q^{34} - 6q^{35} + 2q^{36} - 35q^{37} + 2q^{38} + 8q^{39} - 19q^{40} - 4q^{41} + 4q^{42} - 20q^{43} + 9q^{44} + 2q^{45} - q^{46} + 7q^{47} - 27q^{49} + 16q^{50} + 9q^{51} - 7q^{52} - 24q^{53} + 17q^{54} - 11q^{55} + 12q^{56} - 23q^{57} + 2q^{58} + 17q^{59} - 4q^{61} + 8q^{62} + 10q^{63} + 3q^{64} - 16q^{65} + 46q^{66} - 6q^{67} + 28q^{68} - 2q^{69} + 26q^{70} - q^{71} - q^{72} - 31q^{73} + 11q^{74} + 30q^{75} + 20q^{76} + 3q^{77} + 11q^{78} - 10q^{79} + 24q^{80} - 6q^{81} - 9q^{82} + 22q^{83} + 22q^{84} - 6q^{85} + 38q^{86} + 25q^{87} - 3q^{88} + q^{89} + 2q^{90} + 10q^{91} + 27q^{92} - 6q^{93} + 33q^{94} + 39q^{95} + 46q^{96} - 57q^{97} + 40q^{98} + 35q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.756417 −0.534868 −0.267434 0.963576i \(-0.586176\pi\)
−0.267434 + 0.963576i \(0.586176\pi\)
\(3\) −3.07227 −1.77378 −0.886888 0.461985i \(-0.847137\pi\)
−0.886888 + 0.461985i \(0.847137\pi\)
\(4\) −1.42783 −0.713916
\(5\) 0.386144 0.172689 0.0863445 0.996265i \(-0.472481\pi\)
0.0863445 + 0.996265i \(0.472481\pi\)
\(6\) 2.32392 0.948736
\(7\) 0.194914 0.0736704 0.0368352 0.999321i \(-0.488272\pi\)
0.0368352 + 0.999321i \(0.488272\pi\)
\(8\) 2.59287 0.916719
\(9\) 6.43884 2.14628
\(10\) −0.292086 −0.0923657
\(11\) 2.36326 0.712549 0.356275 0.934381i \(-0.384047\pi\)
0.356275 + 0.934381i \(0.384047\pi\)
\(12\) 4.38669 1.26633
\(13\) −1.22636 −0.340131 −0.170065 0.985433i \(-0.554398\pi\)
−0.170065 + 0.985433i \(0.554398\pi\)
\(14\) −0.147436 −0.0394039
\(15\) −1.18634 −0.306311
\(16\) 0.894372 0.223593
\(17\) −5.04830 −1.22439 −0.612197 0.790705i \(-0.709714\pi\)
−0.612197 + 0.790705i \(0.709714\pi\)
\(18\) −4.87045 −1.14798
\(19\) 4.24460 0.973779 0.486890 0.873464i \(-0.338131\pi\)
0.486890 + 0.873464i \(0.338131\pi\)
\(20\) −0.551349 −0.123285
\(21\) −0.598827 −0.130675
\(22\) −1.78761 −0.381120
\(23\) 1.53457 0.319980 0.159990 0.987119i \(-0.448854\pi\)
0.159990 + 0.987119i \(0.448854\pi\)
\(24\) −7.96600 −1.62605
\(25\) −4.85089 −0.970179
\(26\) 0.927639 0.181925
\(27\) −10.5650 −2.03324
\(28\) −0.278304 −0.0525945
\(29\) −7.31602 −1.35855 −0.679275 0.733883i \(-0.737706\pi\)
−0.679275 + 0.733883i \(0.737706\pi\)
\(30\) 0.897367 0.163836
\(31\) 3.33893 0.599690 0.299845 0.953988i \(-0.403065\pi\)
0.299845 + 0.953988i \(0.403065\pi\)
\(32\) −5.86226 −1.03631
\(33\) −7.26057 −1.26390
\(34\) 3.81863 0.654889
\(35\) 0.0752647 0.0127221
\(36\) −9.19358 −1.53226
\(37\) −2.17138 −0.356973 −0.178487 0.983942i \(-0.557120\pi\)
−0.178487 + 0.983942i \(0.557120\pi\)
\(38\) −3.21069 −0.520843
\(39\) 3.76771 0.603316
\(40\) 1.00122 0.158307
\(41\) −1.04840 −0.163733 −0.0818663 0.996643i \(-0.526088\pi\)
−0.0818663 + 0.996643i \(0.526088\pi\)
\(42\) 0.452963 0.0698937
\(43\) −1.53067 −0.233425 −0.116712 0.993166i \(-0.537236\pi\)
−0.116712 + 0.993166i \(0.537236\pi\)
\(44\) −3.37434 −0.508701
\(45\) 2.48632 0.370639
\(46\) −1.16077 −0.171147
\(47\) −1.08422 −0.158150 −0.0790748 0.996869i \(-0.525197\pi\)
−0.0790748 + 0.996869i \(0.525197\pi\)
\(48\) −2.74775 −0.396604
\(49\) −6.96201 −0.994573
\(50\) 3.66930 0.518917
\(51\) 15.5098 2.17180
\(52\) 1.75104 0.242825
\(53\) 1.55308 0.213332 0.106666 0.994295i \(-0.465983\pi\)
0.106666 + 0.994295i \(0.465983\pi\)
\(54\) 7.99158 1.08752
\(55\) 0.912558 0.123049
\(56\) 0.505386 0.0675350
\(57\) −13.0406 −1.72727
\(58\) 5.53396 0.726645
\(59\) 14.8451 1.93267 0.966334 0.257293i \(-0.0828304\pi\)
0.966334 + 0.257293i \(0.0828304\pi\)
\(60\) 1.69389 0.218681
\(61\) −5.45056 −0.697873 −0.348936 0.937146i \(-0.613457\pi\)
−0.348936 + 0.937146i \(0.613457\pi\)
\(62\) −2.52563 −0.320755
\(63\) 1.25502 0.158117
\(64\) 2.64557 0.330697
\(65\) −0.473551 −0.0587368
\(66\) 5.49202 0.676021
\(67\) −11.3202 −1.38298 −0.691490 0.722386i \(-0.743045\pi\)
−0.691490 + 0.722386i \(0.743045\pi\)
\(68\) 7.20814 0.874115
\(69\) −4.71461 −0.567572
\(70\) −0.0569315 −0.00680462
\(71\) −11.4236 −1.35573 −0.677864 0.735187i \(-0.737094\pi\)
−0.677864 + 0.735187i \(0.737094\pi\)
\(72\) 16.6951 1.96753
\(73\) −0.902229 −0.105598 −0.0527990 0.998605i \(-0.516814\pi\)
−0.0527990 + 0.998605i \(0.516814\pi\)
\(74\) 1.64247 0.190933
\(75\) 14.9032 1.72088
\(76\) −6.06059 −0.695197
\(77\) 0.460631 0.0524938
\(78\) −2.84996 −0.322694
\(79\) −12.7431 −1.43371 −0.716853 0.697225i \(-0.754418\pi\)
−0.716853 + 0.697225i \(0.754418\pi\)
\(80\) 0.345356 0.0386120
\(81\) 13.1421 1.46024
\(82\) 0.793029 0.0875753
\(83\) 6.10241 0.669826 0.334913 0.942249i \(-0.391293\pi\)
0.334913 + 0.942249i \(0.391293\pi\)
\(84\) 0.855025 0.0932908
\(85\) −1.94937 −0.211439
\(86\) 1.15782 0.124851
\(87\) 22.4768 2.40976
\(88\) 6.12763 0.653207
\(89\) 6.44432 0.683097 0.341548 0.939864i \(-0.389049\pi\)
0.341548 + 0.939864i \(0.389049\pi\)
\(90\) −1.88070 −0.198243
\(91\) −0.239034 −0.0250576
\(92\) −2.19111 −0.228439
\(93\) −10.2581 −1.06372
\(94\) 0.820122 0.0845892
\(95\) 1.63903 0.168161
\(96\) 18.0105 1.83818
\(97\) −12.9886 −1.31879 −0.659394 0.751797i \(-0.729187\pi\)
−0.659394 + 0.751797i \(0.729187\pi\)
\(98\) 5.26618 0.531965
\(99\) 15.2166 1.52933
\(100\) 6.92626 0.692626
\(101\) 6.18139 0.615072 0.307536 0.951537i \(-0.400496\pi\)
0.307536 + 0.951537i \(0.400496\pi\)
\(102\) −11.7318 −1.16163
\(103\) 7.78250 0.766833 0.383416 0.923576i \(-0.374747\pi\)
0.383416 + 0.923576i \(0.374747\pi\)
\(104\) −3.17979 −0.311804
\(105\) −0.231233 −0.0225661
\(106\) −1.17477 −0.114104
\(107\) −8.18217 −0.791000 −0.395500 0.918466i \(-0.629429\pi\)
−0.395500 + 0.918466i \(0.629429\pi\)
\(108\) 15.0851 1.45157
\(109\) −5.32486 −0.510029 −0.255015 0.966937i \(-0.582080\pi\)
−0.255015 + 0.966937i \(0.582080\pi\)
\(110\) −0.690275 −0.0658151
\(111\) 6.67107 0.633190
\(112\) 0.174325 0.0164722
\(113\) −3.60573 −0.339199 −0.169599 0.985513i \(-0.554247\pi\)
−0.169599 + 0.985513i \(0.554247\pi\)
\(114\) 9.86411 0.923859
\(115\) 0.592565 0.0552570
\(116\) 10.4461 0.969892
\(117\) −7.89633 −0.730016
\(118\) −11.2291 −1.03372
\(119\) −0.983983 −0.0902016
\(120\) −3.07602 −0.280801
\(121\) −5.41501 −0.492274
\(122\) 4.12290 0.373270
\(123\) 3.22097 0.290425
\(124\) −4.76744 −0.428129
\(125\) −3.80386 −0.340228
\(126\) −0.949316 −0.0845718
\(127\) −17.8352 −1.58262 −0.791308 0.611418i \(-0.790599\pi\)
−0.791308 + 0.611418i \(0.790599\pi\)
\(128\) 9.72337 0.859432
\(129\) 4.70262 0.414043
\(130\) 0.358202 0.0314164
\(131\) −12.1841 −1.06453 −0.532267 0.846577i \(-0.678660\pi\)
−0.532267 + 0.846577i \(0.678660\pi\)
\(132\) 10.3669 0.902321
\(133\) 0.827331 0.0717387
\(134\) 8.56278 0.739711
\(135\) −4.07963 −0.351118
\(136\) −13.0896 −1.12242
\(137\) −13.9009 −1.18764 −0.593819 0.804599i \(-0.702380\pi\)
−0.593819 + 0.804599i \(0.702380\pi\)
\(138\) 3.56621 0.303576
\(139\) 4.99920 0.424027 0.212013 0.977267i \(-0.431998\pi\)
0.212013 + 0.977267i \(0.431998\pi\)
\(140\) −0.107465 −0.00908249
\(141\) 3.33101 0.280522
\(142\) 8.64099 0.725135
\(143\) −2.89820 −0.242360
\(144\) 5.75872 0.479893
\(145\) −2.82504 −0.234607
\(146\) 0.682462 0.0564809
\(147\) 21.3892 1.76415
\(148\) 3.10037 0.254849
\(149\) 17.4937 1.43314 0.716570 0.697515i \(-0.245711\pi\)
0.716570 + 0.697515i \(0.245711\pi\)
\(150\) −11.2731 −0.920443
\(151\) 0.480306 0.0390867 0.0195434 0.999809i \(-0.493779\pi\)
0.0195434 + 0.999809i \(0.493779\pi\)
\(152\) 11.0057 0.892682
\(153\) −32.5052 −2.62789
\(154\) −0.348429 −0.0280772
\(155\) 1.28931 0.103560
\(156\) −5.37965 −0.430717
\(157\) 0.117389 0.00936869 0.00468434 0.999989i \(-0.498509\pi\)
0.00468434 + 0.999989i \(0.498509\pi\)
\(158\) 9.63907 0.766843
\(159\) −4.77147 −0.378402
\(160\) −2.26368 −0.178959
\(161\) 0.299108 0.0235730
\(162\) −9.94093 −0.781033
\(163\) −12.5273 −0.981210 −0.490605 0.871382i \(-0.663224\pi\)
−0.490605 + 0.871382i \(0.663224\pi\)
\(164\) 1.49694 0.116891
\(165\) −2.80362 −0.218262
\(166\) −4.61597 −0.358268
\(167\) −0.744061 −0.0575772 −0.0287886 0.999586i \(-0.509165\pi\)
−0.0287886 + 0.999586i \(0.509165\pi\)
\(168\) −1.55268 −0.119792
\(169\) −11.4960 −0.884311
\(170\) 1.47454 0.113092
\(171\) 27.3303 2.09000
\(172\) 2.18554 0.166646
\(173\) 11.6073 0.882484 0.441242 0.897388i \(-0.354538\pi\)
0.441242 + 0.897388i \(0.354538\pi\)
\(174\) −17.0018 −1.28891
\(175\) −0.945505 −0.0714734
\(176\) 2.11363 0.159321
\(177\) −45.6081 −3.42812
\(178\) −4.87460 −0.365367
\(179\) −21.1974 −1.58437 −0.792183 0.610284i \(-0.791055\pi\)
−0.792183 + 0.610284i \(0.791055\pi\)
\(180\) −3.55005 −0.264605
\(181\) 11.9508 0.888299 0.444150 0.895953i \(-0.353506\pi\)
0.444150 + 0.895953i \(0.353506\pi\)
\(182\) 0.180809 0.0134025
\(183\) 16.7456 1.23787
\(184\) 3.97894 0.293332
\(185\) −0.838467 −0.0616453
\(186\) 7.75941 0.568948
\(187\) −11.9304 −0.872441
\(188\) 1.54808 0.112906
\(189\) −2.05927 −0.149790
\(190\) −1.23979 −0.0899438
\(191\) 24.2243 1.75281 0.876406 0.481574i \(-0.159935\pi\)
0.876406 + 0.481574i \(0.159935\pi\)
\(192\) −8.12792 −0.586582
\(193\) −5.78545 −0.416446 −0.208223 0.978081i \(-0.566768\pi\)
−0.208223 + 0.978081i \(0.566768\pi\)
\(194\) 9.82478 0.705378
\(195\) 1.45488 0.104186
\(196\) 9.94058 0.710042
\(197\) 4.06793 0.289828 0.144914 0.989444i \(-0.453709\pi\)
0.144914 + 0.989444i \(0.453709\pi\)
\(198\) −11.5101 −0.817989
\(199\) −11.4204 −0.809571 −0.404785 0.914412i \(-0.632654\pi\)
−0.404785 + 0.914412i \(0.632654\pi\)
\(200\) −12.5777 −0.889381
\(201\) 34.7786 2.45310
\(202\) −4.67571 −0.328982
\(203\) −1.42599 −0.100085
\(204\) −22.1453 −1.55048
\(205\) −0.404834 −0.0282748
\(206\) −5.88682 −0.410154
\(207\) 9.88084 0.686766
\(208\) −1.09682 −0.0760509
\(209\) 10.0311 0.693866
\(210\) 0.174909 0.0120699
\(211\) 18.0941 1.24565 0.622824 0.782362i \(-0.285985\pi\)
0.622824 + 0.782362i \(0.285985\pi\)
\(212\) −2.21753 −0.152301
\(213\) 35.0963 2.40476
\(214\) 6.18913 0.423080
\(215\) −0.591058 −0.0403098
\(216\) −27.3938 −1.86391
\(217\) 0.650804 0.0441794
\(218\) 4.02782 0.272798
\(219\) 2.77189 0.187307
\(220\) −1.30298 −0.0878469
\(221\) 6.19103 0.416454
\(222\) −5.04611 −0.338673
\(223\) −21.1665 −1.41741 −0.708706 0.705504i \(-0.750721\pi\)
−0.708706 + 0.705504i \(0.750721\pi\)
\(224\) −1.14263 −0.0763455
\(225\) −31.2341 −2.08227
\(226\) 2.72744 0.181426
\(227\) −13.3828 −0.888248 −0.444124 0.895965i \(-0.646485\pi\)
−0.444124 + 0.895965i \(0.646485\pi\)
\(228\) 18.6198 1.23312
\(229\) −12.5484 −0.829225 −0.414612 0.909998i \(-0.636083\pi\)
−0.414612 + 0.909998i \(0.636083\pi\)
\(230\) −0.448226 −0.0295552
\(231\) −1.41518 −0.0931122
\(232\) −18.9695 −1.24541
\(233\) 15.6538 1.02551 0.512757 0.858534i \(-0.328624\pi\)
0.512757 + 0.858534i \(0.328624\pi\)
\(234\) 5.97292 0.390462
\(235\) −0.418665 −0.0273107
\(236\) −21.1963 −1.37976
\(237\) 39.1501 2.54307
\(238\) 0.744302 0.0482459
\(239\) 3.45659 0.223588 0.111794 0.993731i \(-0.464340\pi\)
0.111794 + 0.993731i \(0.464340\pi\)
\(240\) −1.06103 −0.0684891
\(241\) 28.7843 1.85416 0.927080 0.374864i \(-0.122311\pi\)
0.927080 + 0.374864i \(0.122311\pi\)
\(242\) 4.09601 0.263301
\(243\) −8.68104 −0.556889
\(244\) 7.78249 0.498223
\(245\) −2.68834 −0.171752
\(246\) −2.43640 −0.155339
\(247\) −5.20541 −0.331212
\(248\) 8.65743 0.549747
\(249\) −18.7482 −1.18812
\(250\) 2.87731 0.181977
\(251\) −1.82824 −0.115397 −0.0576986 0.998334i \(-0.518376\pi\)
−0.0576986 + 0.998334i \(0.518376\pi\)
\(252\) −1.79195 −0.112882
\(253\) 3.62658 0.228001
\(254\) 13.4908 0.846490
\(255\) 5.98900 0.375046
\(256\) −12.6461 −0.790380
\(257\) 3.65882 0.228231 0.114116 0.993467i \(-0.463597\pi\)
0.114116 + 0.993467i \(0.463597\pi\)
\(258\) −3.55714 −0.221458
\(259\) −0.423232 −0.0262983
\(260\) 0.676152 0.0419332
\(261\) −47.1067 −2.91583
\(262\) 9.21630 0.569385
\(263\) −13.1581 −0.811362 −0.405681 0.914015i \(-0.632966\pi\)
−0.405681 + 0.914015i \(0.632966\pi\)
\(264\) −18.8257 −1.15864
\(265\) 0.599712 0.0368400
\(266\) −0.625807 −0.0383707
\(267\) −19.7987 −1.21166
\(268\) 16.1633 0.987332
\(269\) −30.0225 −1.83050 −0.915251 0.402885i \(-0.868008\pi\)
−0.915251 + 0.402885i \(0.868008\pi\)
\(270\) 3.08590 0.187802
\(271\) −9.04876 −0.549673 −0.274837 0.961491i \(-0.588624\pi\)
−0.274837 + 0.961491i \(0.588624\pi\)
\(272\) −4.51506 −0.273766
\(273\) 0.734377 0.0444465
\(274\) 10.5149 0.635229
\(275\) −11.4639 −0.691300
\(276\) 6.73168 0.405199
\(277\) 10.3482 0.621762 0.310881 0.950449i \(-0.399376\pi\)
0.310881 + 0.950449i \(0.399376\pi\)
\(278\) −3.78148 −0.226798
\(279\) 21.4989 1.28710
\(280\) 0.195152 0.0116625
\(281\) −19.3686 −1.15543 −0.577716 0.816238i \(-0.696056\pi\)
−0.577716 + 0.816238i \(0.696056\pi\)
\(282\) −2.51964 −0.150042
\(283\) 9.72949 0.578358 0.289179 0.957275i \(-0.406618\pi\)
0.289179 + 0.957275i \(0.406618\pi\)
\(284\) 16.3109 0.967876
\(285\) −5.03554 −0.298280
\(286\) 2.19225 0.129631
\(287\) −0.204347 −0.0120622
\(288\) −37.7462 −2.22421
\(289\) 8.48538 0.499140
\(290\) 2.13691 0.125484
\(291\) 39.9044 2.33924
\(292\) 1.28823 0.0753881
\(293\) 26.2321 1.53249 0.766247 0.642546i \(-0.222122\pi\)
0.766247 + 0.642546i \(0.222122\pi\)
\(294\) −16.1791 −0.943586
\(295\) 5.73235 0.333750
\(296\) −5.63012 −0.327244
\(297\) −24.9679 −1.44879
\(298\) −13.2325 −0.766540
\(299\) −1.88193 −0.108835
\(300\) −21.2793 −1.22856
\(301\) −0.298348 −0.0171965
\(302\) −0.363312 −0.0209062
\(303\) −18.9909 −1.09100
\(304\) 3.79626 0.217730
\(305\) −2.10470 −0.120515
\(306\) 24.5875 1.40557
\(307\) −13.7097 −0.782455 −0.391228 0.920294i \(-0.627950\pi\)
−0.391228 + 0.920294i \(0.627950\pi\)
\(308\) −0.657704 −0.0374762
\(309\) −23.9099 −1.36019
\(310\) −0.975257 −0.0553909
\(311\) 30.9043 1.75242 0.876212 0.481926i \(-0.160063\pi\)
0.876212 + 0.481926i \(0.160063\pi\)
\(312\) 9.76918 0.553071
\(313\) −28.2287 −1.59558 −0.797790 0.602935i \(-0.793998\pi\)
−0.797790 + 0.602935i \(0.793998\pi\)
\(314\) −0.0887953 −0.00501101
\(315\) 0.484617 0.0273051
\(316\) 18.1949 1.02355
\(317\) −7.55793 −0.424496 −0.212248 0.977216i \(-0.568078\pi\)
−0.212248 + 0.977216i \(0.568078\pi\)
\(318\) 3.60922 0.202395
\(319\) −17.2896 −0.968034
\(320\) 1.02157 0.0571077
\(321\) 25.1378 1.40306
\(322\) −0.226251 −0.0126085
\(323\) −21.4281 −1.19229
\(324\) −18.7648 −1.04249
\(325\) 5.94894 0.329988
\(326\) 9.47583 0.524818
\(327\) 16.3594 0.904678
\(328\) −2.71837 −0.150097
\(329\) −0.211329 −0.0116509
\(330\) 2.12071 0.116741
\(331\) 20.4225 1.12252 0.561260 0.827640i \(-0.310317\pi\)
0.561260 + 0.827640i \(0.310317\pi\)
\(332\) −8.71322 −0.478200
\(333\) −13.9812 −0.766164
\(334\) 0.562821 0.0307962
\(335\) −4.37122 −0.238825
\(336\) −0.535574 −0.0292179
\(337\) 16.7603 0.912994 0.456497 0.889725i \(-0.349104\pi\)
0.456497 + 0.889725i \(0.349104\pi\)
\(338\) 8.69581 0.472990
\(339\) 11.0778 0.601662
\(340\) 2.78338 0.150950
\(341\) 7.89077 0.427309
\(342\) −20.6731 −1.11788
\(343\) −2.72138 −0.146941
\(344\) −3.96882 −0.213985
\(345\) −1.82052 −0.0980135
\(346\) −8.77994 −0.472012
\(347\) 14.1418 0.759171 0.379586 0.925157i \(-0.376067\pi\)
0.379586 + 0.925157i \(0.376067\pi\)
\(348\) −32.0931 −1.72037
\(349\) −22.1523 −1.18579 −0.592893 0.805282i \(-0.702014\pi\)
−0.592893 + 0.805282i \(0.702014\pi\)
\(350\) 0.715196 0.0382288
\(351\) 12.9565 0.691568
\(352\) −13.8540 −0.738423
\(353\) 17.2209 0.916574 0.458287 0.888804i \(-0.348463\pi\)
0.458287 + 0.888804i \(0.348463\pi\)
\(354\) 34.4988 1.83359
\(355\) −4.41114 −0.234119
\(356\) −9.20141 −0.487674
\(357\) 3.02306 0.159997
\(358\) 16.0341 0.847426
\(359\) 21.9672 1.15938 0.579692 0.814836i \(-0.303173\pi\)
0.579692 + 0.814836i \(0.303173\pi\)
\(360\) 6.44671 0.339771
\(361\) −0.983329 −0.0517541
\(362\) −9.03983 −0.475123
\(363\) 16.6364 0.873183
\(364\) 0.341300 0.0178890
\(365\) −0.348390 −0.0182356
\(366\) −12.6667 −0.662097
\(367\) −13.1841 −0.688202 −0.344101 0.938933i \(-0.611816\pi\)
−0.344101 + 0.938933i \(0.611816\pi\)
\(368\) 1.37248 0.0715452
\(369\) −6.75048 −0.351416
\(370\) 0.634231 0.0329721
\(371\) 0.302716 0.0157162
\(372\) 14.6469 0.759404
\(373\) −35.2516 −1.82526 −0.912630 0.408787i \(-0.865952\pi\)
−0.912630 + 0.408787i \(0.865952\pi\)
\(374\) 9.02440 0.466641
\(375\) 11.6865 0.603488
\(376\) −2.81124 −0.144979
\(377\) 8.97207 0.462085
\(378\) 1.55767 0.0801177
\(379\) 23.6003 1.21227 0.606134 0.795363i \(-0.292720\pi\)
0.606134 + 0.795363i \(0.292720\pi\)
\(380\) −2.34026 −0.120053
\(381\) 54.7945 2.80721
\(382\) −18.3237 −0.937522
\(383\) −13.1311 −0.670970 −0.335485 0.942046i \(-0.608900\pi\)
−0.335485 + 0.942046i \(0.608900\pi\)
\(384\) −29.8728 −1.52444
\(385\) 0.177870 0.00906509
\(386\) 4.37622 0.222744
\(387\) −9.85572 −0.500994
\(388\) 18.5455 0.941505
\(389\) 29.2216 1.48159 0.740797 0.671729i \(-0.234448\pi\)
0.740797 + 0.671729i \(0.234448\pi\)
\(390\) −1.10049 −0.0557257
\(391\) −7.74697 −0.391781
\(392\) −18.0516 −0.911743
\(393\) 37.4330 1.88824
\(394\) −3.07706 −0.155020
\(395\) −4.92065 −0.247585
\(396\) −21.7268 −1.09181
\(397\) 26.2129 1.31559 0.657795 0.753197i \(-0.271489\pi\)
0.657795 + 0.753197i \(0.271489\pi\)
\(398\) 8.63859 0.433013
\(399\) −2.54178 −0.127248
\(400\) −4.33850 −0.216925
\(401\) −24.3910 −1.21803 −0.609014 0.793160i \(-0.708435\pi\)
−0.609014 + 0.793160i \(0.708435\pi\)
\(402\) −26.3072 −1.31208
\(403\) −4.09473 −0.203973
\(404\) −8.82599 −0.439110
\(405\) 5.07476 0.252167
\(406\) 1.07864 0.0535322
\(407\) −5.13154 −0.254361
\(408\) 40.2148 1.99093
\(409\) 13.2071 0.653051 0.326526 0.945188i \(-0.394122\pi\)
0.326526 + 0.945188i \(0.394122\pi\)
\(410\) 0.306223 0.0151233
\(411\) 42.7074 2.10660
\(412\) −11.1121 −0.547454
\(413\) 2.89351 0.142380
\(414\) −7.47404 −0.367329
\(415\) 2.35641 0.115672
\(416\) 7.18924 0.352481
\(417\) −15.3589 −0.752129
\(418\) −7.58770 −0.371126
\(419\) 15.8597 0.774799 0.387400 0.921912i \(-0.373373\pi\)
0.387400 + 0.921912i \(0.373373\pi\)
\(420\) 0.330163 0.0161103
\(421\) −4.53957 −0.221245 −0.110622 0.993863i \(-0.535284\pi\)
−0.110622 + 0.993863i \(0.535284\pi\)
\(422\) −13.6867 −0.666257
\(423\) −6.98111 −0.339433
\(424\) 4.02693 0.195565
\(425\) 24.4888 1.18788
\(426\) −26.5474 −1.28623
\(427\) −1.06239 −0.0514126
\(428\) 11.6828 0.564708
\(429\) 8.90406 0.429892
\(430\) 0.447087 0.0215604
\(431\) −24.0460 −1.15825 −0.579126 0.815238i \(-0.696606\pi\)
−0.579126 + 0.815238i \(0.696606\pi\)
\(432\) −9.44907 −0.454619
\(433\) −20.8069 −0.999914 −0.499957 0.866050i \(-0.666651\pi\)
−0.499957 + 0.866050i \(0.666651\pi\)
\(434\) −0.492279 −0.0236302
\(435\) 8.67928 0.416140
\(436\) 7.60301 0.364118
\(437\) 6.51364 0.311590
\(438\) −2.09671 −0.100185
\(439\) −27.0822 −1.29256 −0.646282 0.763098i \(-0.723677\pi\)
−0.646282 + 0.763098i \(0.723677\pi\)
\(440\) 2.36615 0.112802
\(441\) −44.8273 −2.13463
\(442\) −4.68301 −0.222748
\(443\) 26.1108 1.24056 0.620282 0.784379i \(-0.287018\pi\)
0.620282 + 0.784379i \(0.287018\pi\)
\(444\) −9.52517 −0.452045
\(445\) 2.48844 0.117963
\(446\) 16.0107 0.758129
\(447\) −53.7454 −2.54207
\(448\) 0.515658 0.0243626
\(449\) 18.4224 0.869407 0.434703 0.900574i \(-0.356853\pi\)
0.434703 + 0.900574i \(0.356853\pi\)
\(450\) 23.6260 1.11374
\(451\) −2.47764 −0.116668
\(452\) 5.14838 0.242159
\(453\) −1.47563 −0.0693311
\(454\) 10.1230 0.475095
\(455\) −0.0923016 −0.00432716
\(456\) −33.8125 −1.58342
\(457\) −3.55147 −0.166131 −0.0830654 0.996544i \(-0.526471\pi\)
−0.0830654 + 0.996544i \(0.526471\pi\)
\(458\) 9.49186 0.443526
\(459\) 53.3355 2.48949
\(460\) −0.846084 −0.0394489
\(461\) −2.86665 −0.133513 −0.0667565 0.997769i \(-0.521265\pi\)
−0.0667565 + 0.997769i \(0.521265\pi\)
\(462\) 1.07047 0.0498027
\(463\) 30.5287 1.41879 0.709395 0.704811i \(-0.248968\pi\)
0.709395 + 0.704811i \(0.248968\pi\)
\(464\) −6.54324 −0.303762
\(465\) −3.96111 −0.183692
\(466\) −11.8408 −0.548515
\(467\) 18.7000 0.865332 0.432666 0.901554i \(-0.357573\pi\)
0.432666 + 0.901554i \(0.357573\pi\)
\(468\) 11.2746 0.521170
\(469\) −2.20646 −0.101885
\(470\) 0.316685 0.0146076
\(471\) −0.360652 −0.0166179
\(472\) 38.4914 1.77171
\(473\) −3.61736 −0.166326
\(474\) −29.6138 −1.36021
\(475\) −20.5901 −0.944740
\(476\) 1.40496 0.0643964
\(477\) 10.0000 0.457869
\(478\) −2.61463 −0.119590
\(479\) 7.50976 0.343130 0.171565 0.985173i \(-0.445118\pi\)
0.171565 + 0.985173i \(0.445118\pi\)
\(480\) 6.95463 0.317434
\(481\) 2.66289 0.121418
\(482\) −21.7729 −0.991730
\(483\) −0.918941 −0.0418133
\(484\) 7.73173 0.351442
\(485\) −5.01546 −0.227740
\(486\) 6.56649 0.297862
\(487\) 32.6786 1.48081 0.740404 0.672162i \(-0.234634\pi\)
0.740404 + 0.672162i \(0.234634\pi\)
\(488\) −14.1326 −0.639753
\(489\) 38.4871 1.74045
\(490\) 2.03351 0.0918644
\(491\) 2.81716 0.127137 0.0635683 0.997977i \(-0.479752\pi\)
0.0635683 + 0.997977i \(0.479752\pi\)
\(492\) −4.59901 −0.207339
\(493\) 36.9335 1.66340
\(494\) 3.93746 0.177155
\(495\) 5.87582 0.264098
\(496\) 2.98625 0.134087
\(497\) −2.22661 −0.0998770
\(498\) 14.1815 0.635488
\(499\) 29.0591 1.30086 0.650432 0.759564i \(-0.274588\pi\)
0.650432 + 0.759564i \(0.274588\pi\)
\(500\) 5.43128 0.242894
\(501\) 2.28596 0.102129
\(502\) 1.38291 0.0617223
\(503\) −1.00000 −0.0445878
\(504\) 3.25410 0.144949
\(505\) 2.38691 0.106216
\(506\) −2.74321 −0.121951
\(507\) 35.3189 1.56857
\(508\) 25.4656 1.12986
\(509\) 7.58042 0.335996 0.167998 0.985787i \(-0.446270\pi\)
0.167998 + 0.985787i \(0.446270\pi\)
\(510\) −4.53018 −0.200600
\(511\) −0.175857 −0.00777944
\(512\) −9.88103 −0.436684
\(513\) −44.8444 −1.97993
\(514\) −2.76760 −0.122074
\(515\) 3.00517 0.132423
\(516\) −6.71456 −0.295592
\(517\) −2.56229 −0.112689
\(518\) 0.320140 0.0140661
\(519\) −35.6607 −1.56533
\(520\) −1.22786 −0.0538451
\(521\) −8.90148 −0.389981 −0.194990 0.980805i \(-0.562468\pi\)
−0.194990 + 0.980805i \(0.562468\pi\)
\(522\) 35.6323 1.55958
\(523\) 37.7311 1.64986 0.824932 0.565232i \(-0.191213\pi\)
0.824932 + 0.565232i \(0.191213\pi\)
\(524\) 17.3969 0.759988
\(525\) 2.90484 0.126778
\(526\) 9.95301 0.433972
\(527\) −16.8560 −0.734257
\(528\) −6.49365 −0.282600
\(529\) −20.6451 −0.897613
\(530\) −0.453632 −0.0197045
\(531\) 95.5852 4.14804
\(532\) −1.18129 −0.0512154
\(533\) 1.28572 0.0556905
\(534\) 14.9761 0.648078
\(535\) −3.15950 −0.136597
\(536\) −29.3518 −1.26780
\(537\) 65.1240 2.81031
\(538\) 22.7095 0.979076
\(539\) −16.4530 −0.708682
\(540\) 5.82503 0.250669
\(541\) 11.6293 0.499981 0.249990 0.968248i \(-0.419573\pi\)
0.249990 + 0.968248i \(0.419573\pi\)
\(542\) 6.84464 0.294003
\(543\) −36.7162 −1.57564
\(544\) 29.5945 1.26885
\(545\) −2.05616 −0.0880764
\(546\) −0.555495 −0.0237730
\(547\) −13.4958 −0.577040 −0.288520 0.957474i \(-0.593163\pi\)
−0.288520 + 0.957474i \(0.593163\pi\)
\(548\) 19.8482 0.847874
\(549\) −35.0953 −1.49783
\(550\) 8.67150 0.369754
\(551\) −31.0536 −1.32293
\(552\) −12.2244 −0.520304
\(553\) −2.48379 −0.105622
\(554\) −7.82754 −0.332560
\(555\) 2.57599 0.109345
\(556\) −7.13803 −0.302720
\(557\) −34.2002 −1.44911 −0.724554 0.689218i \(-0.757954\pi\)
−0.724554 + 0.689218i \(0.757954\pi\)
\(558\) −16.2621 −0.688430
\(559\) 1.87715 0.0793949
\(560\) 0.0673146 0.00284456
\(561\) 36.6536 1.54751
\(562\) 14.6507 0.618003
\(563\) 25.6005 1.07893 0.539467 0.842007i \(-0.318626\pi\)
0.539467 + 0.842007i \(0.318626\pi\)
\(564\) −4.75613 −0.200269
\(565\) −1.39233 −0.0585758
\(566\) −7.35956 −0.309345
\(567\) 2.56158 0.107576
\(568\) −29.6199 −1.24282
\(569\) −17.2311 −0.722363 −0.361182 0.932495i \(-0.617627\pi\)
−0.361182 + 0.932495i \(0.617627\pi\)
\(570\) 3.80897 0.159540
\(571\) 2.79325 0.116894 0.0584470 0.998291i \(-0.481385\pi\)
0.0584470 + 0.998291i \(0.481385\pi\)
\(572\) 4.13815 0.173025
\(573\) −74.4237 −3.10909
\(574\) 0.154572 0.00645171
\(575\) −7.44403 −0.310438
\(576\) 17.0344 0.709768
\(577\) −40.1143 −1.66998 −0.834991 0.550264i \(-0.814527\pi\)
−0.834991 + 0.550264i \(0.814527\pi\)
\(578\) −6.41849 −0.266974
\(579\) 17.7745 0.738682
\(580\) 4.03368 0.167490
\(581\) 1.18944 0.0493464
\(582\) −30.1844 −1.25118
\(583\) 3.67032 0.152009
\(584\) −2.33936 −0.0968036
\(585\) −3.04912 −0.126066
\(586\) −19.8424 −0.819682
\(587\) −38.5445 −1.59090 −0.795451 0.606018i \(-0.792766\pi\)
−0.795451 + 0.606018i \(0.792766\pi\)
\(588\) −30.5402 −1.25945
\(589\) 14.1725 0.583966
\(590\) −4.33605 −0.178512
\(591\) −12.4978 −0.514090
\(592\) −1.94202 −0.0798167
\(593\) −25.0372 −1.02816 −0.514078 0.857744i \(-0.671866\pi\)
−0.514078 + 0.857744i \(0.671866\pi\)
\(594\) 18.8862 0.774909
\(595\) −0.379959 −0.0155768
\(596\) −24.9781 −1.02314
\(597\) 35.0865 1.43600
\(598\) 1.42353 0.0582123
\(599\) −23.0220 −0.940655 −0.470328 0.882492i \(-0.655864\pi\)
−0.470328 + 0.882492i \(0.655864\pi\)
\(600\) 38.6422 1.57756
\(601\) −33.9451 −1.38465 −0.692326 0.721585i \(-0.743414\pi\)
−0.692326 + 0.721585i \(0.743414\pi\)
\(602\) 0.225675 0.00919784
\(603\) −72.8888 −2.96826
\(604\) −0.685796 −0.0279046
\(605\) −2.09097 −0.0850102
\(606\) 14.3650 0.583540
\(607\) 24.3039 0.986467 0.493233 0.869897i \(-0.335815\pi\)
0.493233 + 0.869897i \(0.335815\pi\)
\(608\) −24.8830 −1.00914
\(609\) 4.38103 0.177528
\(610\) 1.59203 0.0644595
\(611\) 1.32964 0.0537916
\(612\) 46.4120 1.87609
\(613\) 34.5604 1.39588 0.697940 0.716156i \(-0.254100\pi\)
0.697940 + 0.716156i \(0.254100\pi\)
\(614\) 10.3703 0.418510
\(615\) 1.24376 0.0501532
\(616\) 1.19436 0.0481220
\(617\) 34.3607 1.38331 0.691655 0.722228i \(-0.256882\pi\)
0.691655 + 0.722228i \(0.256882\pi\)
\(618\) 18.0859 0.727521
\(619\) 1.08451 0.0435902 0.0217951 0.999762i \(-0.493062\pi\)
0.0217951 + 0.999762i \(0.493062\pi\)
\(620\) −1.84092 −0.0739331
\(621\) −16.2128 −0.650597
\(622\) −23.3766 −0.937315
\(623\) 1.25609 0.0503240
\(624\) 3.36973 0.134897
\(625\) 22.7856 0.911425
\(626\) 21.3527 0.853425
\(627\) −30.8182 −1.23076
\(628\) −0.167612 −0.00668846
\(629\) 10.9618 0.437076
\(630\) −0.366573 −0.0146046
\(631\) −6.03426 −0.240220 −0.120110 0.992761i \(-0.538325\pi\)
−0.120110 + 0.992761i \(0.538325\pi\)
\(632\) −33.0411 −1.31430
\(633\) −55.5899 −2.20950
\(634\) 5.71695 0.227049
\(635\) −6.88695 −0.273300
\(636\) 6.81286 0.270148
\(637\) 8.53792 0.338285
\(638\) 13.0782 0.517770
\(639\) −73.5545 −2.90977
\(640\) 3.75462 0.148414
\(641\) −22.0716 −0.871778 −0.435889 0.900001i \(-0.643566\pi\)
−0.435889 + 0.900001i \(0.643566\pi\)
\(642\) −19.0147 −0.750450
\(643\) −40.6373 −1.60258 −0.801290 0.598277i \(-0.795852\pi\)
−0.801290 + 0.598277i \(0.795852\pi\)
\(644\) −0.427077 −0.0168292
\(645\) 1.81589 0.0715006
\(646\) 16.2086 0.637717
\(647\) 22.9719 0.903118 0.451559 0.892241i \(-0.350868\pi\)
0.451559 + 0.892241i \(0.350868\pi\)
\(648\) 34.0759 1.33863
\(649\) 35.0828 1.37712
\(650\) −4.49988 −0.176500
\(651\) −1.99944 −0.0783644
\(652\) 17.8868 0.700502
\(653\) −39.0459 −1.52798 −0.763992 0.645226i \(-0.776763\pi\)
−0.763992 + 0.645226i \(0.776763\pi\)
\(654\) −12.3745 −0.483883
\(655\) −4.70483 −0.183833
\(656\) −0.937660 −0.0366095
\(657\) −5.80931 −0.226643
\(658\) 0.159853 0.00623172
\(659\) −26.6054 −1.03640 −0.518199 0.855260i \(-0.673397\pi\)
−0.518199 + 0.855260i \(0.673397\pi\)
\(660\) 4.00311 0.155821
\(661\) −39.3661 −1.53116 −0.765582 0.643338i \(-0.777549\pi\)
−0.765582 + 0.643338i \(0.777549\pi\)
\(662\) −15.4479 −0.600400
\(663\) −19.0205 −0.738696
\(664\) 15.8228 0.614042
\(665\) 0.319469 0.0123885
\(666\) 10.5756 0.409796
\(667\) −11.2269 −0.434709
\(668\) 1.06239 0.0411053
\(669\) 65.0291 2.51417
\(670\) 3.30647 0.127740
\(671\) −12.8811 −0.497269
\(672\) 3.51048 0.135420
\(673\) −13.4145 −0.517090 −0.258545 0.965999i \(-0.583243\pi\)
−0.258545 + 0.965999i \(0.583243\pi\)
\(674\) −12.6778 −0.488331
\(675\) 51.2499 1.97261
\(676\) 16.4144 0.631324
\(677\) −2.13435 −0.0820299 −0.0410149 0.999159i \(-0.513059\pi\)
−0.0410149 + 0.999159i \(0.513059\pi\)
\(678\) −8.37942 −0.321810
\(679\) −2.53165 −0.0971557
\(680\) −5.05448 −0.193830
\(681\) 41.1156 1.57555
\(682\) −5.96871 −0.228554
\(683\) 21.3544 0.817105 0.408552 0.912735i \(-0.366034\pi\)
0.408552 + 0.912735i \(0.366034\pi\)
\(684\) −39.0231 −1.49209
\(685\) −5.36777 −0.205092
\(686\) 2.05850 0.0785940
\(687\) 38.5522 1.47086
\(688\) −1.36899 −0.0521921
\(689\) −1.90463 −0.0725606
\(690\) 1.37707 0.0524242
\(691\) 13.8167 0.525614 0.262807 0.964848i \(-0.415352\pi\)
0.262807 + 0.964848i \(0.415352\pi\)
\(692\) −16.5732 −0.630020
\(693\) 2.96593 0.112666
\(694\) −10.6971 −0.406056
\(695\) 1.93041 0.0732247
\(696\) 58.2794 2.20908
\(697\) 5.29265 0.200473
\(698\) 16.7564 0.634238
\(699\) −48.0927 −1.81903
\(700\) 1.35002 0.0510260
\(701\) 45.1209 1.70419 0.852097 0.523385i \(-0.175331\pi\)
0.852097 + 0.523385i \(0.175331\pi\)
\(702\) −9.80054 −0.369898
\(703\) −9.21666 −0.347613
\(704\) 6.25217 0.235638
\(705\) 1.28625 0.0484430
\(706\) −13.0262 −0.490246
\(707\) 1.20484 0.0453126
\(708\) 65.1208 2.44739
\(709\) −37.6350 −1.41341 −0.706706 0.707507i \(-0.749820\pi\)
−0.706706 + 0.707507i \(0.749820\pi\)
\(710\) 3.33667 0.125223
\(711\) −82.0505 −3.07713
\(712\) 16.7093 0.626208
\(713\) 5.12383 0.191889
\(714\) −2.28670 −0.0855774
\(715\) −1.11912 −0.0418529
\(716\) 30.2663 1.13110
\(717\) −10.6196 −0.396596
\(718\) −16.6164 −0.620117
\(719\) −34.9860 −1.30476 −0.652380 0.757892i \(-0.726229\pi\)
−0.652380 + 0.757892i \(0.726229\pi\)
\(720\) 2.22369 0.0828722
\(721\) 1.51691 0.0564929
\(722\) 0.743807 0.0276816
\(723\) −88.4331 −3.28886
\(724\) −17.0638 −0.634171
\(725\) 35.4892 1.31804
\(726\) −12.5840 −0.467038
\(727\) 0.0594046 0.00220319 0.00110160 0.999999i \(-0.499649\pi\)
0.00110160 + 0.999999i \(0.499649\pi\)
\(728\) −0.619784 −0.0229707
\(729\) −12.7559 −0.472440
\(730\) 0.263529 0.00975363
\(731\) 7.72727 0.285804
\(732\) −23.9099 −0.883736
\(733\) −23.6914 −0.875060 −0.437530 0.899204i \(-0.644147\pi\)
−0.437530 + 0.899204i \(0.644147\pi\)
\(734\) 9.97265 0.368097
\(735\) 8.25930 0.304649
\(736\) −8.99605 −0.331599
\(737\) −26.7525 −0.985441
\(738\) 5.10618 0.187961
\(739\) −13.2539 −0.487551 −0.243776 0.969832i \(-0.578386\pi\)
−0.243776 + 0.969832i \(0.578386\pi\)
\(740\) 1.19719 0.0440096
\(741\) 15.9924 0.587496
\(742\) −0.228979 −0.00840610
\(743\) −13.7983 −0.506210 −0.253105 0.967439i \(-0.581452\pi\)
−0.253105 + 0.967439i \(0.581452\pi\)
\(744\) −26.5980 −0.975129
\(745\) 6.75509 0.247487
\(746\) 26.6649 0.976273
\(747\) 39.2924 1.43763
\(748\) 17.0347 0.622850
\(749\) −1.59481 −0.0582733
\(750\) −8.83987 −0.322786
\(751\) 3.82043 0.139410 0.0697048 0.997568i \(-0.477794\pi\)
0.0697048 + 0.997568i \(0.477794\pi\)
\(752\) −0.969695 −0.0353611
\(753\) 5.61683 0.204689
\(754\) −6.78663 −0.247154
\(755\) 0.185467 0.00674984
\(756\) 2.94029 0.106937
\(757\) 19.6215 0.713157 0.356578 0.934265i \(-0.383943\pi\)
0.356578 + 0.934265i \(0.383943\pi\)
\(758\) −17.8517 −0.648403
\(759\) −11.1418 −0.404423
\(760\) 4.24979 0.154156
\(761\) 47.5141 1.72238 0.861192 0.508280i \(-0.169719\pi\)
0.861192 + 0.508280i \(0.169719\pi\)
\(762\) −41.4475 −1.50148
\(763\) −1.03789 −0.0375741
\(764\) −34.5883 −1.25136
\(765\) −12.5517 −0.453808
\(766\) 9.93262 0.358880
\(767\) −18.2054 −0.657360
\(768\) 38.8521 1.40196
\(769\) −10.0139 −0.361111 −0.180556 0.983565i \(-0.557790\pi\)
−0.180556 + 0.983565i \(0.557790\pi\)
\(770\) −0.134544 −0.00484863
\(771\) −11.2409 −0.404831
\(772\) 8.26066 0.297308
\(773\) −35.8956 −1.29108 −0.645538 0.763728i \(-0.723367\pi\)
−0.645538 + 0.763728i \(0.723367\pi\)
\(774\) 7.45504 0.267966
\(775\) −16.1968 −0.581807
\(776\) −33.6777 −1.20896
\(777\) 1.30028 0.0466474
\(778\) −22.1037 −0.792457
\(779\) −4.45005 −0.159439
\(780\) −2.07732 −0.0743800
\(781\) −26.9968 −0.966023
\(782\) 5.85995 0.209551
\(783\) 77.2940 2.76226
\(784\) −6.22662 −0.222379
\(785\) 0.0453292 0.00161787
\(786\) −28.3149 −1.00996
\(787\) 1.89211 0.0674464 0.0337232 0.999431i \(-0.489264\pi\)
0.0337232 + 0.999431i \(0.489264\pi\)
\(788\) −5.80833 −0.206913
\(789\) 40.4252 1.43917
\(790\) 3.72207 0.132425
\(791\) −0.702806 −0.0249889
\(792\) 39.4548 1.40197
\(793\) 6.68434 0.237368
\(794\) −19.8279 −0.703667
\(795\) −1.84248 −0.0653459
\(796\) 16.3064 0.577966
\(797\) 24.4028 0.864393 0.432197 0.901779i \(-0.357739\pi\)
0.432197 + 0.901779i \(0.357739\pi\)
\(798\) 1.92265 0.0680610
\(799\) 5.47347 0.193637
\(800\) 28.4372 1.00541
\(801\) 41.4940 1.46612
\(802\) 18.4498 0.651484
\(803\) −2.13220 −0.0752437
\(804\) −49.6581 −1.75131
\(805\) 0.115499 0.00407080
\(806\) 3.09733 0.109099
\(807\) 92.2371 3.24690
\(808\) 16.0276 0.563848
\(809\) −32.9341 −1.15790 −0.578951 0.815363i \(-0.696538\pi\)
−0.578951 + 0.815363i \(0.696538\pi\)
\(810\) −3.83863 −0.134876
\(811\) −42.8145 −1.50342 −0.751711 0.659493i \(-0.770771\pi\)
−0.751711 + 0.659493i \(0.770771\pi\)
\(812\) 2.03608 0.0714523
\(813\) 27.8002 0.974997
\(814\) 3.88158 0.136049
\(815\) −4.83732 −0.169444
\(816\) 13.8715 0.485599
\(817\) −6.49708 −0.227304
\(818\) −9.99011 −0.349296
\(819\) −1.53910 −0.0537805
\(820\) 0.578035 0.0201859
\(821\) −28.2632 −0.986393 −0.493196 0.869918i \(-0.664172\pi\)
−0.493196 + 0.869918i \(0.664172\pi\)
\(822\) −32.3046 −1.12675
\(823\) 10.4127 0.362962 0.181481 0.983394i \(-0.441911\pi\)
0.181481 + 0.983394i \(0.441911\pi\)
\(824\) 20.1790 0.702970
\(825\) 35.2202 1.22621
\(826\) −2.18870 −0.0761547
\(827\) −0.0841446 −0.00292600 −0.00146300 0.999999i \(-0.500466\pi\)
−0.00146300 + 0.999999i \(0.500466\pi\)
\(828\) −14.1082 −0.490294
\(829\) 46.3470 1.60970 0.804850 0.593479i \(-0.202246\pi\)
0.804850 + 0.593479i \(0.202246\pi\)
\(830\) −1.78243 −0.0618690
\(831\) −31.7924 −1.10287
\(832\) −3.24442 −0.112480
\(833\) 35.1463 1.21775
\(834\) 11.6177 0.402289
\(835\) −0.287315 −0.00994294
\(836\) −14.3227 −0.495362
\(837\) −35.2760 −1.21932
\(838\) −11.9966 −0.414415
\(839\) −13.9611 −0.481990 −0.240995 0.970526i \(-0.577474\pi\)
−0.240995 + 0.970526i \(0.577474\pi\)
\(840\) −0.599559 −0.0206867
\(841\) 24.5241 0.845660
\(842\) 3.43381 0.118337
\(843\) 59.5054 2.04948
\(844\) −25.8353 −0.889289
\(845\) −4.43913 −0.152711
\(846\) 5.28063 0.181552
\(847\) −1.05546 −0.0362660
\(848\) 1.38903 0.0476994
\(849\) −29.8916 −1.02588
\(850\) −18.5237 −0.635359
\(851\) −3.33214 −0.114224
\(852\) −50.1116 −1.71680
\(853\) −54.5527 −1.86785 −0.933924 0.357472i \(-0.883639\pi\)
−0.933924 + 0.357472i \(0.883639\pi\)
\(854\) 0.803609 0.0274989
\(855\) 10.5534 0.360920
\(856\) −21.2153 −0.725124
\(857\) 27.6798 0.945525 0.472762 0.881190i \(-0.343257\pi\)
0.472762 + 0.881190i \(0.343257\pi\)
\(858\) −6.73519 −0.229935
\(859\) 27.1497 0.926336 0.463168 0.886270i \(-0.346713\pi\)
0.463168 + 0.886270i \(0.346713\pi\)
\(860\) 0.843932 0.0287778
\(861\) 0.627810 0.0213957
\(862\) 18.1888 0.619512
\(863\) 25.9641 0.883827 0.441913 0.897058i \(-0.354300\pi\)
0.441913 + 0.897058i \(0.354300\pi\)
\(864\) 61.9350 2.10707
\(865\) 4.48208 0.152395
\(866\) 15.7387 0.534822
\(867\) −26.0694 −0.885363
\(868\) −0.929239 −0.0315404
\(869\) −30.1151 −1.02159
\(870\) −6.56516 −0.222580
\(871\) 13.8826 0.470394
\(872\) −13.8067 −0.467554
\(873\) −83.6313 −2.83049
\(874\) −4.92703 −0.166659
\(875\) −0.741425 −0.0250647
\(876\) −3.95780 −0.133722
\(877\) 37.2176 1.25675 0.628374 0.777911i \(-0.283721\pi\)
0.628374 + 0.777911i \(0.283721\pi\)
\(878\) 20.4855 0.691351
\(879\) −80.5920 −2.71830
\(880\) 0.816166 0.0275130
\(881\) −6.44185 −0.217031 −0.108516 0.994095i \(-0.534610\pi\)
−0.108516 + 0.994095i \(0.534610\pi\)
\(882\) 33.9081 1.14175
\(883\) 45.9727 1.54711 0.773553 0.633732i \(-0.218478\pi\)
0.773553 + 0.633732i \(0.218478\pi\)
\(884\) −8.83976 −0.297313
\(885\) −17.6113 −0.591998
\(886\) −19.7507 −0.663538
\(887\) −28.3187 −0.950849 −0.475425 0.879756i \(-0.657706\pi\)
−0.475425 + 0.879756i \(0.657706\pi\)
\(888\) 17.2972 0.580457
\(889\) −3.47632 −0.116592
\(890\) −1.88230 −0.0630947
\(891\) 31.0582 1.04049
\(892\) 30.2222 1.01191
\(893\) −4.60208 −0.154003
\(894\) 40.6539 1.35967
\(895\) −8.18524 −0.273602
\(896\) 1.89522 0.0633147
\(897\) 5.78181 0.193049
\(898\) −13.9350 −0.465018
\(899\) −24.4277 −0.814710
\(900\) 44.5971 1.48657
\(901\) −7.84041 −0.261202
\(902\) 1.87413 0.0624017
\(903\) 0.916604 0.0305027
\(904\) −9.34920 −0.310950
\(905\) 4.61475 0.153399
\(906\) 1.11619 0.0370830
\(907\) −12.5099 −0.415384 −0.207692 0.978194i \(-0.566595\pi\)
−0.207692 + 0.978194i \(0.566595\pi\)
\(908\) 19.1084 0.634135
\(909\) 39.8010 1.32012
\(910\) 0.0698185 0.00231446
\(911\) 2.41940 0.0801584 0.0400792 0.999197i \(-0.487239\pi\)
0.0400792 + 0.999197i \(0.487239\pi\)
\(912\) −11.6631 −0.386204
\(913\) 14.4216 0.477284
\(914\) 2.68639 0.0888580
\(915\) 6.46621 0.213766
\(916\) 17.9171 0.591997
\(917\) −2.37485 −0.0784246
\(918\) −40.3439 −1.33155
\(919\) 50.3462 1.66077 0.830384 0.557192i \(-0.188121\pi\)
0.830384 + 0.557192i \(0.188121\pi\)
\(920\) 1.53645 0.0506551
\(921\) 42.1200 1.38790
\(922\) 2.16838 0.0714118
\(923\) 14.0094 0.461125
\(924\) 2.02064 0.0664743
\(925\) 10.5331 0.346328
\(926\) −23.0924 −0.758865
\(927\) 50.1103 1.64584
\(928\) 42.8884 1.40788
\(929\) −32.9858 −1.08223 −0.541115 0.840948i \(-0.681998\pi\)
−0.541115 + 0.840948i \(0.681998\pi\)
\(930\) 2.99625 0.0982509
\(931\) −29.5510 −0.968494
\(932\) −22.3510 −0.732131
\(933\) −94.9464 −3.10841
\(934\) −14.1450 −0.462838
\(935\) −4.60687 −0.150661
\(936\) −20.4742 −0.669219
\(937\) 47.9023 1.56490 0.782450 0.622714i \(-0.213970\pi\)
0.782450 + 0.622714i \(0.213970\pi\)
\(938\) 1.66900 0.0544948
\(939\) 86.7262 2.83020
\(940\) 0.597783 0.0194975
\(941\) −4.42362 −0.144206 −0.0721030 0.997397i \(-0.522971\pi\)
−0.0721030 + 0.997397i \(0.522971\pi\)
\(942\) 0.272803 0.00888841
\(943\) −1.60884 −0.0523912
\(944\) 13.2770 0.432131
\(945\) −0.795175 −0.0258670
\(946\) 2.73624 0.0889627
\(947\) 19.3253 0.627987 0.313994 0.949425i \(-0.398333\pi\)
0.313994 + 0.949425i \(0.398333\pi\)
\(948\) −55.8998 −1.81554
\(949\) 1.10646 0.0359171
\(950\) 15.5747 0.505311
\(951\) 23.2200 0.752960
\(952\) −2.55134 −0.0826895
\(953\) 30.0968 0.974931 0.487465 0.873142i \(-0.337922\pi\)
0.487465 + 0.873142i \(0.337922\pi\)
\(954\) −7.56418 −0.244899
\(955\) 9.35408 0.302691
\(956\) −4.93544 −0.159623
\(957\) 53.1184 1.71708
\(958\) −5.68051 −0.183529
\(959\) −2.70948 −0.0874937
\(960\) −3.13855 −0.101296
\(961\) −19.8515 −0.640371
\(962\) −2.01426 −0.0649423
\(963\) −52.6836 −1.69771
\(964\) −41.0992 −1.32372
\(965\) −2.23402 −0.0719156
\(966\) 0.695103 0.0223646
\(967\) −42.6607 −1.37187 −0.685937 0.727661i \(-0.740608\pi\)
−0.685937 + 0.727661i \(0.740608\pi\)
\(968\) −14.0404 −0.451277
\(969\) 65.8328 2.11485
\(970\) 3.79378 0.121811
\(971\) 33.0050 1.05918 0.529591 0.848253i \(-0.322345\pi\)
0.529591 + 0.848253i \(0.322345\pi\)
\(972\) 12.3951 0.397572
\(973\) 0.974412 0.0312382
\(974\) −24.7187 −0.792037
\(975\) −18.2767 −0.585324
\(976\) −4.87483 −0.156039
\(977\) −5.79949 −0.185542 −0.0927710 0.995687i \(-0.529572\pi\)
−0.0927710 + 0.995687i \(0.529572\pi\)
\(978\) −29.1123 −0.930909
\(979\) 15.2296 0.486740
\(980\) 3.83850 0.122616
\(981\) −34.2859 −1.09467
\(982\) −2.13095 −0.0680013
\(983\) −35.2304 −1.12367 −0.561837 0.827248i \(-0.689905\pi\)
−0.561837 + 0.827248i \(0.689905\pi\)
\(984\) 8.35156 0.266238
\(985\) 1.57081 0.0500501
\(986\) −27.9371 −0.889700
\(987\) 0.649260 0.0206662
\(988\) 7.43245 0.236458
\(989\) −2.34891 −0.0746911
\(990\) −4.44457 −0.141258
\(991\) 9.08023 0.288443 0.144221 0.989545i \(-0.453932\pi\)
0.144221 + 0.989545i \(0.453932\pi\)
\(992\) −19.5737 −0.621466
\(993\) −62.7433 −1.99110
\(994\) 1.68424 0.0534210
\(995\) −4.40992 −0.139804
\(996\) 26.7693 0.848219
\(997\) 59.6440 1.88894 0.944472 0.328591i \(-0.106574\pi\)
0.944472 + 0.328591i \(0.106574\pi\)
\(998\) −21.9808 −0.695791
\(999\) 22.9407 0.725813
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 503.2.a.e.1.5 10
3.2 odd 2 4527.2.a.k.1.6 10
4.3 odd 2 8048.2.a.p.1.10 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
503.2.a.e.1.5 10 1.1 even 1 trivial
4527.2.a.k.1.6 10 3.2 odd 2
8048.2.a.p.1.10 10 4.3 odd 2