Properties

Label 507.4.a.o.1.5
Level $507$
Weight $4$
Character 507.1
Self dual yes
Analytic conductor $29.914$
Analytic rank $1$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 507.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.9139683729\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \( x^{9} - 56x^{7} - 27x^{6} + 945x^{5} + 763x^{4} - 4139x^{3} - 2478x^{2} + 63x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 13^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(0.107680\) of defining polynomial
Character \(\chi\) \(=\) 507.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.447278 q^{2} +3.00000 q^{3} -7.79994 q^{4} -1.93073 q^{5} -1.34183 q^{6} +8.14537 q^{7} +7.06697 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-0.447278 q^{2} +3.00000 q^{3} -7.79994 q^{4} -1.93073 q^{5} -1.34183 q^{6} +8.14537 q^{7} +7.06697 q^{8} +9.00000 q^{9} +0.863573 q^{10} -8.40842 q^{11} -23.3998 q^{12} -3.64325 q^{14} -5.79219 q^{15} +59.2386 q^{16} -52.1271 q^{17} -4.02550 q^{18} -48.8304 q^{19} +15.0596 q^{20} +24.4361 q^{21} +3.76090 q^{22} +88.9229 q^{23} +21.2009 q^{24} -121.272 q^{25} +27.0000 q^{27} -63.5334 q^{28} +191.979 q^{29} +2.59072 q^{30} -115.257 q^{31} -83.0319 q^{32} -25.2252 q^{33} +23.3153 q^{34} -15.7265 q^{35} -70.1995 q^{36} +136.716 q^{37} +21.8408 q^{38} -13.6444 q^{40} -436.077 q^{41} -10.9297 q^{42} +202.048 q^{43} +65.5852 q^{44} -17.3766 q^{45} -39.7733 q^{46} -618.160 q^{47} +177.716 q^{48} -276.653 q^{49} +54.2425 q^{50} -156.381 q^{51} -453.170 q^{53} -12.0765 q^{54} +16.2344 q^{55} +57.5631 q^{56} -146.491 q^{57} -85.8679 q^{58} -500.044 q^{59} +45.1787 q^{60} +480.502 q^{61} +51.5522 q^{62} +73.3083 q^{63} -436.771 q^{64} +11.2827 q^{66} -886.769 q^{67} +406.589 q^{68} +266.769 q^{69} +7.03412 q^{70} -123.732 q^{71} +63.6027 q^{72} -673.168 q^{73} -61.1501 q^{74} -363.817 q^{75} +380.875 q^{76} -68.4897 q^{77} -681.298 q^{79} -114.374 q^{80} +81.0000 q^{81} +195.048 q^{82} +939.418 q^{83} -190.600 q^{84} +100.643 q^{85} -90.3716 q^{86} +575.936 q^{87} -59.4220 q^{88} +754.979 q^{89} +7.77215 q^{90} -693.594 q^{92} -345.772 q^{93} +276.489 q^{94} +94.2783 q^{95} -249.096 q^{96} -1051.10 q^{97} +123.741 q^{98} -75.6757 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 6 q^{2} + 27 q^{3} + 44 q^{4} - 33 q^{5} - 18 q^{6} - 83 q^{7} - 87 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 6 q^{2} + 27 q^{3} + 44 q^{4} - 33 q^{5} - 18 q^{6} - 83 q^{7} - 87 q^{8} + 81 q^{9} - 54 q^{10} - 85 q^{11} + 132 q^{12} + 158 q^{14} - 99 q^{15} + 216 q^{16} + 178 q^{17} - 54 q^{18} - 352 q^{19} - 402 q^{20} - 249 q^{21} - 630 q^{22} + 150 q^{23} - 261 q^{24} - 20 q^{25} + 243 q^{27} - 940 q^{28} - 97 q^{29} - 162 q^{30} - 717 q^{31} - 707 q^{32} - 255 q^{33} - 632 q^{34} - 418 q^{35} + 396 q^{36} - 1108 q^{37} - 660 q^{38} - 1506 q^{40} - 334 q^{41} + 474 q^{42} + 242 q^{43} + 307 q^{44} - 297 q^{45} - 979 q^{46} + 184 q^{47} + 648 q^{48} - 38 q^{49} + 2031 q^{50} + 534 q^{51} - 151 q^{53} - 162 q^{54} + 2064 q^{55} + 2276 q^{56} - 1056 q^{57} - 1161 q^{58} - 537 q^{59} - 1206 q^{60} - 1340 q^{61} + 347 q^{62} - 747 q^{63} + 893 q^{64} - 1890 q^{66} - 2308 q^{67} + 2785 q^{68} + 450 q^{69} + 1420 q^{70} - 96 q^{71} - 783 q^{72} - 2505 q^{73} - 1191 q^{74} - 60 q^{75} - 2409 q^{76} - 2142 q^{77} - 1591 q^{79} + 2671 q^{80} + 729 q^{81} + 1517 q^{82} - 1539 q^{83} - 2820 q^{84} - 4296 q^{85} + 3763 q^{86} - 291 q^{87} - 3716 q^{88} + 592 q^{89} - 486 q^{90} + 515 q^{92} - 2151 q^{93} - 692 q^{94} + 4158 q^{95} - 2121 q^{96} - 1445 q^{97} - 1457 q^{98} - 765 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.447278 −0.158137 −0.0790684 0.996869i \(-0.525195\pi\)
−0.0790684 + 0.996869i \(0.525195\pi\)
\(3\) 3.00000 0.577350
\(4\) −7.79994 −0.974993
\(5\) −1.93073 −0.172690 −0.0863448 0.996265i \(-0.527519\pi\)
−0.0863448 + 0.996265i \(0.527519\pi\)
\(6\) −1.34183 −0.0913003
\(7\) 8.14537 0.439809 0.219904 0.975521i \(-0.429425\pi\)
0.219904 + 0.975521i \(0.429425\pi\)
\(8\) 7.06697 0.312319
\(9\) 9.00000 0.333333
\(10\) 0.863573 0.0273086
\(11\) −8.40842 −0.230476 −0.115238 0.993338i \(-0.536763\pi\)
−0.115238 + 0.993338i \(0.536763\pi\)
\(12\) −23.3998 −0.562912
\(13\) 0 0
\(14\) −3.64325 −0.0695499
\(15\) −5.79219 −0.0997024
\(16\) 59.2386 0.925604
\(17\) −52.1271 −0.743688 −0.371844 0.928295i \(-0.621274\pi\)
−0.371844 + 0.928295i \(0.621274\pi\)
\(18\) −4.02550 −0.0527122
\(19\) −48.8304 −0.589604 −0.294802 0.955558i \(-0.595254\pi\)
−0.294802 + 0.955558i \(0.595254\pi\)
\(20\) 15.0596 0.168371
\(21\) 24.4361 0.253924
\(22\) 3.76090 0.0364467
\(23\) 88.9229 0.806162 0.403081 0.915164i \(-0.367939\pi\)
0.403081 + 0.915164i \(0.367939\pi\)
\(24\) 21.2009 0.180317
\(25\) −121.272 −0.970178
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) −63.5334 −0.428810
\(29\) 191.979 1.22930 0.614648 0.788802i \(-0.289298\pi\)
0.614648 + 0.788802i \(0.289298\pi\)
\(30\) 2.59072 0.0157666
\(31\) −115.257 −0.667769 −0.333885 0.942614i \(-0.608360\pi\)
−0.333885 + 0.942614i \(0.608360\pi\)
\(32\) −83.0319 −0.458691
\(33\) −25.2252 −0.133065
\(34\) 23.3153 0.117604
\(35\) −15.7265 −0.0759504
\(36\) −70.1995 −0.324998
\(37\) 136.716 0.607459 0.303729 0.952758i \(-0.401768\pi\)
0.303729 + 0.952758i \(0.401768\pi\)
\(38\) 21.8408 0.0932380
\(39\) 0 0
\(40\) −13.6444 −0.0539342
\(41\) −436.077 −1.66107 −0.830535 0.556967i \(-0.811965\pi\)
−0.830535 + 0.556967i \(0.811965\pi\)
\(42\) −10.9297 −0.0401547
\(43\) 202.048 0.716559 0.358279 0.933614i \(-0.383364\pi\)
0.358279 + 0.933614i \(0.383364\pi\)
\(44\) 65.5852 0.224712
\(45\) −17.3766 −0.0575632
\(46\) −39.7733 −0.127484
\(47\) −618.160 −1.91847 −0.959233 0.282617i \(-0.908798\pi\)
−0.959233 + 0.282617i \(0.908798\pi\)
\(48\) 177.716 0.534398
\(49\) −276.653 −0.806568
\(50\) 54.2425 0.153421
\(51\) −156.381 −0.429368
\(52\) 0 0
\(53\) −453.170 −1.17448 −0.587242 0.809411i \(-0.699786\pi\)
−0.587242 + 0.809411i \(0.699786\pi\)
\(54\) −12.0765 −0.0304334
\(55\) 16.2344 0.0398008
\(56\) 57.5631 0.137361
\(57\) −146.491 −0.340408
\(58\) −85.8679 −0.194397
\(59\) −500.044 −1.10339 −0.551697 0.834045i \(-0.686019\pi\)
−0.551697 + 0.834045i \(0.686019\pi\)
\(60\) 45.1787 0.0972091
\(61\) 480.502 1.00856 0.504279 0.863541i \(-0.331758\pi\)
0.504279 + 0.863541i \(0.331758\pi\)
\(62\) 51.5522 0.105599
\(63\) 73.3083 0.146603
\(64\) −436.771 −0.853068
\(65\) 0 0
\(66\) 11.2827 0.0210425
\(67\) −886.769 −1.61696 −0.808478 0.588526i \(-0.799708\pi\)
−0.808478 + 0.588526i \(0.799708\pi\)
\(68\) 406.589 0.725090
\(69\) 266.769 0.465438
\(70\) 7.03412 0.0120105
\(71\) −123.732 −0.206821 −0.103410 0.994639i \(-0.532975\pi\)
−0.103410 + 0.994639i \(0.532975\pi\)
\(72\) 63.6027 0.104106
\(73\) −673.168 −1.07929 −0.539646 0.841892i \(-0.681442\pi\)
−0.539646 + 0.841892i \(0.681442\pi\)
\(74\) −61.1501 −0.0960616
\(75\) −363.817 −0.560133
\(76\) 380.875 0.574859
\(77\) −68.4897 −0.101365
\(78\) 0 0
\(79\) −681.298 −0.970279 −0.485139 0.874437i \(-0.661231\pi\)
−0.485139 + 0.874437i \(0.661231\pi\)
\(80\) −114.374 −0.159842
\(81\) 81.0000 0.111111
\(82\) 195.048 0.262676
\(83\) 939.418 1.24234 0.621172 0.783675i \(-0.286657\pi\)
0.621172 + 0.783675i \(0.286657\pi\)
\(84\) −190.600 −0.247574
\(85\) 100.643 0.128427
\(86\) −90.3716 −0.113314
\(87\) 575.936 0.709734
\(88\) −59.4220 −0.0719819
\(89\) 754.979 0.899187 0.449593 0.893233i \(-0.351569\pi\)
0.449593 + 0.893233i \(0.351569\pi\)
\(90\) 7.77215 0.00910286
\(91\) 0 0
\(92\) −693.594 −0.786002
\(93\) −345.772 −0.385537
\(94\) 276.489 0.303380
\(95\) 94.2783 0.101818
\(96\) −249.096 −0.264825
\(97\) −1051.10 −1.10024 −0.550118 0.835087i \(-0.685417\pi\)
−0.550118 + 0.835087i \(0.685417\pi\)
\(98\) 123.741 0.127548
\(99\) −75.6757 −0.0768252
\(100\) 945.917 0.945917
\(101\) 599.873 0.590986 0.295493 0.955345i \(-0.404516\pi\)
0.295493 + 0.955345i \(0.404516\pi\)
\(102\) 69.9460 0.0678989
\(103\) −293.312 −0.280591 −0.140295 0.990110i \(-0.544805\pi\)
−0.140295 + 0.990110i \(0.544805\pi\)
\(104\) 0 0
\(105\) −47.1795 −0.0438500
\(106\) 202.693 0.185729
\(107\) 1533.69 1.38568 0.692839 0.721092i \(-0.256360\pi\)
0.692839 + 0.721092i \(0.256360\pi\)
\(108\) −210.598 −0.187637
\(109\) −590.718 −0.519088 −0.259544 0.965731i \(-0.583572\pi\)
−0.259544 + 0.965731i \(0.583572\pi\)
\(110\) −7.26128 −0.00629396
\(111\) 410.148 0.350717
\(112\) 482.521 0.407089
\(113\) −653.785 −0.544274 −0.272137 0.962259i \(-0.587730\pi\)
−0.272137 + 0.962259i \(0.587730\pi\)
\(114\) 65.5224 0.0538310
\(115\) −171.686 −0.139216
\(116\) −1497.42 −1.19855
\(117\) 0 0
\(118\) 223.659 0.174487
\(119\) −424.595 −0.327080
\(120\) −40.9332 −0.0311389
\(121\) −1260.30 −0.946881
\(122\) −214.918 −0.159490
\(123\) −1308.23 −0.959019
\(124\) 899.002 0.651070
\(125\) 475.485 0.340229
\(126\) −32.7892 −0.0231833
\(127\) −141.203 −0.0986595 −0.0493297 0.998783i \(-0.515709\pi\)
−0.0493297 + 0.998783i \(0.515709\pi\)
\(128\) 859.613 0.593592
\(129\) 606.144 0.413705
\(130\) 0 0
\(131\) −731.910 −0.488147 −0.244074 0.969757i \(-0.578484\pi\)
−0.244074 + 0.969757i \(0.578484\pi\)
\(132\) 196.755 0.129738
\(133\) −397.742 −0.259313
\(134\) 396.633 0.255700
\(135\) −52.1297 −0.0332341
\(136\) −368.381 −0.232268
\(137\) −1762.27 −1.09898 −0.549492 0.835499i \(-0.685179\pi\)
−0.549492 + 0.835499i \(0.685179\pi\)
\(138\) −119.320 −0.0736028
\(139\) −664.776 −0.405652 −0.202826 0.979215i \(-0.565013\pi\)
−0.202826 + 0.979215i \(0.565013\pi\)
\(140\) 122.666 0.0740511
\(141\) −1854.48 −1.10763
\(142\) 55.3425 0.0327059
\(143\) 0 0
\(144\) 533.148 0.308535
\(145\) −370.659 −0.212286
\(146\) 301.094 0.170676
\(147\) −829.959 −0.465672
\(148\) −1066.38 −0.592268
\(149\) 3300.71 1.81480 0.907399 0.420270i \(-0.138064\pi\)
0.907399 + 0.420270i \(0.138064\pi\)
\(150\) 162.727 0.0885776
\(151\) −1464.15 −0.789079 −0.394540 0.918879i \(-0.629096\pi\)
−0.394540 + 0.918879i \(0.629096\pi\)
\(152\) −345.083 −0.184144
\(153\) −469.144 −0.247896
\(154\) 30.6339 0.0160296
\(155\) 222.531 0.115317
\(156\) 0 0
\(157\) 1535.32 0.780456 0.390228 0.920718i \(-0.372396\pi\)
0.390228 + 0.920718i \(0.372396\pi\)
\(158\) 304.730 0.153437
\(159\) −1359.51 −0.678089
\(160\) 160.312 0.0792111
\(161\) 724.310 0.354557
\(162\) −36.2295 −0.0175707
\(163\) −793.554 −0.381325 −0.190663 0.981656i \(-0.561064\pi\)
−0.190663 + 0.981656i \(0.561064\pi\)
\(164\) 3401.38 1.61953
\(165\) 48.7031 0.0229790
\(166\) −420.181 −0.196460
\(167\) 1336.87 0.619460 0.309730 0.950825i \(-0.399761\pi\)
0.309730 + 0.950825i \(0.399761\pi\)
\(168\) 172.689 0.0793052
\(169\) 0 0
\(170\) −45.0156 −0.0203090
\(171\) −439.474 −0.196535
\(172\) −1575.96 −0.698640
\(173\) 568.460 0.249822 0.124911 0.992168i \(-0.460135\pi\)
0.124911 + 0.992168i \(0.460135\pi\)
\(174\) −257.604 −0.112235
\(175\) −987.808 −0.426693
\(176\) −498.103 −0.213329
\(177\) −1500.13 −0.637044
\(178\) −337.686 −0.142194
\(179\) 1546.00 0.645552 0.322776 0.946475i \(-0.395384\pi\)
0.322776 + 0.946475i \(0.395384\pi\)
\(180\) 135.536 0.0561237
\(181\) 3408.96 1.39992 0.699960 0.714182i \(-0.253201\pi\)
0.699960 + 0.714182i \(0.253201\pi\)
\(182\) 0 0
\(183\) 1441.51 0.582291
\(184\) 628.416 0.251780
\(185\) −263.962 −0.104902
\(186\) 154.656 0.0609675
\(187\) 438.307 0.171402
\(188\) 4821.61 1.87049
\(189\) 219.925 0.0846412
\(190\) −42.1686 −0.0161012
\(191\) −3464.71 −1.31255 −0.656277 0.754520i \(-0.727870\pi\)
−0.656277 + 0.754520i \(0.727870\pi\)
\(192\) −1310.31 −0.492519
\(193\) −4652.24 −1.73510 −0.867552 0.497346i \(-0.834308\pi\)
−0.867552 + 0.497346i \(0.834308\pi\)
\(194\) 470.134 0.173988
\(195\) 0 0
\(196\) 2157.88 0.786398
\(197\) 2870.98 1.03832 0.519160 0.854677i \(-0.326245\pi\)
0.519160 + 0.854677i \(0.326245\pi\)
\(198\) 33.8481 0.0121489
\(199\) 25.0330 0.00891730 0.00445865 0.999990i \(-0.498581\pi\)
0.00445865 + 0.999990i \(0.498581\pi\)
\(200\) −857.028 −0.303005
\(201\) −2660.31 −0.933550
\(202\) −268.310 −0.0934565
\(203\) 1563.74 0.540655
\(204\) 1219.77 0.418631
\(205\) 841.947 0.286849
\(206\) 131.192 0.0443717
\(207\) 800.307 0.268721
\(208\) 0 0
\(209\) 410.587 0.135889
\(210\) 21.1024 0.00693429
\(211\) −3605.91 −1.17650 −0.588249 0.808679i \(-0.700183\pi\)
−0.588249 + 0.808679i \(0.700183\pi\)
\(212\) 3534.70 1.14511
\(213\) −371.195 −0.119408
\(214\) −685.987 −0.219127
\(215\) −390.100 −0.123742
\(216\) 190.808 0.0601058
\(217\) −938.815 −0.293691
\(218\) 264.215 0.0820868
\(219\) −2019.50 −0.623130
\(220\) −126.627 −0.0388054
\(221\) 0 0
\(222\) −183.450 −0.0554612
\(223\) 4304.37 1.29256 0.646282 0.763098i \(-0.276323\pi\)
0.646282 + 0.763098i \(0.276323\pi\)
\(224\) −676.326 −0.201736
\(225\) −1091.45 −0.323393
\(226\) 292.424 0.0860697
\(227\) 5475.19 1.60089 0.800443 0.599409i \(-0.204598\pi\)
0.800443 + 0.599409i \(0.204598\pi\)
\(228\) 1142.62 0.331895
\(229\) −378.108 −0.109110 −0.0545548 0.998511i \(-0.517374\pi\)
−0.0545548 + 0.998511i \(0.517374\pi\)
\(230\) 76.7914 0.0220151
\(231\) −205.469 −0.0585232
\(232\) 1356.71 0.383932
\(233\) 2547.96 0.716405 0.358202 0.933644i \(-0.383390\pi\)
0.358202 + 0.933644i \(0.383390\pi\)
\(234\) 0 0
\(235\) 1193.50 0.331299
\(236\) 3900.32 1.07580
\(237\) −2043.89 −0.560191
\(238\) 189.912 0.0517234
\(239\) 6313.91 1.70884 0.854420 0.519583i \(-0.173913\pi\)
0.854420 + 0.519583i \(0.173913\pi\)
\(240\) −343.121 −0.0922849
\(241\) 6763.73 1.80784 0.903921 0.427699i \(-0.140676\pi\)
0.903921 + 0.427699i \(0.140676\pi\)
\(242\) 563.704 0.149737
\(243\) 243.000 0.0641500
\(244\) −3747.89 −0.983336
\(245\) 534.142 0.139286
\(246\) 585.144 0.151656
\(247\) 0 0
\(248\) −814.521 −0.208557
\(249\) 2818.25 0.717267
\(250\) −212.674 −0.0538027
\(251\) −3416.00 −0.859028 −0.429514 0.903060i \(-0.641315\pi\)
−0.429514 + 0.903060i \(0.641315\pi\)
\(252\) −571.801 −0.142937
\(253\) −747.701 −0.185801
\(254\) 63.1571 0.0156017
\(255\) 301.930 0.0741474
\(256\) 3109.68 0.759199
\(257\) 7002.36 1.69959 0.849796 0.527112i \(-0.176725\pi\)
0.849796 + 0.527112i \(0.176725\pi\)
\(258\) −271.115 −0.0654220
\(259\) 1113.60 0.267166
\(260\) 0 0
\(261\) 1727.81 0.409765
\(262\) 327.367 0.0771940
\(263\) 3369.76 0.790071 0.395035 0.918666i \(-0.370732\pi\)
0.395035 + 0.918666i \(0.370732\pi\)
\(264\) −178.266 −0.0415588
\(265\) 874.948 0.202821
\(266\) 177.901 0.0410069
\(267\) 2264.94 0.519146
\(268\) 6916.75 1.57652
\(269\) −7801.70 −1.76832 −0.884160 0.467185i \(-0.845268\pi\)
−0.884160 + 0.467185i \(0.845268\pi\)
\(270\) 23.3165 0.00525554
\(271\) −4410.05 −0.988528 −0.494264 0.869312i \(-0.664562\pi\)
−0.494264 + 0.869312i \(0.664562\pi\)
\(272\) −3087.94 −0.688360
\(273\) 0 0
\(274\) 788.225 0.173790
\(275\) 1019.71 0.223603
\(276\) −2080.78 −0.453798
\(277\) −4636.44 −1.00569 −0.502846 0.864376i \(-0.667714\pi\)
−0.502846 + 0.864376i \(0.667714\pi\)
\(278\) 297.340 0.0641485
\(279\) −1037.32 −0.222590
\(280\) −111.139 −0.0237207
\(281\) −1090.50 −0.231508 −0.115754 0.993278i \(-0.536928\pi\)
−0.115754 + 0.993278i \(0.536928\pi\)
\(282\) 829.468 0.175156
\(283\) 1140.52 0.239565 0.119782 0.992800i \(-0.461780\pi\)
0.119782 + 0.992800i \(0.461780\pi\)
\(284\) 965.101 0.201649
\(285\) 282.835 0.0587849
\(286\) 0 0
\(287\) −3552.01 −0.730553
\(288\) −747.287 −0.152897
\(289\) −2195.76 −0.446929
\(290\) 165.788 0.0335703
\(291\) −3153.30 −0.635222
\(292\) 5250.67 1.05230
\(293\) 335.079 0.0668106 0.0334053 0.999442i \(-0.489365\pi\)
0.0334053 + 0.999442i \(0.489365\pi\)
\(294\) 371.223 0.0736399
\(295\) 965.449 0.190545
\(296\) 966.168 0.189721
\(297\) −227.027 −0.0443551
\(298\) −1476.34 −0.286986
\(299\) 0 0
\(300\) 2837.75 0.546125
\(301\) 1645.76 0.315149
\(302\) 654.883 0.124782
\(303\) 1799.62 0.341206
\(304\) −2892.65 −0.545739
\(305\) −927.719 −0.174167
\(306\) 209.838 0.0392014
\(307\) −2540.04 −0.472207 −0.236104 0.971728i \(-0.575870\pi\)
−0.236104 + 0.971728i \(0.575870\pi\)
\(308\) 534.215 0.0988303
\(309\) −879.935 −0.161999
\(310\) −99.5332 −0.0182358
\(311\) 2376.36 0.433283 0.216642 0.976251i \(-0.430490\pi\)
0.216642 + 0.976251i \(0.430490\pi\)
\(312\) 0 0
\(313\) 1315.78 0.237612 0.118806 0.992917i \(-0.462093\pi\)
0.118806 + 0.992917i \(0.462093\pi\)
\(314\) −686.713 −0.123419
\(315\) −141.538 −0.0253168
\(316\) 5314.09 0.946015
\(317\) 6043.13 1.07071 0.535357 0.844626i \(-0.320177\pi\)
0.535357 + 0.844626i \(0.320177\pi\)
\(318\) 608.079 0.107231
\(319\) −1614.24 −0.283323
\(320\) 843.286 0.147316
\(321\) 4601.08 0.800022
\(322\) −323.968 −0.0560685
\(323\) 2545.39 0.438481
\(324\) −631.795 −0.108333
\(325\) 0 0
\(326\) 354.940 0.0603015
\(327\) −1772.16 −0.299695
\(328\) −3081.75 −0.518784
\(329\) −5035.14 −0.843758
\(330\) −21.7838 −0.00363382
\(331\) −8168.18 −1.35639 −0.678193 0.734884i \(-0.737237\pi\)
−0.678193 + 0.734884i \(0.737237\pi\)
\(332\) −7327.40 −1.21128
\(333\) 1230.44 0.202486
\(334\) −597.951 −0.0979594
\(335\) 1712.11 0.279232
\(336\) 1447.56 0.235033
\(337\) 3076.32 0.497263 0.248631 0.968598i \(-0.420019\pi\)
0.248631 + 0.968598i \(0.420019\pi\)
\(338\) 0 0
\(339\) −1961.36 −0.314237
\(340\) −785.012 −0.125215
\(341\) 969.133 0.153905
\(342\) 196.567 0.0310793
\(343\) −5047.30 −0.794544
\(344\) 1427.87 0.223795
\(345\) −515.058 −0.0803762
\(346\) −254.260 −0.0395061
\(347\) −8754.07 −1.35430 −0.677152 0.735844i \(-0.736786\pi\)
−0.677152 + 0.735844i \(0.736786\pi\)
\(348\) −4492.27 −0.691985
\(349\) −1064.97 −0.163343 −0.0816715 0.996659i \(-0.526026\pi\)
−0.0816715 + 0.996659i \(0.526026\pi\)
\(350\) 441.825 0.0674758
\(351\) 0 0
\(352\) 698.167 0.105717
\(353\) −5047.62 −0.761070 −0.380535 0.924767i \(-0.624260\pi\)
−0.380535 + 0.924767i \(0.624260\pi\)
\(354\) 670.977 0.100740
\(355\) 238.893 0.0357158
\(356\) −5888.79 −0.876700
\(357\) −1273.78 −0.188840
\(358\) −691.494 −0.102085
\(359\) −8152.39 −1.19851 −0.599257 0.800557i \(-0.704537\pi\)
−0.599257 + 0.800557i \(0.704537\pi\)
\(360\) −122.800 −0.0179781
\(361\) −4474.59 −0.652367
\(362\) −1524.75 −0.221379
\(363\) −3780.90 −0.546682
\(364\) 0 0
\(365\) 1299.71 0.186383
\(366\) −644.754 −0.0920816
\(367\) 11636.3 1.65507 0.827537 0.561412i \(-0.189742\pi\)
0.827537 + 0.561412i \(0.189742\pi\)
\(368\) 5267.67 0.746186
\(369\) −3924.70 −0.553690
\(370\) 118.064 0.0165888
\(371\) −3691.24 −0.516548
\(372\) 2697.00 0.375896
\(373\) 4720.59 0.655289 0.327644 0.944801i \(-0.393745\pi\)
0.327644 + 0.944801i \(0.393745\pi\)
\(374\) −196.045 −0.0271049
\(375\) 1426.45 0.196431
\(376\) −4368.52 −0.599173
\(377\) 0 0
\(378\) −98.3677 −0.0133849
\(379\) −12933.7 −1.75293 −0.876464 0.481467i \(-0.840104\pi\)
−0.876464 + 0.481467i \(0.840104\pi\)
\(380\) −735.365 −0.0992722
\(381\) −423.609 −0.0569611
\(382\) 1549.69 0.207563
\(383\) −554.945 −0.0740376 −0.0370188 0.999315i \(-0.511786\pi\)
−0.0370188 + 0.999315i \(0.511786\pi\)
\(384\) 2578.84 0.342711
\(385\) 132.235 0.0175047
\(386\) 2080.84 0.274384
\(387\) 1818.43 0.238853
\(388\) 8198.51 1.07272
\(389\) 5091.35 0.663604 0.331802 0.943349i \(-0.392343\pi\)
0.331802 + 0.943349i \(0.392343\pi\)
\(390\) 0 0
\(391\) −4635.30 −0.599532
\(392\) −1955.10 −0.251907
\(393\) −2195.73 −0.281832
\(394\) −1284.13 −0.164196
\(395\) 1315.40 0.167557
\(396\) 590.266 0.0749040
\(397\) −10254.6 −1.29638 −0.648188 0.761480i \(-0.724473\pi\)
−0.648188 + 0.761480i \(0.724473\pi\)
\(398\) −11.1967 −0.00141015
\(399\) −1193.23 −0.149714
\(400\) −7184.00 −0.898001
\(401\) −505.816 −0.0629906 −0.0314953 0.999504i \(-0.510027\pi\)
−0.0314953 + 0.999504i \(0.510027\pi\)
\(402\) 1189.90 0.147629
\(403\) 0 0
\(404\) −4678.97 −0.576207
\(405\) −156.389 −0.0191877
\(406\) −699.426 −0.0854974
\(407\) −1149.57 −0.140005
\(408\) −1105.14 −0.134100
\(409\) −2219.21 −0.268295 −0.134148 0.990961i \(-0.542830\pi\)
−0.134148 + 0.990961i \(0.542830\pi\)
\(410\) −376.585 −0.0453614
\(411\) −5286.81 −0.634499
\(412\) 2287.82 0.273574
\(413\) −4073.04 −0.485282
\(414\) −357.960 −0.0424946
\(415\) −1813.76 −0.214540
\(416\) 0 0
\(417\) −1994.33 −0.234203
\(418\) −183.646 −0.0214891
\(419\) 10249.3 1.19501 0.597506 0.801864i \(-0.296158\pi\)
0.597506 + 0.801864i \(0.296158\pi\)
\(420\) 367.997 0.0427534
\(421\) −97.9194 −0.0113356 −0.00566781 0.999984i \(-0.501804\pi\)
−0.00566781 + 0.999984i \(0.501804\pi\)
\(422\) 1612.85 0.186048
\(423\) −5563.44 −0.639489
\(424\) −3202.54 −0.366814
\(425\) 6321.58 0.721510
\(426\) 166.028 0.0188828
\(427\) 3913.87 0.443572
\(428\) −11962.7 −1.35103
\(429\) 0 0
\(430\) 174.483 0.0195682
\(431\) −11727.2 −1.31062 −0.655310 0.755360i \(-0.727462\pi\)
−0.655310 + 0.755360i \(0.727462\pi\)
\(432\) 1599.44 0.178133
\(433\) 6945.13 0.770812 0.385406 0.922747i \(-0.374061\pi\)
0.385406 + 0.922747i \(0.374061\pi\)
\(434\) 419.911 0.0464433
\(435\) −1111.98 −0.122564
\(436\) 4607.57 0.506107
\(437\) −4342.15 −0.475316
\(438\) 903.281 0.0985398
\(439\) 11446.3 1.24442 0.622210 0.782851i \(-0.286235\pi\)
0.622210 + 0.782851i \(0.286235\pi\)
\(440\) 114.728 0.0124305
\(441\) −2489.88 −0.268856
\(442\) 0 0
\(443\) −10513.6 −1.12758 −0.563789 0.825919i \(-0.690657\pi\)
−0.563789 + 0.825919i \(0.690657\pi\)
\(444\) −3199.13 −0.341946
\(445\) −1457.66 −0.155280
\(446\) −1925.25 −0.204402
\(447\) 9902.14 1.04777
\(448\) −3557.66 −0.375187
\(449\) 9862.37 1.03660 0.518300 0.855199i \(-0.326565\pi\)
0.518300 + 0.855199i \(0.326565\pi\)
\(450\) 488.182 0.0511403
\(451\) 3666.72 0.382836
\(452\) 5099.49 0.530663
\(453\) −4392.45 −0.455575
\(454\) −2448.93 −0.253159
\(455\) 0 0
\(456\) −1035.25 −0.106316
\(457\) 5990.48 0.613179 0.306590 0.951842i \(-0.400812\pi\)
0.306590 + 0.951842i \(0.400812\pi\)
\(458\) 169.120 0.0172542
\(459\) −1407.43 −0.143123
\(460\) 1339.14 0.135734
\(461\) −16579.5 −1.67502 −0.837510 0.546423i \(-0.815989\pi\)
−0.837510 + 0.546423i \(0.815989\pi\)
\(462\) 91.9018 0.00925467
\(463\) −3606.65 −0.362020 −0.181010 0.983481i \(-0.557937\pi\)
−0.181010 + 0.983481i \(0.557937\pi\)
\(464\) 11372.6 1.13784
\(465\) 667.593 0.0665782
\(466\) −1139.65 −0.113290
\(467\) −6789.61 −0.672775 −0.336387 0.941724i \(-0.609205\pi\)
−0.336387 + 0.941724i \(0.609205\pi\)
\(468\) 0 0
\(469\) −7223.06 −0.711152
\(470\) −533.826 −0.0523906
\(471\) 4605.95 0.450596
\(472\) −3533.80 −0.344611
\(473\) −1698.90 −0.165149
\(474\) 914.189 0.0885867
\(475\) 5921.78 0.572021
\(476\) 3311.82 0.318901
\(477\) −4078.53 −0.391495
\(478\) −2824.07 −0.270230
\(479\) 12688.3 1.21032 0.605161 0.796103i \(-0.293109\pi\)
0.605161 + 0.796103i \(0.293109\pi\)
\(480\) 480.936 0.0457326
\(481\) 0 0
\(482\) −3025.27 −0.285886
\(483\) 2172.93 0.204703
\(484\) 9830.26 0.923202
\(485\) 2029.39 0.189999
\(486\) −108.689 −0.0101445
\(487\) 1217.28 0.113265 0.0566325 0.998395i \(-0.481964\pi\)
0.0566325 + 0.998395i \(0.481964\pi\)
\(488\) 3395.69 0.314991
\(489\) −2380.66 −0.220158
\(490\) −238.910 −0.0220262
\(491\) −10496.3 −0.964747 −0.482374 0.875966i \(-0.660225\pi\)
−0.482374 + 0.875966i \(0.660225\pi\)
\(492\) 10204.1 0.935037
\(493\) −10007.3 −0.914211
\(494\) 0 0
\(495\) 146.109 0.0132669
\(496\) −6827.69 −0.618090
\(497\) −1007.84 −0.0909615
\(498\) −1260.54 −0.113426
\(499\) 10516.5 0.943450 0.471725 0.881746i \(-0.343632\pi\)
0.471725 + 0.881746i \(0.343632\pi\)
\(500\) −3708.75 −0.331721
\(501\) 4010.60 0.357645
\(502\) 1527.90 0.135844
\(503\) 13400.9 1.18790 0.593952 0.804500i \(-0.297567\pi\)
0.593952 + 0.804500i \(0.297567\pi\)
\(504\) 518.068 0.0457869
\(505\) −1158.19 −0.102057
\(506\) 334.430 0.0293819
\(507\) 0 0
\(508\) 1101.38 0.0961923
\(509\) −4683.90 −0.407878 −0.203939 0.978984i \(-0.565374\pi\)
−0.203939 + 0.978984i \(0.565374\pi\)
\(510\) −135.047 −0.0117254
\(511\) −5483.20 −0.474682
\(512\) −8267.80 −0.713649
\(513\) −1318.42 −0.113469
\(514\) −3132.00 −0.268768
\(515\) 566.305 0.0484551
\(516\) −4727.89 −0.403360
\(517\) 5197.75 0.442160
\(518\) −498.090 −0.0422487
\(519\) 1705.38 0.144235
\(520\) 0 0
\(521\) 15034.6 1.26426 0.632128 0.774864i \(-0.282182\pi\)
0.632128 + 0.774864i \(0.282182\pi\)
\(522\) −772.811 −0.0647989
\(523\) 8009.33 0.669643 0.334822 0.942282i \(-0.391324\pi\)
0.334822 + 0.942282i \(0.391324\pi\)
\(524\) 5708.86 0.475940
\(525\) −2963.42 −0.246351
\(526\) −1507.22 −0.124939
\(527\) 6008.04 0.496612
\(528\) −1494.31 −0.123166
\(529\) −4259.71 −0.350103
\(530\) −391.345 −0.0320735
\(531\) −4500.40 −0.367798
\(532\) 3102.36 0.252828
\(533\) 0 0
\(534\) −1013.06 −0.0820960
\(535\) −2961.14 −0.239292
\(536\) −6266.77 −0.505006
\(537\) 4638.01 0.372710
\(538\) 3489.53 0.279636
\(539\) 2326.21 0.185894
\(540\) 406.608 0.0324030
\(541\) −20833.2 −1.65561 −0.827807 0.561013i \(-0.810412\pi\)
−0.827807 + 0.561013i \(0.810412\pi\)
\(542\) 1972.52 0.156323
\(543\) 10226.9 0.808245
\(544\) 4328.22 0.341123
\(545\) 1140.52 0.0896411
\(546\) 0 0
\(547\) −14247.2 −1.11365 −0.556825 0.830630i \(-0.687981\pi\)
−0.556825 + 0.830630i \(0.687981\pi\)
\(548\) 13745.6 1.07150
\(549\) 4324.52 0.336186
\(550\) −456.093 −0.0353598
\(551\) −9374.41 −0.724797
\(552\) 1885.25 0.145365
\(553\) −5549.43 −0.426737
\(554\) 2073.78 0.159037
\(555\) −791.885 −0.0605651
\(556\) 5185.22 0.395508
\(557\) −2007.94 −0.152745 −0.0763725 0.997079i \(-0.524334\pi\)
−0.0763725 + 0.997079i \(0.524334\pi\)
\(558\) 463.969 0.0351996
\(559\) 0 0
\(560\) −931.616 −0.0703000
\(561\) 1314.92 0.0989589
\(562\) 487.758 0.0366100
\(563\) 10562.1 0.790659 0.395330 0.918539i \(-0.370630\pi\)
0.395330 + 0.918539i \(0.370630\pi\)
\(564\) 14464.8 1.07993
\(565\) 1262.28 0.0939905
\(566\) −510.129 −0.0378840
\(567\) 659.775 0.0488676
\(568\) −874.409 −0.0645940
\(569\) −23207.2 −1.70983 −0.854916 0.518766i \(-0.826392\pi\)
−0.854916 + 0.518766i \(0.826392\pi\)
\(570\) −126.506 −0.00929605
\(571\) 7987.78 0.585426 0.292713 0.956200i \(-0.405442\pi\)
0.292713 + 0.956200i \(0.405442\pi\)
\(572\) 0 0
\(573\) −10394.1 −0.757803
\(574\) 1588.74 0.115527
\(575\) −10783.9 −0.782120
\(576\) −3930.94 −0.284356
\(577\) −3224.41 −0.232641 −0.116321 0.993212i \(-0.537110\pi\)
−0.116321 + 0.993212i \(0.537110\pi\)
\(578\) 982.116 0.0706759
\(579\) −13956.7 −1.00176
\(580\) 2891.12 0.206978
\(581\) 7651.90 0.546393
\(582\) 1410.40 0.100452
\(583\) 3810.44 0.270690
\(584\) −4757.26 −0.337084
\(585\) 0 0
\(586\) −149.873 −0.0105652
\(587\) 661.034 0.0464800 0.0232400 0.999730i \(-0.492602\pi\)
0.0232400 + 0.999730i \(0.492602\pi\)
\(588\) 6473.63 0.454027
\(589\) 5628.07 0.393719
\(590\) −431.824 −0.0301321
\(591\) 8612.94 0.599474
\(592\) 8098.87 0.562266
\(593\) 10172.5 0.704443 0.352221 0.935917i \(-0.385426\pi\)
0.352221 + 0.935917i \(0.385426\pi\)
\(594\) 101.544 0.00701417
\(595\) 819.777 0.0564834
\(596\) −25745.4 −1.76942
\(597\) 75.0990 0.00514840
\(598\) 0 0
\(599\) 23462.2 1.60040 0.800200 0.599733i \(-0.204726\pi\)
0.800200 + 0.599733i \(0.204726\pi\)
\(600\) −2571.08 −0.174940
\(601\) −10482.6 −0.711474 −0.355737 0.934586i \(-0.615770\pi\)
−0.355737 + 0.934586i \(0.615770\pi\)
\(602\) −736.111 −0.0498366
\(603\) −7980.92 −0.538986
\(604\) 11420.3 0.769347
\(605\) 2433.29 0.163516
\(606\) −804.930 −0.0539572
\(607\) −24949.8 −1.66834 −0.834169 0.551508i \(-0.814053\pi\)
−0.834169 + 0.551508i \(0.814053\pi\)
\(608\) 4054.48 0.270446
\(609\) 4691.21 0.312147
\(610\) 414.949 0.0275423
\(611\) 0 0
\(612\) 3659.30 0.241697
\(613\) −8238.86 −0.542846 −0.271423 0.962460i \(-0.587494\pi\)
−0.271423 + 0.962460i \(0.587494\pi\)
\(614\) 1136.10 0.0746733
\(615\) 2525.84 0.165613
\(616\) −484.014 −0.0316583
\(617\) −5248.85 −0.342481 −0.171241 0.985229i \(-0.554778\pi\)
−0.171241 + 0.985229i \(0.554778\pi\)
\(618\) 393.576 0.0256180
\(619\) 17820.9 1.15716 0.578580 0.815626i \(-0.303607\pi\)
0.578580 + 0.815626i \(0.303607\pi\)
\(620\) −1735.73 −0.112433
\(621\) 2400.92 0.155146
\(622\) −1062.89 −0.0685180
\(623\) 6149.58 0.395470
\(624\) 0 0
\(625\) 14241.0 0.911424
\(626\) −588.522 −0.0375752
\(627\) 1231.76 0.0784557
\(628\) −11975.4 −0.760938
\(629\) −7126.62 −0.451760
\(630\) 63.3071 0.00400351
\(631\) −25445.0 −1.60531 −0.802653 0.596446i \(-0.796579\pi\)
−0.802653 + 0.596446i \(0.796579\pi\)
\(632\) −4814.71 −0.303036
\(633\) −10817.7 −0.679252
\(634\) −2702.96 −0.169319
\(635\) 272.625 0.0170375
\(636\) 10604.1 0.661132
\(637\) 0 0
\(638\) 722.013 0.0448037
\(639\) −1113.59 −0.0689402
\(640\) −1659.68 −0.102507
\(641\) −18701.0 −1.15233 −0.576167 0.817332i \(-0.695452\pi\)
−0.576167 + 0.817332i \(0.695452\pi\)
\(642\) −2057.96 −0.126513
\(643\) 28465.9 1.74586 0.872928 0.487849i \(-0.162218\pi\)
0.872928 + 0.487849i \(0.162218\pi\)
\(644\) −5649.58 −0.345690
\(645\) −1170.30 −0.0714426
\(646\) −1138.50 −0.0693399
\(647\) −10924.6 −0.663818 −0.331909 0.943311i \(-0.607693\pi\)
−0.331909 + 0.943311i \(0.607693\pi\)
\(648\) 572.425 0.0347021
\(649\) 4204.58 0.254305
\(650\) 0 0
\(651\) −2816.44 −0.169562
\(652\) 6189.68 0.371789
\(653\) −27110.9 −1.62470 −0.812351 0.583169i \(-0.801813\pi\)
−0.812351 + 0.583169i \(0.801813\pi\)
\(654\) 792.646 0.0473929
\(655\) 1413.12 0.0842980
\(656\) −25832.6 −1.53749
\(657\) −6058.51 −0.359764
\(658\) 2252.11 0.133429
\(659\) 1727.08 0.102090 0.0510452 0.998696i \(-0.483745\pi\)
0.0510452 + 0.998696i \(0.483745\pi\)
\(660\) −379.881 −0.0224043
\(661\) −20651.1 −1.21518 −0.607592 0.794249i \(-0.707864\pi\)
−0.607592 + 0.794249i \(0.707864\pi\)
\(662\) 3653.45 0.214494
\(663\) 0 0
\(664\) 6638.84 0.388007
\(665\) 767.932 0.0447806
\(666\) −550.351 −0.0320205
\(667\) 17071.3 0.991010
\(668\) −10427.5 −0.603969
\(669\) 12913.1 0.746263
\(670\) −765.790 −0.0441568
\(671\) −4040.26 −0.232448
\(672\) −2028.98 −0.116472
\(673\) 10858.0 0.621907 0.310954 0.950425i \(-0.399352\pi\)
0.310954 + 0.950425i \(0.399352\pi\)
\(674\) −1375.97 −0.0786355
\(675\) −3274.35 −0.186711
\(676\) 0 0
\(677\) 29132.9 1.65387 0.826933 0.562301i \(-0.190084\pi\)
0.826933 + 0.562301i \(0.190084\pi\)
\(678\) 877.272 0.0496924
\(679\) −8561.59 −0.483894
\(680\) 711.244 0.0401102
\(681\) 16425.6 0.924272
\(682\) −433.472 −0.0243380
\(683\) 9443.65 0.529065 0.264532 0.964377i \(-0.414782\pi\)
0.264532 + 0.964377i \(0.414782\pi\)
\(684\) 3427.87 0.191620
\(685\) 3402.46 0.189783
\(686\) 2257.55 0.125647
\(687\) −1134.33 −0.0629945
\(688\) 11969.0 0.663249
\(689\) 0 0
\(690\) 230.374 0.0127104
\(691\) 23700.1 1.30477 0.652384 0.757889i \(-0.273769\pi\)
0.652384 + 0.757889i \(0.273769\pi\)
\(692\) −4433.96 −0.243575
\(693\) −616.407 −0.0337884
\(694\) 3915.51 0.214165
\(695\) 1283.50 0.0700519
\(696\) 4070.12 0.221663
\(697\) 22731.5 1.23532
\(698\) 476.340 0.0258305
\(699\) 7643.88 0.413616
\(700\) 7704.84 0.416022
\(701\) 21643.1 1.16612 0.583059 0.812430i \(-0.301856\pi\)
0.583059 + 0.812430i \(0.301856\pi\)
\(702\) 0 0
\(703\) −6675.91 −0.358160
\(704\) 3672.55 0.196611
\(705\) 3580.50 0.191276
\(706\) 2257.69 0.120353
\(707\) 4886.18 0.259921
\(708\) 11700.9 0.621114
\(709\) 28965.3 1.53429 0.767146 0.641472i \(-0.221676\pi\)
0.767146 + 0.641472i \(0.221676\pi\)
\(710\) −106.851 −0.00564798
\(711\) −6131.68 −0.323426
\(712\) 5335.41 0.280833
\(713\) −10249.0 −0.538330
\(714\) 569.736 0.0298625
\(715\) 0 0
\(716\) −12058.7 −0.629408
\(717\) 18941.7 0.986599
\(718\) 3646.39 0.189529
\(719\) −21256.5 −1.10255 −0.551276 0.834323i \(-0.685859\pi\)
−0.551276 + 0.834323i \(0.685859\pi\)
\(720\) −1029.36 −0.0532807
\(721\) −2389.13 −0.123406
\(722\) 2001.39 0.103163
\(723\) 20291.2 1.04376
\(724\) −26589.7 −1.36491
\(725\) −23281.7 −1.19264
\(726\) 1691.11 0.0864505
\(727\) −11868.9 −0.605494 −0.302747 0.953071i \(-0.597904\pi\)
−0.302747 + 0.953071i \(0.597904\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) −581.330 −0.0294739
\(731\) −10532.2 −0.532896
\(732\) −11243.7 −0.567729
\(733\) −27296.4 −1.37546 −0.687732 0.725965i \(-0.741393\pi\)
−0.687732 + 0.725965i \(0.741393\pi\)
\(734\) −5204.68 −0.261728
\(735\) 1602.43 0.0804168
\(736\) −7383.44 −0.369779
\(737\) 7456.32 0.372669
\(738\) 1755.43 0.0875587
\(739\) 26424.6 1.31535 0.657674 0.753302i \(-0.271540\pi\)
0.657674 + 0.753302i \(0.271540\pi\)
\(740\) 2058.89 0.102279
\(741\) 0 0
\(742\) 1651.01 0.0816853
\(743\) 25974.8 1.28253 0.641267 0.767318i \(-0.278409\pi\)
0.641267 + 0.767318i \(0.278409\pi\)
\(744\) −2443.56 −0.120410
\(745\) −6372.78 −0.313397
\(746\) −2111.42 −0.103625
\(747\) 8454.76 0.414114
\(748\) −3418.77 −0.167116
\(749\) 12492.5 0.609433
\(750\) −638.022 −0.0310630
\(751\) 20234.8 0.983195 0.491598 0.870823i \(-0.336413\pi\)
0.491598 + 0.870823i \(0.336413\pi\)
\(752\) −36619.0 −1.77574
\(753\) −10248.0 −0.495960
\(754\) 0 0
\(755\) 2826.88 0.136266
\(756\) −1715.40 −0.0825246
\(757\) −3363.48 −0.161490 −0.0807449 0.996735i \(-0.525730\pi\)
−0.0807449 + 0.996735i \(0.525730\pi\)
\(758\) 5784.97 0.277202
\(759\) −2243.10 −0.107272
\(760\) 666.262 0.0317998
\(761\) 21311.3 1.01516 0.507579 0.861605i \(-0.330541\pi\)
0.507579 + 0.861605i \(0.330541\pi\)
\(762\) 189.471 0.00900764
\(763\) −4811.62 −0.228299
\(764\) 27024.5 1.27973
\(765\) 905.790 0.0428090
\(766\) 248.215 0.0117081
\(767\) 0 0
\(768\) 9329.04 0.438324
\(769\) 34989.6 1.64078 0.820388 0.571807i \(-0.193758\pi\)
0.820388 + 0.571807i \(0.193758\pi\)
\(770\) −59.1458 −0.00276814
\(771\) 21007.1 0.981260
\(772\) 36287.2 1.69171
\(773\) −20034.1 −0.932181 −0.466091 0.884737i \(-0.654338\pi\)
−0.466091 + 0.884737i \(0.654338\pi\)
\(774\) −813.345 −0.0377714
\(775\) 13977.5 0.647855
\(776\) −7428.09 −0.343625
\(777\) 3340.81 0.154248
\(778\) −2277.25 −0.104940
\(779\) 21293.9 0.979373
\(780\) 0 0
\(781\) 1040.39 0.0476671
\(782\) 2073.27 0.0948081
\(783\) 5183.43 0.236578
\(784\) −16388.5 −0.746563
\(785\) −2964.28 −0.134777
\(786\) 982.102 0.0445680
\(787\) −5591.98 −0.253282 −0.126641 0.991949i \(-0.540420\pi\)
−0.126641 + 0.991949i \(0.540420\pi\)
\(788\) −22393.5 −1.01235
\(789\) 10109.3 0.456147
\(790\) −588.351 −0.0264969
\(791\) −5325.32 −0.239376
\(792\) −534.798 −0.0239940
\(793\) 0 0
\(794\) 4586.64 0.205005
\(795\) 2624.84 0.117099
\(796\) −195.256 −0.00869430
\(797\) −12738.6 −0.566152 −0.283076 0.959098i \(-0.591355\pi\)
−0.283076 + 0.959098i \(0.591355\pi\)
\(798\) 533.704 0.0236753
\(799\) 32222.9 1.42674
\(800\) 10069.5 0.445012
\(801\) 6794.81 0.299729
\(802\) 226.240 0.00996113
\(803\) 5660.28 0.248751
\(804\) 20750.2 0.910205
\(805\) −1398.45 −0.0612283
\(806\) 0 0
\(807\) −23405.1 −1.02094
\(808\) 4239.28 0.184576
\(809\) −8806.48 −0.382719 −0.191359 0.981520i \(-0.561290\pi\)
−0.191359 + 0.981520i \(0.561290\pi\)
\(810\) 69.9494 0.00303429
\(811\) 10565.8 0.457478 0.228739 0.973488i \(-0.426540\pi\)
0.228739 + 0.973488i \(0.426540\pi\)
\(812\) −12197.1 −0.527134
\(813\) −13230.1 −0.570727
\(814\) 514.176 0.0221399
\(815\) 1532.14 0.0658509
\(816\) −9263.82 −0.397425
\(817\) −9866.09 −0.422486
\(818\) 992.604 0.0424273
\(819\) 0 0
\(820\) −6567.14 −0.279676
\(821\) −422.966 −0.0179800 −0.00899002 0.999960i \(-0.502862\pi\)
−0.00899002 + 0.999960i \(0.502862\pi\)
\(822\) 2364.67 0.100338
\(823\) 11148.8 0.472202 0.236101 0.971728i \(-0.424130\pi\)
0.236101 + 0.971728i \(0.424130\pi\)
\(824\) −2072.83 −0.0876339
\(825\) 3059.12 0.129097
\(826\) 1821.78 0.0767409
\(827\) 813.158 0.0341914 0.0170957 0.999854i \(-0.494558\pi\)
0.0170957 + 0.999854i \(0.494558\pi\)
\(828\) −6242.34 −0.262001
\(829\) −28320.8 −1.18652 −0.593258 0.805012i \(-0.702159\pi\)
−0.593258 + 0.805012i \(0.702159\pi\)
\(830\) 811.255 0.0339266
\(831\) −13909.3 −0.580637
\(832\) 0 0
\(833\) 14421.1 0.599835
\(834\) 892.020 0.0370361
\(835\) −2581.13 −0.106974
\(836\) −3202.55 −0.132491
\(837\) −3111.95 −0.128512
\(838\) −4584.28 −0.188975
\(839\) 22139.3 0.911006 0.455503 0.890234i \(-0.349459\pi\)
0.455503 + 0.890234i \(0.349459\pi\)
\(840\) −333.416 −0.0136952
\(841\) 12466.8 0.511166
\(842\) 43.7972 0.00179258
\(843\) −3271.50 −0.133661
\(844\) 28125.9 1.14708
\(845\) 0 0
\(846\) 2488.41 0.101127
\(847\) −10265.6 −0.416446
\(848\) −26845.2 −1.08711
\(849\) 3421.56 0.138313
\(850\) −2827.50 −0.114097
\(851\) 12157.2 0.489710
\(852\) 2895.30 0.116422
\(853\) 14080.5 0.565191 0.282595 0.959239i \(-0.408805\pi\)
0.282595 + 0.959239i \(0.408805\pi\)
\(854\) −1750.59 −0.0701451
\(855\) 848.505 0.0339395
\(856\) 10838.6 0.432774
\(857\) 10907.7 0.434772 0.217386 0.976086i \(-0.430247\pi\)
0.217386 + 0.976086i \(0.430247\pi\)
\(858\) 0 0
\(859\) 7739.24 0.307403 0.153702 0.988117i \(-0.450881\pi\)
0.153702 + 0.988117i \(0.450881\pi\)
\(860\) 3042.76 0.120648
\(861\) −10656.0 −0.421785
\(862\) 5245.30 0.207257
\(863\) 29072.5 1.14674 0.573372 0.819295i \(-0.305635\pi\)
0.573372 + 0.819295i \(0.305635\pi\)
\(864\) −2241.86 −0.0882751
\(865\) −1097.54 −0.0431417
\(866\) −3106.40 −0.121894
\(867\) −6587.28 −0.258034
\(868\) 7322.70 0.286346
\(869\) 5728.64 0.223626
\(870\) 497.363 0.0193818
\(871\) 0 0
\(872\) −4174.59 −0.162121
\(873\) −9459.89 −0.366746
\(874\) 1942.15 0.0751649
\(875\) 3873.00 0.149636
\(876\) 15752.0 0.607547
\(877\) −2391.96 −0.0920988 −0.0460494 0.998939i \(-0.514663\pi\)
−0.0460494 + 0.998939i \(0.514663\pi\)
\(878\) −5119.66 −0.196788
\(879\) 1005.24 0.0385731
\(880\) 961.702 0.0368397
\(881\) −11975.8 −0.457974 −0.228987 0.973430i \(-0.573541\pi\)
−0.228987 + 0.973430i \(0.573541\pi\)
\(882\) 1113.67 0.0425160
\(883\) −44712.4 −1.70407 −0.852035 0.523485i \(-0.824632\pi\)
−0.852035 + 0.523485i \(0.824632\pi\)
\(884\) 0 0
\(885\) 2896.35 0.110011
\(886\) 4702.51 0.178311
\(887\) 17023.3 0.644403 0.322202 0.946671i \(-0.395577\pi\)
0.322202 + 0.946671i \(0.395577\pi\)
\(888\) 2898.51 0.109535
\(889\) −1150.15 −0.0433913
\(890\) 651.979 0.0245555
\(891\) −681.082 −0.0256084
\(892\) −33573.8 −1.26024
\(893\) 30185.0 1.13113
\(894\) −4429.01 −0.165692
\(895\) −2984.91 −0.111480
\(896\) 7001.87 0.261067
\(897\) 0 0
\(898\) −4411.22 −0.163925
\(899\) −22127.0 −0.820886
\(900\) 8513.25 0.315306
\(901\) 23622.5 0.873449
\(902\) −1640.04 −0.0605405
\(903\) 4937.27 0.181951
\(904\) −4620.28 −0.169987
\(905\) −6581.77 −0.241752
\(906\) 1964.65 0.0720432
\(907\) 2303.59 0.0843324 0.0421662 0.999111i \(-0.486574\pi\)
0.0421662 + 0.999111i \(0.486574\pi\)
\(908\) −42706.2 −1.56085
\(909\) 5398.85 0.196995
\(910\) 0 0
\(911\) −12897.0 −0.469042 −0.234521 0.972111i \(-0.575352\pi\)
−0.234521 + 0.972111i \(0.575352\pi\)
\(912\) −8677.95 −0.315083
\(913\) −7899.01 −0.286330
\(914\) −2679.41 −0.0969662
\(915\) −2783.16 −0.100556
\(916\) 2949.22 0.106381
\(917\) −5961.68 −0.214691
\(918\) 629.514 0.0226330
\(919\) 16785.2 0.602496 0.301248 0.953546i \(-0.402597\pi\)
0.301248 + 0.953546i \(0.402597\pi\)
\(920\) −1213.30 −0.0434797
\(921\) −7620.11 −0.272629
\(922\) 7415.64 0.264882
\(923\) 0 0
\(924\) 1602.65 0.0570597
\(925\) −16579.9 −0.589344
\(926\) 1613.18 0.0572487
\(927\) −2639.81 −0.0935303
\(928\) −15940.4 −0.563866
\(929\) −16348.0 −0.577352 −0.288676 0.957427i \(-0.593215\pi\)
−0.288676 + 0.957427i \(0.593215\pi\)
\(930\) −298.600 −0.0105285
\(931\) 13509.1 0.475556
\(932\) −19873.9 −0.698489
\(933\) 7129.08 0.250156
\(934\) 3036.85 0.106390
\(935\) −846.251 −0.0295993
\(936\) 0 0
\(937\) −26247.5 −0.915121 −0.457560 0.889179i \(-0.651277\pi\)
−0.457560 + 0.889179i \(0.651277\pi\)
\(938\) 3230.72 0.112459
\(939\) 3947.35 0.137185
\(940\) −9309.22 −0.323014
\(941\) −43838.4 −1.51869 −0.759347 0.650686i \(-0.774481\pi\)
−0.759347 + 0.650686i \(0.774481\pi\)
\(942\) −2060.14 −0.0712558
\(943\) −38777.3 −1.33909
\(944\) −29621.9 −1.02130
\(945\) −424.615 −0.0146167
\(946\) 759.882 0.0261162
\(947\) 45077.2 1.54679 0.773395 0.633924i \(-0.218557\pi\)
0.773395 + 0.633924i \(0.218557\pi\)
\(948\) 15942.3 0.546182
\(949\) 0 0
\(950\) −2648.68 −0.0904575
\(951\) 18129.4 0.618177
\(952\) −3000.60 −0.102153
\(953\) 26479.1 0.900046 0.450023 0.893017i \(-0.351416\pi\)
0.450023 + 0.893017i \(0.351416\pi\)
\(954\) 1824.24 0.0619097
\(955\) 6689.41 0.226664
\(956\) −49248.1 −1.66611
\(957\) −4842.71 −0.163576
\(958\) −5675.21 −0.191396
\(959\) −14354.3 −0.483343
\(960\) 2529.86 0.0850529
\(961\) −16506.7 −0.554084
\(962\) 0 0
\(963\) 13803.2 0.461893
\(964\) −52756.7 −1.76263
\(965\) 8982.20 0.299635
\(966\) −971.905 −0.0323711
\(967\) 602.475 0.0200355 0.0100177 0.999950i \(-0.496811\pi\)
0.0100177 + 0.999950i \(0.496811\pi\)
\(968\) −8906.49 −0.295729
\(969\) 7636.17 0.253157
\(970\) −907.701 −0.0300459
\(971\) −11330.2 −0.374464 −0.187232 0.982316i \(-0.559952\pi\)
−0.187232 + 0.982316i \(0.559952\pi\)
\(972\) −1895.39 −0.0625458
\(973\) −5414.85 −0.178409
\(974\) −544.461 −0.0179114
\(975\) 0 0
\(976\) 28464.3 0.933524
\(977\) −55633.6 −1.82178 −0.910888 0.412653i \(-0.864602\pi\)
−0.910888 + 0.412653i \(0.864602\pi\)
\(978\) 1064.82 0.0348151
\(979\) −6348.18 −0.207241
\(980\) −4166.27 −0.135803
\(981\) −5316.47 −0.173029
\(982\) 4694.76 0.152562
\(983\) 6884.90 0.223392 0.111696 0.993742i \(-0.464372\pi\)
0.111696 + 0.993742i \(0.464372\pi\)
\(984\) −9245.24 −0.299520
\(985\) −5543.08 −0.179307
\(986\) 4476.05 0.144570
\(987\) −15105.4 −0.487144
\(988\) 0 0
\(989\) 17966.7 0.577662
\(990\) −65.3515 −0.00209799
\(991\) 52646.1 1.68755 0.843773 0.536700i \(-0.180329\pi\)
0.843773 + 0.536700i \(0.180329\pi\)
\(992\) 9570.05 0.306300
\(993\) −24504.5 −0.783109
\(994\) 450.786 0.0143844
\(995\) −48.3319 −0.00153992
\(996\) −21982.2 −0.699330
\(997\) 26100.7 0.829106 0.414553 0.910025i \(-0.363938\pi\)
0.414553 + 0.910025i \(0.363938\pi\)
\(998\) −4703.78 −0.149194
\(999\) 3691.33 0.116906
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 507.4.a.o.1.5 9
3.2 odd 2 1521.4.a.bi.1.5 9
13.5 odd 4 507.4.b.k.337.11 18
13.8 odd 4 507.4.b.k.337.8 18
13.12 even 2 507.4.a.p.1.5 yes 9
39.38 odd 2 1521.4.a.bf.1.5 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.4.a.o.1.5 9 1.1 even 1 trivial
507.4.a.p.1.5 yes 9 13.12 even 2
507.4.b.k.337.8 18 13.8 odd 4
507.4.b.k.337.11 18 13.5 odd 4
1521.4.a.bf.1.5 9 39.38 odd 2
1521.4.a.bi.1.5 9 3.2 odd 2