Properties

Label 169.4.a.h.1.2
Level $169$
Weight $4$
Character 169.1
Self dual yes
Analytic conductor $9.971$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 169.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.97132279097\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 169.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.46410 q^{2} -7.00000 q^{3} +4.00000 q^{4} +13.8564 q^{5} -24.2487 q^{6} -22.5167 q^{7} -13.8564 q^{8} +22.0000 q^{9} +O(q^{10})\) \(q+3.46410 q^{2} -7.00000 q^{3} +4.00000 q^{4} +13.8564 q^{5} -24.2487 q^{6} -22.5167 q^{7} -13.8564 q^{8} +22.0000 q^{9} +48.0000 q^{10} -22.5167 q^{11} -28.0000 q^{12} -78.0000 q^{14} -96.9948 q^{15} -80.0000 q^{16} -27.0000 q^{17} +76.2102 q^{18} -88.3346 q^{19} +55.4256 q^{20} +157.617 q^{21} -78.0000 q^{22} -57.0000 q^{23} +96.9948 q^{24} +67.0000 q^{25} +35.0000 q^{27} -90.0666 q^{28} -69.0000 q^{29} -336.000 q^{30} -72.7461 q^{31} -166.277 q^{32} +157.617 q^{33} -93.5307 q^{34} -312.000 q^{35} +88.0000 q^{36} -39.8372 q^{37} -306.000 q^{38} -192.000 q^{40} +393.176 q^{41} +546.000 q^{42} +85.0000 q^{43} -90.0666 q^{44} +304.841 q^{45} -197.454 q^{46} +342.946 q^{47} +560.000 q^{48} +164.000 q^{49} +232.095 q^{50} +189.000 q^{51} +426.000 q^{53} +121.244 q^{54} -312.000 q^{55} +312.000 q^{56} +618.342 q^{57} -239.023 q^{58} +19.0526 q^{59} -387.979 q^{60} -17.0000 q^{61} -252.000 q^{62} -495.367 q^{63} +64.0000 q^{64} +546.000 q^{66} -164.545 q^{67} -108.000 q^{68} +399.000 q^{69} -1080.80 q^{70} +583.701 q^{71} -304.841 q^{72} -1004.59 q^{73} -138.000 q^{74} -469.000 q^{75} -353.338 q^{76} +507.000 q^{77} -1244.00 q^{79} -1108.51 q^{80} -839.000 q^{81} +1362.00 q^{82} -426.084 q^{83} +630.466 q^{84} -374.123 q^{85} +294.449 q^{86} +483.000 q^{87} +312.000 q^{88} +306.573 q^{89} +1056.00 q^{90} -228.000 q^{92} +509.223 q^{93} +1188.00 q^{94} -1224.00 q^{95} +1163.94 q^{96} +1234.95 q^{97} +568.113 q^{98} -495.367 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 14 q^{3} + 8 q^{4} + 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 14 q^{3} + 8 q^{4} + 44 q^{9} + 96 q^{10} - 56 q^{12} - 156 q^{14} - 160 q^{16} - 54 q^{17} - 156 q^{22} - 114 q^{23} + 134 q^{25} + 70 q^{27} - 138 q^{29} - 672 q^{30} - 624 q^{35} + 176 q^{36} - 612 q^{38} - 384 q^{40} + 1092 q^{42} + 170 q^{43} + 1120 q^{48} + 328 q^{49} + 378 q^{51} + 852 q^{53} - 624 q^{55} + 624 q^{56} - 34 q^{61} - 504 q^{62} + 128 q^{64} + 1092 q^{66} - 216 q^{68} + 798 q^{69} - 276 q^{74} - 938 q^{75} + 1014 q^{77} - 2488 q^{79} - 1678 q^{81} + 2724 q^{82} + 966 q^{87} + 624 q^{88} + 2112 q^{90} - 456 q^{92} + 2376 q^{94} - 2448 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.46410 1.22474 0.612372 0.790569i \(-0.290215\pi\)
0.612372 + 0.790569i \(0.290215\pi\)
\(3\) −7.00000 −1.34715 −0.673575 0.739119i \(-0.735242\pi\)
−0.673575 + 0.739119i \(0.735242\pi\)
\(4\) 4.00000 0.500000
\(5\) 13.8564 1.23935 0.619677 0.784857i \(-0.287263\pi\)
0.619677 + 0.784857i \(0.287263\pi\)
\(6\) −24.2487 −1.64992
\(7\) −22.5167 −1.21579 −0.607893 0.794019i \(-0.707985\pi\)
−0.607893 + 0.794019i \(0.707985\pi\)
\(8\) −13.8564 −0.612372
\(9\) 22.0000 0.814815
\(10\) 48.0000 1.51789
\(11\) −22.5167 −0.617184 −0.308592 0.951194i \(-0.599858\pi\)
−0.308592 + 0.951194i \(0.599858\pi\)
\(12\) −28.0000 −0.673575
\(13\) 0 0
\(14\) −78.0000 −1.48903
\(15\) −96.9948 −1.66960
\(16\) −80.0000 −1.25000
\(17\) −27.0000 −0.385204 −0.192602 0.981277i \(-0.561693\pi\)
−0.192602 + 0.981277i \(0.561693\pi\)
\(18\) 76.2102 0.997940
\(19\) −88.3346 −1.06660 −0.533299 0.845927i \(-0.679048\pi\)
−0.533299 + 0.845927i \(0.679048\pi\)
\(20\) 55.4256 0.619677
\(21\) 157.617 1.63785
\(22\) −78.0000 −0.755893
\(23\) −57.0000 −0.516753 −0.258377 0.966044i \(-0.583188\pi\)
−0.258377 + 0.966044i \(0.583188\pi\)
\(24\) 96.9948 0.824958
\(25\) 67.0000 0.536000
\(26\) 0 0
\(27\) 35.0000 0.249472
\(28\) −90.0666 −0.607893
\(29\) −69.0000 −0.441827 −0.220913 0.975293i \(-0.570904\pi\)
−0.220913 + 0.975293i \(0.570904\pi\)
\(30\) −336.000 −2.04483
\(31\) −72.7461 −0.421471 −0.210735 0.977543i \(-0.567586\pi\)
−0.210735 + 0.977543i \(0.567586\pi\)
\(32\) −166.277 −0.918559
\(33\) 157.617 0.831440
\(34\) −93.5307 −0.471776
\(35\) −312.000 −1.50679
\(36\) 88.0000 0.407407
\(37\) −39.8372 −0.177005 −0.0885026 0.996076i \(-0.528208\pi\)
−0.0885026 + 0.996076i \(0.528208\pi\)
\(38\) −306.000 −1.30631
\(39\) 0 0
\(40\) −192.000 −0.758947
\(41\) 393.176 1.49765 0.748826 0.662767i \(-0.230618\pi\)
0.748826 + 0.662767i \(0.230618\pi\)
\(42\) 546.000 2.00594
\(43\) 85.0000 0.301451 0.150725 0.988576i \(-0.451839\pi\)
0.150725 + 0.988576i \(0.451839\pi\)
\(44\) −90.0666 −0.308592
\(45\) 304.841 1.00984
\(46\) −197.454 −0.632891
\(47\) 342.946 1.06434 0.532168 0.846639i \(-0.321377\pi\)
0.532168 + 0.846639i \(0.321377\pi\)
\(48\) 560.000 1.68394
\(49\) 164.000 0.478134
\(50\) 232.095 0.656463
\(51\) 189.000 0.518927
\(52\) 0 0
\(53\) 426.000 1.10407 0.552034 0.833822i \(-0.313852\pi\)
0.552034 + 0.833822i \(0.313852\pi\)
\(54\) 121.244 0.305540
\(55\) −312.000 −0.764910
\(56\) 312.000 0.744513
\(57\) 618.342 1.43687
\(58\) −239.023 −0.541125
\(59\) 19.0526 0.0420412 0.0210206 0.999779i \(-0.493308\pi\)
0.0210206 + 0.999779i \(0.493308\pi\)
\(60\) −387.979 −0.834799
\(61\) −17.0000 −0.0356824 −0.0178412 0.999841i \(-0.505679\pi\)
−0.0178412 + 0.999841i \(0.505679\pi\)
\(62\) −252.000 −0.516194
\(63\) −495.367 −0.990640
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) 546.000 1.01830
\(67\) −164.545 −0.300035 −0.150018 0.988683i \(-0.547933\pi\)
−0.150018 + 0.988683i \(0.547933\pi\)
\(68\) −108.000 −0.192602
\(69\) 399.000 0.696144
\(70\) −1080.80 −1.84543
\(71\) 583.701 0.975670 0.487835 0.872936i \(-0.337787\pi\)
0.487835 + 0.872936i \(0.337787\pi\)
\(72\) −304.841 −0.498970
\(73\) −1004.59 −1.61066 −0.805331 0.592826i \(-0.798012\pi\)
−0.805331 + 0.592826i \(0.798012\pi\)
\(74\) −138.000 −0.216786
\(75\) −469.000 −0.722073
\(76\) −353.338 −0.533299
\(77\) 507.000 0.750364
\(78\) 0 0
\(79\) −1244.00 −1.77166 −0.885829 0.464012i \(-0.846409\pi\)
−0.885829 + 0.464012i \(0.846409\pi\)
\(80\) −1108.51 −1.54919
\(81\) −839.000 −1.15089
\(82\) 1362.00 1.83424
\(83\) −426.084 −0.563480 −0.281740 0.959491i \(-0.590912\pi\)
−0.281740 + 0.959491i \(0.590912\pi\)
\(84\) 630.466 0.818923
\(85\) −374.123 −0.477404
\(86\) 294.449 0.369200
\(87\) 483.000 0.595207
\(88\) 312.000 0.377947
\(89\) 306.573 0.365131 0.182566 0.983194i \(-0.441560\pi\)
0.182566 + 0.983194i \(0.441560\pi\)
\(90\) 1056.00 1.23680
\(91\) 0 0
\(92\) −228.000 −0.258377
\(93\) 509.223 0.567785
\(94\) 1188.00 1.30354
\(95\) −1224.00 −1.32189
\(96\) 1163.94 1.23744
\(97\) 1234.95 1.29268 0.646342 0.763048i \(-0.276298\pi\)
0.646342 + 0.763048i \(0.276298\pi\)
\(98\) 568.113 0.585592
\(99\) −495.367 −0.502891
\(100\) 268.000 0.268000
\(101\) −1959.00 −1.92998 −0.964989 0.262290i \(-0.915522\pi\)
−0.964989 + 0.262290i \(0.915522\pi\)
\(102\) 654.715 0.635554
\(103\) −1856.00 −1.77551 −0.887753 0.460320i \(-0.847735\pi\)
−0.887753 + 0.460320i \(0.847735\pi\)
\(104\) 0 0
\(105\) 2184.00 2.02987
\(106\) 1475.71 1.35220
\(107\) −255.000 −0.230390 −0.115195 0.993343i \(-0.536749\pi\)
−0.115195 + 0.993343i \(0.536749\pi\)
\(108\) 140.000 0.124736
\(109\) −609.682 −0.535752 −0.267876 0.963453i \(-0.586322\pi\)
−0.267876 + 0.963453i \(0.586322\pi\)
\(110\) −1080.80 −0.936820
\(111\) 278.860 0.238453
\(112\) 1801.33 1.51973
\(113\) 411.000 0.342156 0.171078 0.985257i \(-0.445275\pi\)
0.171078 + 0.985257i \(0.445275\pi\)
\(114\) 2142.00 1.75980
\(115\) −789.815 −0.640440
\(116\) −276.000 −0.220913
\(117\) 0 0
\(118\) 66.0000 0.0514898
\(119\) 607.950 0.468325
\(120\) 1344.00 1.02242
\(121\) −824.000 −0.619083
\(122\) −58.8897 −0.0437018
\(123\) −2752.23 −2.01756
\(124\) −290.985 −0.210735
\(125\) −803.672 −0.575061
\(126\) −1716.00 −1.21328
\(127\) 2243.00 1.56720 0.783599 0.621267i \(-0.213382\pi\)
0.783599 + 0.621267i \(0.213382\pi\)
\(128\) 1551.92 1.07165
\(129\) −595.000 −0.406099
\(130\) 0 0
\(131\) −372.000 −0.248105 −0.124053 0.992276i \(-0.539589\pi\)
−0.124053 + 0.992276i \(0.539589\pi\)
\(132\) 630.466 0.415720
\(133\) 1989.00 1.29675
\(134\) −570.000 −0.367466
\(135\) 484.974 0.309185
\(136\) 374.123 0.235888
\(137\) 1189.92 0.742056 0.371028 0.928622i \(-0.379005\pi\)
0.371028 + 0.928622i \(0.379005\pi\)
\(138\) 1382.18 0.852599
\(139\) −2545.00 −1.55298 −0.776490 0.630130i \(-0.783002\pi\)
−0.776490 + 0.630130i \(0.783002\pi\)
\(140\) −1248.00 −0.753395
\(141\) −2400.62 −1.43382
\(142\) 2022.00 1.19495
\(143\) 0 0
\(144\) −1760.00 −1.01852
\(145\) −956.092 −0.547580
\(146\) −3480.00 −1.97265
\(147\) −1148.00 −0.644119
\(148\) −159.349 −0.0885026
\(149\) 1304.23 0.717094 0.358547 0.933512i \(-0.383272\pi\)
0.358547 + 0.933512i \(0.383272\pi\)
\(150\) −1624.66 −0.884355
\(151\) 86.6025 0.0466729 0.0233365 0.999728i \(-0.492571\pi\)
0.0233365 + 0.999728i \(0.492571\pi\)
\(152\) 1224.00 0.653155
\(153\) −594.000 −0.313870
\(154\) 1756.30 0.919004
\(155\) −1008.00 −0.522352
\(156\) 0 0
\(157\) −1534.00 −0.779787 −0.389893 0.920860i \(-0.627488\pi\)
−0.389893 + 0.920860i \(0.627488\pi\)
\(158\) −4309.34 −2.16983
\(159\) −2982.00 −1.48735
\(160\) −2304.00 −1.13842
\(161\) 1283.45 0.628261
\(162\) −2906.38 −1.40955
\(163\) 1633.32 0.784858 0.392429 0.919782i \(-0.371635\pi\)
0.392429 + 0.919782i \(0.371635\pi\)
\(164\) 1572.70 0.748826
\(165\) 2184.00 1.03045
\(166\) −1476.00 −0.690119
\(167\) 1626.40 0.753618 0.376809 0.926291i \(-0.377021\pi\)
0.376809 + 0.926291i \(0.377021\pi\)
\(168\) −2184.00 −1.00297
\(169\) 0 0
\(170\) −1296.00 −0.584698
\(171\) −1943.36 −0.869079
\(172\) 340.000 0.150725
\(173\) 873.000 0.383659 0.191829 0.981428i \(-0.438558\pi\)
0.191829 + 0.981428i \(0.438558\pi\)
\(174\) 1673.16 0.728977
\(175\) −1508.62 −0.651661
\(176\) 1801.33 0.771481
\(177\) −133.368 −0.0566359
\(178\) 1062.00 0.447193
\(179\) 1287.00 0.537402 0.268701 0.963224i \(-0.413406\pi\)
0.268701 + 0.963224i \(0.413406\pi\)
\(180\) 1219.36 0.504922
\(181\) −2.00000 −0.000821319 0 −0.000410660 1.00000i \(-0.500131\pi\)
−0.000410660 1.00000i \(0.500131\pi\)
\(182\) 0 0
\(183\) 119.000 0.0480696
\(184\) 789.815 0.316445
\(185\) −552.000 −0.219372
\(186\) 1764.00 0.695391
\(187\) 607.950 0.237742
\(188\) 1371.78 0.532168
\(189\) −788.083 −0.303305
\(190\) −4240.06 −1.61898
\(191\) −2841.00 −1.07627 −0.538135 0.842859i \(-0.680871\pi\)
−0.538135 + 0.842859i \(0.680871\pi\)
\(192\) −448.000 −0.168394
\(193\) −4245.26 −1.58332 −0.791659 0.610964i \(-0.790782\pi\)
−0.791659 + 0.610964i \(0.790782\pi\)
\(194\) 4278.00 1.58321
\(195\) 0 0
\(196\) 656.000 0.239067
\(197\) −2752.23 −0.995371 −0.497686 0.867357i \(-0.665817\pi\)
−0.497686 + 0.867357i \(0.665817\pi\)
\(198\) −1716.00 −0.615913
\(199\) 1685.00 0.600234 0.300117 0.953902i \(-0.402974\pi\)
0.300117 + 0.953902i \(0.402974\pi\)
\(200\) −928.379 −0.328232
\(201\) 1151.81 0.404192
\(202\) −6786.18 −2.36373
\(203\) 1553.65 0.537167
\(204\) 756.000 0.259464
\(205\) 5448.00 1.85612
\(206\) −6429.37 −2.17454
\(207\) −1254.00 −0.421058
\(208\) 0 0
\(209\) 1989.00 0.658287
\(210\) 7565.60 2.48608
\(211\) 1681.00 0.548459 0.274229 0.961664i \(-0.411577\pi\)
0.274229 + 0.961664i \(0.411577\pi\)
\(212\) 1704.00 0.552034
\(213\) −4085.91 −1.31437
\(214\) −883.346 −0.282170
\(215\) 1177.79 0.373604
\(216\) −484.974 −0.152770
\(217\) 1638.00 0.512418
\(218\) −2112.00 −0.656159
\(219\) 7032.13 2.16980
\(220\) −1248.00 −0.382455
\(221\) 0 0
\(222\) 966.000 0.292044
\(223\) 4096.30 1.23008 0.615042 0.788495i \(-0.289139\pi\)
0.615042 + 0.788495i \(0.289139\pi\)
\(224\) 3744.00 1.11677
\(225\) 1474.00 0.436741
\(226\) 1423.75 0.419054
\(227\) 438.209 0.128128 0.0640638 0.997946i \(-0.479594\pi\)
0.0640638 + 0.997946i \(0.479594\pi\)
\(228\) 2473.37 0.718433
\(229\) −180.133 −0.0519805 −0.0259903 0.999662i \(-0.508274\pi\)
−0.0259903 + 0.999662i \(0.508274\pi\)
\(230\) −2736.00 −0.784376
\(231\) −3549.00 −1.01085
\(232\) 956.092 0.270563
\(233\) 5778.00 1.62459 0.812295 0.583247i \(-0.198218\pi\)
0.812295 + 0.583247i \(0.198218\pi\)
\(234\) 0 0
\(235\) 4752.00 1.31909
\(236\) 76.2102 0.0210206
\(237\) 8708.00 2.38669
\(238\) 2106.00 0.573579
\(239\) −1860.22 −0.503464 −0.251732 0.967797i \(-0.581000\pi\)
−0.251732 + 0.967797i \(0.581000\pi\)
\(240\) 7759.59 2.08700
\(241\) −2059.41 −0.550449 −0.275224 0.961380i \(-0.588752\pi\)
−0.275224 + 0.961380i \(0.588752\pi\)
\(242\) −2854.42 −0.758219
\(243\) 4928.00 1.30095
\(244\) −68.0000 −0.0178412
\(245\) 2272.45 0.592578
\(246\) −9534.00 −2.47100
\(247\) 0 0
\(248\) 1008.00 0.258097
\(249\) 2982.59 0.759093
\(250\) −2784.00 −0.704302
\(251\) −4491.00 −1.12936 −0.564680 0.825310i \(-0.691000\pi\)
−0.564680 + 0.825310i \(0.691000\pi\)
\(252\) −1981.47 −0.495320
\(253\) 1283.45 0.318932
\(254\) 7769.98 1.91942
\(255\) 2618.86 0.643135
\(256\) 4864.00 1.18750
\(257\) −5451.00 −1.32305 −0.661525 0.749923i \(-0.730091\pi\)
−0.661525 + 0.749923i \(0.730091\pi\)
\(258\) −2061.14 −0.497368
\(259\) 897.000 0.215200
\(260\) 0 0
\(261\) −1518.00 −0.360007
\(262\) −1288.65 −0.303866
\(263\) −783.000 −0.183581 −0.0917906 0.995778i \(-0.529259\pi\)
−0.0917906 + 0.995778i \(0.529259\pi\)
\(264\) −2184.00 −0.509151
\(265\) 5902.83 1.36833
\(266\) 6890.10 1.58819
\(267\) −2146.01 −0.491887
\(268\) −658.179 −0.150018
\(269\) −5085.00 −1.15256 −0.576279 0.817253i \(-0.695496\pi\)
−0.576279 + 0.817253i \(0.695496\pi\)
\(270\) 1680.00 0.378672
\(271\) −1325.02 −0.297008 −0.148504 0.988912i \(-0.547446\pi\)
−0.148504 + 0.988912i \(0.547446\pi\)
\(272\) 2160.00 0.481505
\(273\) 0 0
\(274\) 4122.00 0.908829
\(275\) −1508.62 −0.330811
\(276\) 1596.00 0.348072
\(277\) 3421.00 0.742050 0.371025 0.928623i \(-0.379006\pi\)
0.371025 + 0.928623i \(0.379006\pi\)
\(278\) −8816.14 −1.90200
\(279\) −1600.41 −0.343421
\(280\) 4323.20 0.922716
\(281\) −810.600 −0.172087 −0.0860433 0.996291i \(-0.527422\pi\)
−0.0860433 + 0.996291i \(0.527422\pi\)
\(282\) −8316.00 −1.75607
\(283\) −7177.00 −1.50752 −0.753760 0.657149i \(-0.771762\pi\)
−0.753760 + 0.657149i \(0.771762\pi\)
\(284\) 2334.80 0.487835
\(285\) 8568.00 1.78079
\(286\) 0 0
\(287\) −8853.00 −1.82082
\(288\) −3658.09 −0.748455
\(289\) −4184.00 −0.851618
\(290\) −3312.00 −0.670646
\(291\) −8644.67 −1.74144
\(292\) −4018.36 −0.805331
\(293\) −9313.24 −1.85695 −0.928473 0.371400i \(-0.878878\pi\)
−0.928473 + 0.371400i \(0.878878\pi\)
\(294\) −3976.79 −0.788881
\(295\) 264.000 0.0521040
\(296\) 552.000 0.108393
\(297\) −788.083 −0.153970
\(298\) 4518.00 0.878257
\(299\) 0 0
\(300\) −1876.00 −0.361036
\(301\) −1913.92 −0.366499
\(302\) 300.000 0.0571625
\(303\) 13713.0 2.59997
\(304\) 7066.77 1.33325
\(305\) −235.559 −0.0442232
\(306\) −2057.68 −0.384410
\(307\) 4777.00 0.888070 0.444035 0.896009i \(-0.353547\pi\)
0.444035 + 0.896009i \(0.353547\pi\)
\(308\) 2028.00 0.375182
\(309\) 12992.0 2.39187
\(310\) −3491.81 −0.639748
\(311\) −6192.00 −1.12899 −0.564495 0.825436i \(-0.690929\pi\)
−0.564495 + 0.825436i \(0.690929\pi\)
\(312\) 0 0
\(313\) −770.000 −0.139051 −0.0695255 0.997580i \(-0.522149\pi\)
−0.0695255 + 0.997580i \(0.522149\pi\)
\(314\) −5313.93 −0.955040
\(315\) −6864.00 −1.22775
\(316\) −4976.00 −0.885829
\(317\) 8057.50 1.42762 0.713808 0.700341i \(-0.246969\pi\)
0.713808 + 0.700341i \(0.246969\pi\)
\(318\) −10330.0 −1.82162
\(319\) 1553.65 0.272689
\(320\) 886.810 0.154919
\(321\) 1785.00 0.310371
\(322\) 4446.00 0.769459
\(323\) 2385.03 0.410857
\(324\) −3356.00 −0.575446
\(325\) 0 0
\(326\) 5658.00 0.961250
\(327\) 4267.77 0.721738
\(328\) −5448.00 −0.917120
\(329\) −7722.00 −1.29400
\(330\) 7565.60 1.26204
\(331\) −5277.56 −0.876377 −0.438189 0.898883i \(-0.644380\pi\)
−0.438189 + 0.898883i \(0.644380\pi\)
\(332\) −1704.34 −0.281740
\(333\) −876.418 −0.144226
\(334\) 5634.00 0.922990
\(335\) −2280.00 −0.371850
\(336\) −12609.3 −2.04731
\(337\) −8278.00 −1.33808 −0.669038 0.743228i \(-0.733294\pi\)
−0.669038 + 0.743228i \(0.733294\pi\)
\(338\) 0 0
\(339\) −2877.00 −0.460936
\(340\) −1496.49 −0.238702
\(341\) 1638.00 0.260125
\(342\) −6732.00 −1.06440
\(343\) 4030.48 0.634477
\(344\) −1177.79 −0.184600
\(345\) 5528.71 0.862770
\(346\) 3024.16 0.469884
\(347\) 6867.00 1.06236 0.531181 0.847258i \(-0.321748\pi\)
0.531181 + 0.847258i \(0.321748\pi\)
\(348\) 1932.00 0.297604
\(349\) 12153.8 1.86412 0.932060 0.362303i \(-0.118010\pi\)
0.932060 + 0.362303i \(0.118010\pi\)
\(350\) −5226.00 −0.798118
\(351\) 0 0
\(352\) 3744.00 0.566920
\(353\) −5807.57 −0.875653 −0.437827 0.899059i \(-0.644252\pi\)
−0.437827 + 0.899059i \(0.644252\pi\)
\(354\) −462.000 −0.0693645
\(355\) 8088.00 1.20920
\(356\) 1226.29 0.182566
\(357\) −4255.65 −0.630904
\(358\) 4458.30 0.658180
\(359\) −1340.61 −0.197088 −0.0985439 0.995133i \(-0.531419\pi\)
−0.0985439 + 0.995133i \(0.531419\pi\)
\(360\) −4224.00 −0.618401
\(361\) 944.000 0.137629
\(362\) −6.92820 −0.00100591
\(363\) 5768.00 0.833999
\(364\) 0 0
\(365\) −13920.0 −1.99618
\(366\) 412.228 0.0588730
\(367\) 3665.00 0.521285 0.260642 0.965435i \(-0.416066\pi\)
0.260642 + 0.965435i \(0.416066\pi\)
\(368\) 4560.00 0.645941
\(369\) 8649.86 1.22031
\(370\) −1912.18 −0.268675
\(371\) −9592.10 −1.34231
\(372\) 2036.89 0.283892
\(373\) 5371.00 0.745576 0.372788 0.927917i \(-0.378402\pi\)
0.372788 + 0.927917i \(0.378402\pi\)
\(374\) 2106.00 0.291173
\(375\) 5625.70 0.774693
\(376\) −4752.00 −0.651770
\(377\) 0 0
\(378\) −2730.00 −0.371471
\(379\) 11509.5 1.55990 0.779950 0.625842i \(-0.215244\pi\)
0.779950 + 0.625842i \(0.215244\pi\)
\(380\) −4896.00 −0.660946
\(381\) −15701.0 −2.11125
\(382\) −9841.51 −1.31816
\(383\) 2419.67 0.322819 0.161409 0.986888i \(-0.448396\pi\)
0.161409 + 0.986888i \(0.448396\pi\)
\(384\) −10863.4 −1.44368
\(385\) 7025.20 0.929967
\(386\) −14706.0 −1.93916
\(387\) 1870.00 0.245626
\(388\) 4939.81 0.646342
\(389\) 9858.00 1.28489 0.642443 0.766334i \(-0.277921\pi\)
0.642443 + 0.766334i \(0.277921\pi\)
\(390\) 0 0
\(391\) 1539.00 0.199055
\(392\) −2272.45 −0.292796
\(393\) 2604.00 0.334235
\(394\) −9534.00 −1.21908
\(395\) −17237.4 −2.19571
\(396\) −1981.47 −0.251446
\(397\) 8720.88 1.10249 0.551245 0.834344i \(-0.314153\pi\)
0.551245 + 0.834344i \(0.314153\pi\)
\(398\) 5837.01 0.735133
\(399\) −13923.0 −1.74692
\(400\) −5360.00 −0.670000
\(401\) −7584.65 −0.944537 −0.472269 0.881455i \(-0.656565\pi\)
−0.472269 + 0.881455i \(0.656565\pi\)
\(402\) 3990.00 0.495033
\(403\) 0 0
\(404\) −7836.00 −0.964989
\(405\) −11625.5 −1.42636
\(406\) 5382.00 0.657892
\(407\) 897.000 0.109245
\(408\) −2618.86 −0.317777
\(409\) 4304.15 0.520358 0.260179 0.965560i \(-0.416218\pi\)
0.260179 + 0.965560i \(0.416218\pi\)
\(410\) 18872.4 2.27327
\(411\) −8329.43 −0.999661
\(412\) −7424.00 −0.887753
\(413\) −429.000 −0.0511131
\(414\) −4343.98 −0.515689
\(415\) −5904.00 −0.698352
\(416\) 0 0
\(417\) 17815.0 2.09210
\(418\) 6890.10 0.806234
\(419\) 5397.00 0.629262 0.314631 0.949214i \(-0.398119\pi\)
0.314631 + 0.949214i \(0.398119\pi\)
\(420\) 8736.00 1.01494
\(421\) 7260.76 0.840541 0.420270 0.907399i \(-0.361935\pi\)
0.420270 + 0.907399i \(0.361935\pi\)
\(422\) 5823.15 0.671722
\(423\) 7544.81 0.867237
\(424\) −5902.83 −0.676101
\(425\) −1809.00 −0.206469
\(426\) −14154.0 −1.60977
\(427\) 382.783 0.0433822
\(428\) −1020.00 −0.115195
\(429\) 0 0
\(430\) 4080.00 0.457570
\(431\) 486.706 0.0543940 0.0271970 0.999630i \(-0.491342\pi\)
0.0271970 + 0.999630i \(0.491342\pi\)
\(432\) −2800.00 −0.311840
\(433\) −12139.0 −1.34726 −0.673629 0.739069i \(-0.735266\pi\)
−0.673629 + 0.739069i \(0.735266\pi\)
\(434\) 5674.20 0.627581
\(435\) 6692.64 0.737673
\(436\) −2438.73 −0.267876
\(437\) 5035.07 0.551167
\(438\) 24360.0 2.65746
\(439\) −461.000 −0.0501192 −0.0250596 0.999686i \(-0.507978\pi\)
−0.0250596 + 0.999686i \(0.507978\pi\)
\(440\) 4323.20 0.468410
\(441\) 3608.00 0.389591
\(442\) 0 0
\(443\) 12156.0 1.30372 0.651861 0.758338i \(-0.273988\pi\)
0.651861 + 0.758338i \(0.273988\pi\)
\(444\) 1115.44 0.119226
\(445\) 4248.00 0.452527
\(446\) 14190.0 1.50654
\(447\) −9129.64 −0.966034
\(448\) −1441.07 −0.151973
\(449\) 296.181 0.0311306 0.0155653 0.999879i \(-0.495045\pi\)
0.0155653 + 0.999879i \(0.495045\pi\)
\(450\) 5106.09 0.534896
\(451\) −8853.00 −0.924327
\(452\) 1644.00 0.171078
\(453\) −606.218 −0.0628755
\(454\) 1518.00 0.156924
\(455\) 0 0
\(456\) −8568.00 −0.879898
\(457\) −611.414 −0.0625837 −0.0312918 0.999510i \(-0.509962\pi\)
−0.0312918 + 0.999510i \(0.509962\pi\)
\(458\) −624.000 −0.0636629
\(459\) −945.000 −0.0960977
\(460\) −3159.26 −0.320220
\(461\) −13127.2 −1.32624 −0.663119 0.748514i \(-0.730767\pi\)
−0.663119 + 0.748514i \(0.730767\pi\)
\(462\) −12294.1 −1.23804
\(463\) 834.848 0.0837985 0.0418992 0.999122i \(-0.486659\pi\)
0.0418992 + 0.999122i \(0.486659\pi\)
\(464\) 5520.00 0.552284
\(465\) 7056.00 0.703686
\(466\) 20015.6 1.98971
\(467\) −14496.0 −1.43639 −0.718196 0.695841i \(-0.755032\pi\)
−0.718196 + 0.695841i \(0.755032\pi\)
\(468\) 0 0
\(469\) 3705.00 0.364778
\(470\) 16461.4 1.61555
\(471\) 10738.0 1.05049
\(472\) −264.000 −0.0257449
\(473\) −1913.92 −0.186051
\(474\) 30165.4 2.92309
\(475\) −5918.42 −0.571696
\(476\) 2431.80 0.234162
\(477\) 9372.00 0.899611
\(478\) −6444.00 −0.616614
\(479\) −8897.54 −0.848725 −0.424362 0.905492i \(-0.639502\pi\)
−0.424362 + 0.905492i \(0.639502\pi\)
\(480\) 16128.0 1.53362
\(481\) 0 0
\(482\) −7134.00 −0.674159
\(483\) −8984.15 −0.846362
\(484\) −3296.00 −0.309542
\(485\) 17112.0 1.60209
\(486\) 17071.1 1.59333
\(487\) 4754.48 0.442394 0.221197 0.975229i \(-0.429004\pi\)
0.221197 + 0.975229i \(0.429004\pi\)
\(488\) 235.559 0.0218509
\(489\) −11433.3 −1.05732
\(490\) 7872.00 0.725757
\(491\) 1635.00 0.150278 0.0751390 0.997173i \(-0.476060\pi\)
0.0751390 + 0.997173i \(0.476060\pi\)
\(492\) −11008.9 −1.00878
\(493\) 1863.00 0.170193
\(494\) 0 0
\(495\) −6864.00 −0.623260
\(496\) 5819.69 0.526838
\(497\) −13143.0 −1.18621
\(498\) 10332.0 0.929695
\(499\) −14434.9 −1.29498 −0.647490 0.762074i \(-0.724181\pi\)
−0.647490 + 0.762074i \(0.724181\pi\)
\(500\) −3214.69 −0.287530
\(501\) −11384.8 −1.01524
\(502\) −15557.3 −1.38318
\(503\) 12687.0 1.12462 0.562312 0.826925i \(-0.309912\pi\)
0.562312 + 0.826925i \(0.309912\pi\)
\(504\) 6864.00 0.606641
\(505\) −27144.7 −2.39193
\(506\) 4446.00 0.390610
\(507\) 0 0
\(508\) 8972.00 0.783599
\(509\) 5748.68 0.500600 0.250300 0.968168i \(-0.419471\pi\)
0.250300 + 0.968168i \(0.419471\pi\)
\(510\) 9072.00 0.787676
\(511\) 22620.0 1.95822
\(512\) 4434.05 0.382733
\(513\) −3091.71 −0.266086
\(514\) −18882.8 −1.62040
\(515\) −25717.5 −2.20048
\(516\) −2380.00 −0.203050
\(517\) −7722.00 −0.656892
\(518\) 3107.30 0.263565
\(519\) −6111.00 −0.516846
\(520\) 0 0
\(521\) 6054.00 0.509080 0.254540 0.967062i \(-0.418076\pi\)
0.254540 + 0.967062i \(0.418076\pi\)
\(522\) −5258.51 −0.440917
\(523\) −14803.0 −1.23765 −0.618824 0.785530i \(-0.712391\pi\)
−0.618824 + 0.785530i \(0.712391\pi\)
\(524\) −1488.00 −0.124053
\(525\) 10560.3 0.877885
\(526\) −2712.39 −0.224840
\(527\) 1964.15 0.162352
\(528\) −12609.3 −1.03930
\(529\) −8918.00 −0.732966
\(530\) 20448.0 1.67586
\(531\) 419.156 0.0342558
\(532\) 7956.00 0.648377
\(533\) 0 0
\(534\) −7434.00 −0.602436
\(535\) −3533.38 −0.285536
\(536\) 2280.00 0.183733
\(537\) −9009.00 −0.723961
\(538\) −17615.0 −1.41159
\(539\) −3692.73 −0.295097
\(540\) 1939.90 0.154592
\(541\) 21470.5 1.70626 0.853132 0.521695i \(-0.174700\pi\)
0.853132 + 0.521695i \(0.174700\pi\)
\(542\) −4590.00 −0.363759
\(543\) 14.0000 0.00110644
\(544\) 4489.48 0.353832
\(545\) −8448.00 −0.663986
\(546\) 0 0
\(547\) −13516.0 −1.05649 −0.528247 0.849091i \(-0.677151\pi\)
−0.528247 + 0.849091i \(0.677151\pi\)
\(548\) 4759.68 0.371028
\(549\) −374.000 −0.0290746
\(550\) −5226.00 −0.405159
\(551\) 6095.09 0.471251
\(552\) −5528.71 −0.426300
\(553\) 28010.7 2.15396
\(554\) 11850.7 0.908822
\(555\) 3864.00 0.295527
\(556\) −10180.0 −0.776490
\(557\) 2890.79 0.219905 0.109952 0.993937i \(-0.464930\pi\)
0.109952 + 0.993937i \(0.464930\pi\)
\(558\) −5544.00 −0.420603
\(559\) 0 0
\(560\) 24960.0 1.88349
\(561\) −4255.65 −0.320274
\(562\) −2808.00 −0.210762
\(563\) −11583.0 −0.867079 −0.433539 0.901135i \(-0.642735\pi\)
−0.433539 + 0.901135i \(0.642735\pi\)
\(564\) −9602.49 −0.716911
\(565\) 5694.98 0.424053
\(566\) −24861.9 −1.84633
\(567\) 18891.5 1.39924
\(568\) −8088.00 −0.597473
\(569\) −12879.0 −0.948885 −0.474443 0.880286i \(-0.657350\pi\)
−0.474443 + 0.880286i \(0.657350\pi\)
\(570\) 29680.4 2.18101
\(571\) 11636.0 0.852805 0.426402 0.904534i \(-0.359781\pi\)
0.426402 + 0.904534i \(0.359781\pi\)
\(572\) 0 0
\(573\) 19887.0 1.44990
\(574\) −30667.7 −2.23004
\(575\) −3819.00 −0.276980
\(576\) 1408.00 0.101852
\(577\) 12311.4 0.888269 0.444134 0.895960i \(-0.353511\pi\)
0.444134 + 0.895960i \(0.353511\pi\)
\(578\) −14493.8 −1.04301
\(579\) 29716.8 2.13297
\(580\) −3824.37 −0.273790
\(581\) 9594.00 0.685071
\(582\) −29946.0 −2.13282
\(583\) −9592.10 −0.681414
\(584\) 13920.0 0.986325
\(585\) 0 0
\(586\) −32262.0 −2.27428
\(587\) 15645.6 1.10011 0.550054 0.835129i \(-0.314607\pi\)
0.550054 + 0.835129i \(0.314607\pi\)
\(588\) −4592.00 −0.322059
\(589\) 6426.00 0.449539
\(590\) 914.523 0.0638141
\(591\) 19265.6 1.34092
\(592\) 3186.97 0.221256
\(593\) −25821.4 −1.78813 −0.894063 0.447942i \(-0.852157\pi\)
−0.894063 + 0.447942i \(0.852157\pi\)
\(594\) −2730.00 −0.188575
\(595\) 8424.00 0.580421
\(596\) 5216.94 0.358547
\(597\) −11795.0 −0.808605
\(598\) 0 0
\(599\) 1668.00 0.113777 0.0568887 0.998381i \(-0.481882\pi\)
0.0568887 + 0.998381i \(0.481882\pi\)
\(600\) 6498.65 0.442177
\(601\) 13699.0 0.929773 0.464887 0.885370i \(-0.346095\pi\)
0.464887 + 0.885370i \(0.346095\pi\)
\(602\) −6630.00 −0.448868
\(603\) −3619.99 −0.244473
\(604\) 346.410 0.0233365
\(605\) −11417.7 −0.767264
\(606\) 47503.2 3.18430
\(607\) −23173.0 −1.54953 −0.774764 0.632251i \(-0.782131\pi\)
−0.774764 + 0.632251i \(0.782131\pi\)
\(608\) 14688.0 0.979732
\(609\) −10875.5 −0.723644
\(610\) −816.000 −0.0541621
\(611\) 0 0
\(612\) −2376.00 −0.156935
\(613\) −16615.6 −1.09477 −0.547387 0.836880i \(-0.684377\pi\)
−0.547387 + 0.836880i \(0.684377\pi\)
\(614\) 16548.0 1.08766
\(615\) −38136.0 −2.50047
\(616\) −7025.20 −0.459502
\(617\) 28393.5 1.85264 0.926321 0.376736i \(-0.122954\pi\)
0.926321 + 0.376736i \(0.122954\pi\)
\(618\) 45005.6 2.92944
\(619\) 6245.78 0.405556 0.202778 0.979225i \(-0.435003\pi\)
0.202778 + 0.979225i \(0.435003\pi\)
\(620\) −4032.00 −0.261176
\(621\) −1995.00 −0.128916
\(622\) −21449.7 −1.38273
\(623\) −6903.00 −0.443921
\(624\) 0 0
\(625\) −19511.0 −1.24870
\(626\) −2667.36 −0.170302
\(627\) −13923.0 −0.886812
\(628\) −6136.00 −0.389893
\(629\) 1075.60 0.0681830
\(630\) −23777.6 −1.50369
\(631\) −22379.8 −1.41193 −0.705964 0.708247i \(-0.749486\pi\)
−0.705964 + 0.708247i \(0.749486\pi\)
\(632\) 17237.4 1.08491
\(633\) −11767.0 −0.738857
\(634\) 27912.0 1.74847
\(635\) 31079.9 1.94231
\(636\) −11928.0 −0.743673
\(637\) 0 0
\(638\) 5382.00 0.333974
\(639\) 12841.4 0.794990
\(640\) 21504.0 1.32816
\(641\) −19827.0 −1.22172 −0.610858 0.791740i \(-0.709175\pi\)
−0.610858 + 0.791740i \(0.709175\pi\)
\(642\) 6183.42 0.380125
\(643\) −8450.68 −0.518293 −0.259146 0.965838i \(-0.583441\pi\)
−0.259146 + 0.965838i \(0.583441\pi\)
\(644\) 5133.80 0.314130
\(645\) −8244.56 −0.503301
\(646\) 8262.00 0.503195
\(647\) −2949.00 −0.179192 −0.0895959 0.995978i \(-0.528558\pi\)
−0.0895959 + 0.995978i \(0.528558\pi\)
\(648\) 11625.5 0.704774
\(649\) −429.000 −0.0259472
\(650\) 0 0
\(651\) −11466.0 −0.690304
\(652\) 6533.30 0.392429
\(653\) 12039.0 0.721474 0.360737 0.932668i \(-0.382525\pi\)
0.360737 + 0.932668i \(0.382525\pi\)
\(654\) 14784.0 0.883945
\(655\) −5154.58 −0.307490
\(656\) −31454.0 −1.87206
\(657\) −22101.0 −1.31239
\(658\) −26749.8 −1.58483
\(659\) 3363.00 0.198792 0.0993960 0.995048i \(-0.468309\pi\)
0.0993960 + 0.995048i \(0.468309\pi\)
\(660\) 8736.00 0.515225
\(661\) 10158.5 0.597759 0.298880 0.954291i \(-0.403387\pi\)
0.298880 + 0.954291i \(0.403387\pi\)
\(662\) −18282.0 −1.07334
\(663\) 0 0
\(664\) 5904.00 0.345060
\(665\) 27560.4 1.60714
\(666\) −3036.00 −0.176641
\(667\) 3933.00 0.228315
\(668\) 6505.58 0.376809
\(669\) −28674.1 −1.65711
\(670\) −7898.15 −0.455421
\(671\) 382.783 0.0220226
\(672\) −26208.0 −1.50446
\(673\) 18169.0 1.04066 0.520329 0.853966i \(-0.325809\pi\)
0.520329 + 0.853966i \(0.325809\pi\)
\(674\) −28675.8 −1.63880
\(675\) 2345.00 0.133717
\(676\) 0 0
\(677\) 9042.00 0.513312 0.256656 0.966503i \(-0.417379\pi\)
0.256656 + 0.966503i \(0.417379\pi\)
\(678\) −9966.22 −0.564529
\(679\) −27807.0 −1.57163
\(680\) 5184.00 0.292349
\(681\) −3067.46 −0.172607
\(682\) 5674.20 0.318587
\(683\) 12462.1 0.698169 0.349084 0.937091i \(-0.386493\pi\)
0.349084 + 0.937091i \(0.386493\pi\)
\(684\) −7773.44 −0.434540
\(685\) 16488.0 0.919670
\(686\) 13962.0 0.777072
\(687\) 1260.93 0.0700256
\(688\) −6800.00 −0.376813
\(689\) 0 0
\(690\) 19152.0 1.05667
\(691\) 4318.00 0.237720 0.118860 0.992911i \(-0.462076\pi\)
0.118860 + 0.992911i \(0.462076\pi\)
\(692\) 3492.00 0.191829
\(693\) 11154.0 0.611408
\(694\) 23788.0 1.30112
\(695\) −35264.6 −1.92469
\(696\) −6692.64 −0.364489
\(697\) −10615.7 −0.576901
\(698\) 42102.0 2.28307
\(699\) −40446.0 −2.18857
\(700\) −6034.47 −0.325830
\(701\) 18270.0 0.984377 0.492189 0.870489i \(-0.336197\pi\)
0.492189 + 0.870489i \(0.336197\pi\)
\(702\) 0 0
\(703\) 3519.00 0.188793
\(704\) −1441.07 −0.0771481
\(705\) −33264.0 −1.77701
\(706\) −20118.0 −1.07245
\(707\) 44110.1 2.34644
\(708\) −533.472 −0.0283179
\(709\) −1629.86 −0.0863338 −0.0431669 0.999068i \(-0.513745\pi\)
−0.0431669 + 0.999068i \(0.513745\pi\)
\(710\) 28017.7 1.48096
\(711\) −27368.0 −1.44357
\(712\) −4248.00 −0.223596
\(713\) 4146.53 0.217796
\(714\) −14742.0 −0.772697
\(715\) 0 0
\(716\) 5148.00 0.268701
\(717\) 13021.6 0.678241
\(718\) −4644.00 −0.241382
\(719\) −9831.00 −0.509923 −0.254961 0.966951i \(-0.582063\pi\)
−0.254961 + 0.966951i \(0.582063\pi\)
\(720\) −24387.3 −1.26231
\(721\) 41790.9 2.15863
\(722\) 3270.11 0.168561
\(723\) 14415.9 0.741537
\(724\) −8.00000 −0.000410660 0
\(725\) −4623.00 −0.236819
\(726\) 19980.9 1.02144
\(727\) 15464.0 0.788897 0.394448 0.918918i \(-0.370936\pi\)
0.394448 + 0.918918i \(0.370936\pi\)
\(728\) 0 0
\(729\) −11843.0 −0.601687
\(730\) −48220.3 −2.44481
\(731\) −2295.00 −0.116120
\(732\) 476.000 0.0240348
\(733\) −12616.3 −0.635733 −0.317866 0.948136i \(-0.602966\pi\)
−0.317866 + 0.948136i \(0.602966\pi\)
\(734\) 12695.9 0.638441
\(735\) −15907.2 −0.798291
\(736\) 9477.78 0.474668
\(737\) 3705.00 0.185177
\(738\) 29964.0 1.49457
\(739\) −16283.0 −0.810528 −0.405264 0.914200i \(-0.632820\pi\)
−0.405264 + 0.914200i \(0.632820\pi\)
\(740\) −2208.00 −0.109686
\(741\) 0 0
\(742\) −33228.0 −1.64399
\(743\) −10806.3 −0.533571 −0.266786 0.963756i \(-0.585962\pi\)
−0.266786 + 0.963756i \(0.585962\pi\)
\(744\) −7056.00 −0.347696
\(745\) 18072.0 0.888734
\(746\) 18605.7 0.913140
\(747\) −9373.86 −0.459132
\(748\) 2431.80 0.118871
\(749\) 5741.75 0.280105
\(750\) 19488.0 0.948802
\(751\) 13615.0 0.661542 0.330771 0.943711i \(-0.392691\pi\)
0.330771 + 0.943711i \(0.392691\pi\)
\(752\) −27435.7 −1.33042
\(753\) 31437.0 1.52142
\(754\) 0 0
\(755\) 1200.00 0.0578443
\(756\) −3152.33 −0.151652
\(757\) 5551.00 0.266519 0.133259 0.991081i \(-0.457456\pi\)
0.133259 + 0.991081i \(0.457456\pi\)
\(758\) 39870.0 1.91048
\(759\) −8984.15 −0.429649
\(760\) 16960.2 0.809490
\(761\) −10082.3 −0.480265 −0.240133 0.970740i \(-0.577191\pi\)
−0.240133 + 0.970740i \(0.577191\pi\)
\(762\) −54389.9 −2.58574
\(763\) 13728.0 0.651359
\(764\) −11364.0 −0.538135
\(765\) −8230.71 −0.388996
\(766\) 8382.00 0.395371
\(767\) 0 0
\(768\) −34048.0 −1.59974
\(769\) −29758.4 −1.39547 −0.697733 0.716357i \(-0.745808\pi\)
−0.697733 + 0.716357i \(0.745808\pi\)
\(770\) 24336.0 1.13897
\(771\) 38157.0 1.78235
\(772\) −16981.0 −0.791659
\(773\) 27735.3 1.29052 0.645259 0.763964i \(-0.276749\pi\)
0.645259 + 0.763964i \(0.276749\pi\)
\(774\) 6477.87 0.300830
\(775\) −4873.99 −0.225908
\(776\) −17112.0 −0.791604
\(777\) −6279.00 −0.289907
\(778\) 34149.1 1.57366
\(779\) −34731.0 −1.59739
\(780\) 0 0
\(781\) −13143.0 −0.602168
\(782\) 5331.25 0.243792
\(783\) −2415.00 −0.110224
\(784\) −13120.0 −0.597668
\(785\) −21255.7 −0.966432
\(786\) 9020.52 0.409353
\(787\) 31549.3 1.42899 0.714493 0.699643i \(-0.246658\pi\)
0.714493 + 0.699643i \(0.246658\pi\)
\(788\) −11008.9 −0.497686
\(789\) 5481.00 0.247311
\(790\) −59712.0 −2.68919
\(791\) −9254.35 −0.415988
\(792\) 6864.00 0.307957
\(793\) 0 0
\(794\) 30210.0 1.35027
\(795\) −41319.8 −1.84335
\(796\) 6740.00 0.300117
\(797\) −1455.00 −0.0646659 −0.0323330 0.999477i \(-0.510294\pi\)
−0.0323330 + 0.999477i \(0.510294\pi\)
\(798\) −48230.7 −2.13953
\(799\) −9259.54 −0.409986
\(800\) −11140.6 −0.492347
\(801\) 6744.61 0.297514
\(802\) −26274.0 −1.15682
\(803\) 22620.0 0.994075
\(804\) 4607.26 0.202096
\(805\) 17784.0 0.778638
\(806\) 0 0
\(807\) 35595.0 1.55267
\(808\) 27144.7 1.18187
\(809\) 1659.00 0.0720981 0.0360490 0.999350i \(-0.488523\pi\)
0.0360490 + 0.999350i \(0.488523\pi\)
\(810\) −40272.0 −1.74693
\(811\) 4402.87 0.190636 0.0953180 0.995447i \(-0.469613\pi\)
0.0953180 + 0.995447i \(0.469613\pi\)
\(812\) 6214.60 0.268583
\(813\) 9275.13 0.400114
\(814\) 3107.30 0.133797
\(815\) 22632.0 0.972717
\(816\) −15120.0 −0.648659
\(817\) −7508.44 −0.321526
\(818\) 14910.0 0.637306
\(819\) 0 0
\(820\) 21792.0 0.928061
\(821\) 28701.8 1.22010 0.610049 0.792364i \(-0.291150\pi\)
0.610049 + 0.792364i \(0.291150\pi\)
\(822\) −28854.0 −1.22433
\(823\) −15779.0 −0.668313 −0.334156 0.942518i \(-0.608451\pi\)
−0.334156 + 0.942518i \(0.608451\pi\)
\(824\) 25717.5 1.08727
\(825\) 10560.3 0.445652
\(826\) −1486.10 −0.0626005
\(827\) 7354.29 0.309231 0.154615 0.987975i \(-0.450586\pi\)
0.154615 + 0.987975i \(0.450586\pi\)
\(828\) −5016.00 −0.210529
\(829\) −17371.0 −0.727768 −0.363884 0.931444i \(-0.618550\pi\)
−0.363884 + 0.931444i \(0.618550\pi\)
\(830\) −20452.1 −0.855303
\(831\) −23947.0 −0.999654
\(832\) 0 0
\(833\) −4428.00 −0.184179
\(834\) 61713.0 2.56228
\(835\) 22536.0 0.934001
\(836\) 7956.00 0.329144
\(837\) −2546.11 −0.105145
\(838\) 18695.8 0.770685
\(839\) −29474.3 −1.21283 −0.606416 0.795148i \(-0.707393\pi\)
−0.606416 + 0.795148i \(0.707393\pi\)
\(840\) −30262.4 −1.24304
\(841\) −19628.0 −0.804789
\(842\) 25152.0 1.02945
\(843\) 5674.20 0.231827
\(844\) 6724.00 0.274229
\(845\) 0 0
\(846\) 26136.0 1.06214
\(847\) 18553.7 0.752673
\(848\) −34080.0 −1.38008
\(849\) 50239.0 2.03086
\(850\) −6266.56 −0.252872
\(851\) 2270.72 0.0914680
\(852\) −16343.6 −0.657187
\(853\) −2909.85 −0.116801 −0.0584005 0.998293i \(-0.518600\pi\)
−0.0584005 + 0.998293i \(0.518600\pi\)
\(854\) 1326.00 0.0531321
\(855\) −26928.0 −1.07710
\(856\) 3533.38 0.141085
\(857\) 5346.00 0.213087 0.106544 0.994308i \(-0.466022\pi\)
0.106544 + 0.994308i \(0.466022\pi\)
\(858\) 0 0
\(859\) 24244.0 0.962974 0.481487 0.876453i \(-0.340097\pi\)
0.481487 + 0.876453i \(0.340097\pi\)
\(860\) 4711.18 0.186802
\(861\) 61971.0 2.45292
\(862\) 1686.00 0.0666188
\(863\) 32780.8 1.29301 0.646507 0.762908i \(-0.276229\pi\)
0.646507 + 0.762908i \(0.276229\pi\)
\(864\) −5819.69 −0.229155
\(865\) 12096.6 0.475489
\(866\) −42050.7 −1.65005
\(867\) 29288.0 1.14726
\(868\) 6552.00 0.256209
\(869\) 28010.7 1.09344
\(870\) 23184.0 0.903461
\(871\) 0 0
\(872\) 8448.00 0.328080
\(873\) 27168.9 1.05330
\(874\) 17442.0 0.675039
\(875\) 18096.0 0.699150
\(876\) 28128.5 1.08490
\(877\) −4543.17 −0.174928 −0.0874640 0.996168i \(-0.527876\pi\)
−0.0874640 + 0.996168i \(0.527876\pi\)
\(878\) −1596.95 −0.0613832
\(879\) 65192.7 2.50159
\(880\) 24960.0 0.956138
\(881\) 20517.0 0.784603 0.392302 0.919837i \(-0.371679\pi\)
0.392302 + 0.919837i \(0.371679\pi\)
\(882\) 12498.5 0.477149
\(883\) 23852.0 0.909042 0.454521 0.890736i \(-0.349811\pi\)
0.454521 + 0.890736i \(0.349811\pi\)
\(884\) 0 0
\(885\) −1848.00 −0.0701919
\(886\) 42109.6 1.59673
\(887\) 38757.0 1.46712 0.733558 0.679626i \(-0.237858\pi\)
0.733558 + 0.679626i \(0.237858\pi\)
\(888\) −3864.00 −0.146022
\(889\) −50504.9 −1.90538
\(890\) 14715.5 0.554230
\(891\) 18891.5 0.710312
\(892\) 16385.2 0.615042
\(893\) −30294.0 −1.13522
\(894\) −31626.0 −1.18315
\(895\) 17833.2 0.666031
\(896\) −34944.0 −1.30290
\(897\) 0 0
\(898\) 1026.00 0.0381270
\(899\) 5019.48 0.186217
\(900\) 5896.00 0.218370
\(901\) −11502.0 −0.425291
\(902\) −30667.7 −1.13206
\(903\) 13397.4 0.493730
\(904\) −5694.98 −0.209527
\(905\) −27.7128 −0.00101791
\(906\) −2100.00 −0.0770064
\(907\) 39071.0 1.43035 0.715177 0.698943i \(-0.246346\pi\)
0.715177 + 0.698943i \(0.246346\pi\)
\(908\) 1752.84 0.0640638
\(909\) −43098.0 −1.57257
\(910\) 0 0
\(911\) −53040.0 −1.92897 −0.964486 0.264134i \(-0.914914\pi\)
−0.964486 + 0.264134i \(0.914914\pi\)
\(912\) −49467.4 −1.79608
\(913\) 9594.00 0.347771
\(914\) −2118.00 −0.0766490
\(915\) 1648.91 0.0595753
\(916\) −720.533 −0.0259903
\(917\) 8376.20 0.301643
\(918\) −3273.58 −0.117695
\(919\) 367.000 0.0131732 0.00658662 0.999978i \(-0.497903\pi\)
0.00658662 + 0.999978i \(0.497903\pi\)
\(920\) 10944.0 0.392188
\(921\) −33439.0 −1.19636
\(922\) −45474.0 −1.62430
\(923\) 0 0
\(924\) −14196.0 −0.505427
\(925\) −2669.09 −0.0948748
\(926\) 2892.00 0.102632
\(927\) −40832.0 −1.44671
\(928\) 11473.1 0.405844
\(929\) −29935.0 −1.05720 −0.528599 0.848872i \(-0.677282\pi\)
−0.528599 + 0.848872i \(0.677282\pi\)
\(930\) 24442.7 0.861836
\(931\) −14486.9 −0.509976
\(932\) 23112.0 0.812295
\(933\) 43344.0 1.52092
\(934\) −50215.6 −1.75921
\(935\) 8424.00 0.294646
\(936\) 0 0
\(937\) 42166.0 1.47012 0.735060 0.678002i \(-0.237154\pi\)
0.735060 + 0.678002i \(0.237154\pi\)
\(938\) 12834.5 0.446760
\(939\) 5390.00 0.187323
\(940\) 19008.0 0.659545
\(941\) −35022.1 −1.21327 −0.606635 0.794981i \(-0.707481\pi\)
−0.606635 + 0.794981i \(0.707481\pi\)
\(942\) 37197.5 1.28658
\(943\) −22411.0 −0.773916
\(944\) −1524.20 −0.0525515
\(945\) −10920.0 −0.375902
\(946\) −6630.00 −0.227865
\(947\) −2599.81 −0.0892106 −0.0446053 0.999005i \(-0.514203\pi\)
−0.0446053 + 0.999005i \(0.514203\pi\)
\(948\) 34832.0 1.19334
\(949\) 0 0
\(950\) −20502.0 −0.700182
\(951\) −56402.5 −1.92321
\(952\) −8424.00 −0.286789
\(953\) −10623.0 −0.361084 −0.180542 0.983567i \(-0.557785\pi\)
−0.180542 + 0.983567i \(0.557785\pi\)
\(954\) 32465.6 1.10179
\(955\) −39366.1 −1.33388
\(956\) −7440.89 −0.251732
\(957\) −10875.5 −0.367353
\(958\) −30822.0 −1.03947
\(959\) −26793.0 −0.902180
\(960\) −6207.67 −0.208700
\(961\) −24499.0 −0.822362
\(962\) 0 0
\(963\) −5610.00 −0.187726
\(964\) −8237.63 −0.275224
\(965\) −58824.0 −1.96229
\(966\) −31122.0 −1.03658
\(967\) 20199.2 0.671729 0.335864 0.941910i \(-0.390972\pi\)
0.335864 + 0.941910i \(0.390972\pi\)
\(968\) 11417.7 0.379110
\(969\) −16695.2 −0.553486
\(970\) 59277.7 1.96216
\(971\) −2325.00 −0.0768412 −0.0384206 0.999262i \(-0.512233\pi\)
−0.0384206 + 0.999262i \(0.512233\pi\)
\(972\) 19712.0 0.650476
\(973\) 57304.9 1.88809
\(974\) 16470.0 0.541820
\(975\) 0 0
\(976\) 1360.00 0.0446030
\(977\) −32938.4 −1.07860 −0.539300 0.842113i \(-0.681311\pi\)
−0.539300 + 0.842113i \(0.681311\pi\)
\(978\) −39606.0 −1.29495
\(979\) −6903.00 −0.225353
\(980\) 9089.80 0.296289
\(981\) −13413.0 −0.436538
\(982\) 5663.81 0.184052
\(983\) −42702.0 −1.38554 −0.692768 0.721161i \(-0.743609\pi\)
−0.692768 + 0.721161i \(0.743609\pi\)
\(984\) 38136.0 1.23550
\(985\) −38136.0 −1.23362
\(986\) 6453.62 0.208443
\(987\) 54054.0 1.74322
\(988\) 0 0
\(989\) −4845.00 −0.155776
\(990\) −23777.6 −0.763335
\(991\) −4843.00 −0.155240 −0.0776201 0.996983i \(-0.524732\pi\)
−0.0776201 + 0.996983i \(0.524732\pi\)
\(992\) 12096.0 0.387146
\(993\) 36942.9 1.18061
\(994\) −45528.7 −1.45280
\(995\) 23348.0 0.743902
\(996\) 11930.4 0.379546
\(997\) 10943.0 0.347611 0.173806 0.984780i \(-0.444394\pi\)
0.173806 + 0.984780i \(0.444394\pi\)
\(998\) −50004.0 −1.58602
\(999\) −1394.30 −0.0441579
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 169.4.a.h.1.2 2
3.2 odd 2 1521.4.a.q.1.1 2
13.2 odd 12 169.4.e.b.147.1 2
13.3 even 3 169.4.c.i.22.1 4
13.4 even 6 169.4.c.i.146.2 4
13.5 odd 4 169.4.b.b.168.1 2
13.6 odd 12 13.4.e.a.10.1 yes 2
13.7 odd 12 169.4.e.b.23.1 2
13.8 odd 4 169.4.b.b.168.2 2
13.9 even 3 169.4.c.i.146.1 4
13.10 even 6 169.4.c.i.22.2 4
13.11 odd 12 13.4.e.a.4.1 2
13.12 even 2 inner 169.4.a.h.1.1 2
39.11 even 12 117.4.q.c.82.1 2
39.32 even 12 117.4.q.c.10.1 2
39.38 odd 2 1521.4.a.q.1.2 2
52.11 even 12 208.4.w.a.17.1 2
52.19 even 12 208.4.w.a.49.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.4.e.a.4.1 2 13.11 odd 12
13.4.e.a.10.1 yes 2 13.6 odd 12
117.4.q.c.10.1 2 39.32 even 12
117.4.q.c.82.1 2 39.11 even 12
169.4.a.h.1.1 2 13.12 even 2 inner
169.4.a.h.1.2 2 1.1 even 1 trivial
169.4.b.b.168.1 2 13.5 odd 4
169.4.b.b.168.2 2 13.8 odd 4
169.4.c.i.22.1 4 13.3 even 3
169.4.c.i.22.2 4 13.10 even 6
169.4.c.i.146.1 4 13.9 even 3
169.4.c.i.146.2 4 13.4 even 6
169.4.e.b.23.1 2 13.7 odd 12
169.4.e.b.147.1 2 13.2 odd 12
208.4.w.a.17.1 2 52.11 even 12
208.4.w.a.49.1 2 52.19 even 12
1521.4.a.q.1.1 2 3.2 odd 2
1521.4.a.q.1.2 2 39.38 odd 2