Properties

Label 35.3.d.a.6.1
Level $35$
Weight $3$
Character 35.6
Analytic conductor $0.954$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 35.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.953680925261\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
Defining polynomial: \( x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 6.1
Root \(-2.23607i\) of defining polynomial
Character \(\chi\) \(=\) 35.6
Dual form 35.3.d.a.6.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -4.47214i q^{3} -3.00000 q^{4} -2.23607i q^{5} +4.47214i q^{6} +7.00000 q^{7} +7.00000 q^{8} -11.0000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -4.47214i q^{3} -3.00000 q^{4} -2.23607i q^{5} +4.47214i q^{6} +7.00000 q^{7} +7.00000 q^{8} -11.0000 q^{9} +2.23607i q^{10} +2.00000 q^{11} +13.4164i q^{12} +13.4164i q^{13} -7.00000 q^{14} -10.0000 q^{15} +5.00000 q^{16} -26.8328i q^{17} +11.0000 q^{18} +13.4164i q^{19} +6.70820i q^{20} -31.3050i q^{21} -2.00000 q^{22} +26.0000 q^{23} -31.3050i q^{24} -5.00000 q^{25} -13.4164i q^{26} +8.94427i q^{27} -21.0000 q^{28} -22.0000 q^{29} +10.0000 q^{30} +53.6656i q^{31} -33.0000 q^{32} -8.94427i q^{33} +26.8328i q^{34} -15.6525i q^{35} +33.0000 q^{36} +14.0000 q^{37} -13.4164i q^{38} +60.0000 q^{39} -15.6525i q^{40} +26.8328i q^{41} +31.3050i q^{42} -34.0000 q^{43} -6.00000 q^{44} +24.5967i q^{45} -26.0000 q^{46} +26.8328i q^{47} -22.3607i q^{48} +49.0000 q^{49} +5.00000 q^{50} -120.000 q^{51} -40.2492i q^{52} -34.0000 q^{53} -8.94427i q^{54} -4.47214i q^{55} +49.0000 q^{56} +60.0000 q^{57} +22.0000 q^{58} +40.2492i q^{59} +30.0000 q^{60} -93.9149i q^{61} -53.6656i q^{62} -77.0000 q^{63} +13.0000 q^{64} +30.0000 q^{65} +8.94427i q^{66} +14.0000 q^{67} +80.4984i q^{68} -116.276i q^{69} +15.6525i q^{70} +62.0000 q^{71} -77.0000 q^{72} +53.6656i q^{73} -14.0000 q^{74} +22.3607i q^{75} -40.2492i q^{76} +14.0000 q^{77} -60.0000 q^{78} +38.0000 q^{79} -11.1803i q^{80} -59.0000 q^{81} -26.8328i q^{82} -40.2492i q^{83} +93.9149i q^{84} -60.0000 q^{85} +34.0000 q^{86} +98.3870i q^{87} +14.0000 q^{88} +26.8328i q^{89} -24.5967i q^{90} +93.9149i q^{91} -78.0000 q^{92} +240.000 q^{93} -26.8328i q^{94} +30.0000 q^{95} +147.580i q^{96} +26.8328i q^{97} -49.0000 q^{98} -22.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 6 q^{4} + 14 q^{7} + 14 q^{8} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 6 q^{4} + 14 q^{7} + 14 q^{8} - 22 q^{9} + 4 q^{11} - 14 q^{14} - 20 q^{15} + 10 q^{16} + 22 q^{18} - 4 q^{22} + 52 q^{23} - 10 q^{25} - 42 q^{28} - 44 q^{29} + 20 q^{30} - 66 q^{32} + 66 q^{36} + 28 q^{37} + 120 q^{39} - 68 q^{43} - 12 q^{44} - 52 q^{46} + 98 q^{49} + 10 q^{50} - 240 q^{51} - 68 q^{53} + 98 q^{56} + 120 q^{57} + 44 q^{58} + 60 q^{60} - 154 q^{63} + 26 q^{64} + 60 q^{65} + 28 q^{67} + 124 q^{71} - 154 q^{72} - 28 q^{74} + 28 q^{77} - 120 q^{78} + 76 q^{79} - 118 q^{81} - 120 q^{85} + 68 q^{86} + 28 q^{88} - 156 q^{92} + 480 q^{93} + 60 q^{95} - 98 q^{98} - 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\)
\(\chi(n)\) \(1\) \(-1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.500000 −0.250000 0.968246i \(-0.580431\pi\)
−0.250000 + 0.968246i \(0.580431\pi\)
\(3\) − 4.47214i − 1.49071i −0.666667 0.745356i \(-0.732280\pi\)
0.666667 0.745356i \(-0.267720\pi\)
\(4\) −3.00000 −0.750000
\(5\) − 2.23607i − 0.447214i
\(6\) 4.47214i 0.745356i
\(7\) 7.00000 1.00000
\(8\) 7.00000 0.875000
\(9\) −11.0000 −1.22222
\(10\) 2.23607i 0.223607i
\(11\) 2.00000 0.181818 0.0909091 0.995859i \(-0.471023\pi\)
0.0909091 + 0.995859i \(0.471023\pi\)
\(12\) 13.4164i 1.11803i
\(13\) 13.4164i 1.03203i 0.856579 + 0.516016i \(0.172585\pi\)
−0.856579 + 0.516016i \(0.827415\pi\)
\(14\) −7.00000 −0.500000
\(15\) −10.0000 −0.666667
\(16\) 5.00000 0.312500
\(17\) − 26.8328i − 1.57840i −0.614136 0.789200i \(-0.710495\pi\)
0.614136 0.789200i \(-0.289505\pi\)
\(18\) 11.0000 0.611111
\(19\) 13.4164i 0.706127i 0.935599 + 0.353063i \(0.114860\pi\)
−0.935599 + 0.353063i \(0.885140\pi\)
\(20\) 6.70820i 0.335410i
\(21\) − 31.3050i − 1.49071i
\(22\) −2.00000 −0.0909091
\(23\) 26.0000 1.13043 0.565217 0.824942i \(-0.308792\pi\)
0.565217 + 0.824942i \(0.308792\pi\)
\(24\) − 31.3050i − 1.30437i
\(25\) −5.00000 −0.200000
\(26\) − 13.4164i − 0.516016i
\(27\) 8.94427i 0.331269i
\(28\) −21.0000 −0.750000
\(29\) −22.0000 −0.758621 −0.379310 0.925270i \(-0.623839\pi\)
−0.379310 + 0.925270i \(0.623839\pi\)
\(30\) 10.0000 0.333333
\(31\) 53.6656i 1.73115i 0.500780 + 0.865575i \(0.333047\pi\)
−0.500780 + 0.865575i \(0.666953\pi\)
\(32\) −33.0000 −1.03125
\(33\) − 8.94427i − 0.271039i
\(34\) 26.8328i 0.789200i
\(35\) − 15.6525i − 0.447214i
\(36\) 33.0000 0.916667
\(37\) 14.0000 0.378378 0.189189 0.981941i \(-0.439414\pi\)
0.189189 + 0.981941i \(0.439414\pi\)
\(38\) − 13.4164i − 0.353063i
\(39\) 60.0000 1.53846
\(40\) − 15.6525i − 0.391312i
\(41\) 26.8328i 0.654459i 0.944945 + 0.327229i \(0.106115\pi\)
−0.944945 + 0.327229i \(0.893885\pi\)
\(42\) 31.3050i 0.745356i
\(43\) −34.0000 −0.790698 −0.395349 0.918531i \(-0.629376\pi\)
−0.395349 + 0.918531i \(0.629376\pi\)
\(44\) −6.00000 −0.136364
\(45\) 24.5967i 0.546594i
\(46\) −26.0000 −0.565217
\(47\) 26.8328i 0.570911i 0.958392 + 0.285455i \(0.0921449\pi\)
−0.958392 + 0.285455i \(0.907855\pi\)
\(48\) − 22.3607i − 0.465847i
\(49\) 49.0000 1.00000
\(50\) 5.00000 0.100000
\(51\) −120.000 −2.35294
\(52\) − 40.2492i − 0.774024i
\(53\) −34.0000 −0.641509 −0.320755 0.947162i \(-0.603937\pi\)
−0.320755 + 0.947162i \(0.603937\pi\)
\(54\) − 8.94427i − 0.165635i
\(55\) − 4.47214i − 0.0813116i
\(56\) 49.0000 0.875000
\(57\) 60.0000 1.05263
\(58\) 22.0000 0.379310
\(59\) 40.2492i 0.682190i 0.940029 + 0.341095i \(0.110798\pi\)
−0.940029 + 0.341095i \(0.889202\pi\)
\(60\) 30.0000 0.500000
\(61\) − 93.9149i − 1.53959i −0.638293 0.769794i \(-0.720359\pi\)
0.638293 0.769794i \(-0.279641\pi\)
\(62\) − 53.6656i − 0.865575i
\(63\) −77.0000 −1.22222
\(64\) 13.0000 0.203125
\(65\) 30.0000 0.461538
\(66\) 8.94427i 0.135519i
\(67\) 14.0000 0.208955 0.104478 0.994527i \(-0.466683\pi\)
0.104478 + 0.994527i \(0.466683\pi\)
\(68\) 80.4984i 1.18380i
\(69\) − 116.276i − 1.68515i
\(70\) 15.6525i 0.223607i
\(71\) 62.0000 0.873239 0.436620 0.899646i \(-0.356176\pi\)
0.436620 + 0.899646i \(0.356176\pi\)
\(72\) −77.0000 −1.06944
\(73\) 53.6656i 0.735146i 0.929995 + 0.367573i \(0.119811\pi\)
−0.929995 + 0.367573i \(0.880189\pi\)
\(74\) −14.0000 −0.189189
\(75\) 22.3607i 0.298142i
\(76\) − 40.2492i − 0.529595i
\(77\) 14.0000 0.181818
\(78\) −60.0000 −0.769231
\(79\) 38.0000 0.481013 0.240506 0.970648i \(-0.422687\pi\)
0.240506 + 0.970648i \(0.422687\pi\)
\(80\) − 11.1803i − 0.139754i
\(81\) −59.0000 −0.728395
\(82\) − 26.8328i − 0.327229i
\(83\) − 40.2492i − 0.484930i −0.970160 0.242465i \(-0.922044\pi\)
0.970160 0.242465i \(-0.0779560\pi\)
\(84\) 93.9149i 1.11803i
\(85\) −60.0000 −0.705882
\(86\) 34.0000 0.395349
\(87\) 98.3870i 1.13088i
\(88\) 14.0000 0.159091
\(89\) 26.8328i 0.301492i 0.988573 + 0.150746i \(0.0481676\pi\)
−0.988573 + 0.150746i \(0.951832\pi\)
\(90\) − 24.5967i − 0.273297i
\(91\) 93.9149i 1.03203i
\(92\) −78.0000 −0.847826
\(93\) 240.000 2.58065
\(94\) − 26.8328i − 0.285455i
\(95\) 30.0000 0.315789
\(96\) 147.580i 1.53730i
\(97\) 26.8328i 0.276627i 0.990388 + 0.138313i \(0.0441681\pi\)
−0.990388 + 0.138313i \(0.955832\pi\)
\(98\) −49.0000 −0.500000
\(99\) −22.0000 −0.222222
\(100\) 15.0000 0.150000
\(101\) − 67.0820i − 0.664179i −0.943248 0.332089i \(-0.892246\pi\)
0.943248 0.332089i \(-0.107754\pi\)
\(102\) 120.000 1.17647
\(103\) − 160.997i − 1.56308i −0.623857 0.781538i \(-0.714435\pi\)
0.623857 0.781538i \(-0.285565\pi\)
\(104\) 93.9149i 0.903027i
\(105\) −70.0000 −0.666667
\(106\) 34.0000 0.320755
\(107\) −106.000 −0.990654 −0.495327 0.868707i \(-0.664952\pi\)
−0.495327 + 0.868707i \(0.664952\pi\)
\(108\) − 26.8328i − 0.248452i
\(109\) −142.000 −1.30275 −0.651376 0.758755i \(-0.725808\pi\)
−0.651376 + 0.758755i \(0.725808\pi\)
\(110\) 4.47214i 0.0406558i
\(111\) − 62.6099i − 0.564053i
\(112\) 35.0000 0.312500
\(113\) −34.0000 −0.300885 −0.150442 0.988619i \(-0.548070\pi\)
−0.150442 + 0.988619i \(0.548070\pi\)
\(114\) −60.0000 −0.526316
\(115\) − 58.1378i − 0.505546i
\(116\) 66.0000 0.568966
\(117\) − 147.580i − 1.26137i
\(118\) − 40.2492i − 0.341095i
\(119\) − 187.830i − 1.57840i
\(120\) −70.0000 −0.583333
\(121\) −117.000 −0.966942
\(122\) 93.9149i 0.769794i
\(123\) 120.000 0.975610
\(124\) − 160.997i − 1.29836i
\(125\) 11.1803i 0.0894427i
\(126\) 77.0000 0.611111
\(127\) 194.000 1.52756 0.763780 0.645477i \(-0.223341\pi\)
0.763780 + 0.645477i \(0.223341\pi\)
\(128\) 119.000 0.929688
\(129\) 152.053i 1.17870i
\(130\) −30.0000 −0.230769
\(131\) 120.748i 0.921738i 0.887468 + 0.460869i \(0.152462\pi\)
−0.887468 + 0.460869i \(0.847538\pi\)
\(132\) 26.8328i 0.203279i
\(133\) 93.9149i 0.706127i
\(134\) −14.0000 −0.104478
\(135\) 20.0000 0.148148
\(136\) − 187.830i − 1.38110i
\(137\) −166.000 −1.21168 −0.605839 0.795587i \(-0.707163\pi\)
−0.605839 + 0.795587i \(0.707163\pi\)
\(138\) 116.276i 0.842576i
\(139\) 93.9149i 0.675646i 0.941210 + 0.337823i \(0.109691\pi\)
−0.941210 + 0.337823i \(0.890309\pi\)
\(140\) 46.9574i 0.335410i
\(141\) 120.000 0.851064
\(142\) −62.0000 −0.436620
\(143\) 26.8328i 0.187642i
\(144\) −55.0000 −0.381944
\(145\) 49.1935i 0.339265i
\(146\) − 53.6656i − 0.367573i
\(147\) − 219.135i − 1.49071i
\(148\) −42.0000 −0.283784
\(149\) −142.000 −0.953020 −0.476510 0.879169i \(-0.658098\pi\)
−0.476510 + 0.879169i \(0.658098\pi\)
\(150\) − 22.3607i − 0.149071i
\(151\) 2.00000 0.0132450 0.00662252 0.999978i \(-0.497892\pi\)
0.00662252 + 0.999978i \(0.497892\pi\)
\(152\) 93.9149i 0.617861i
\(153\) 295.161i 1.92916i
\(154\) −14.0000 −0.0909091
\(155\) 120.000 0.774194
\(156\) −180.000 −1.15385
\(157\) 67.0820i 0.427274i 0.976913 + 0.213637i \(0.0685310\pi\)
−0.976913 + 0.213637i \(0.931469\pi\)
\(158\) −38.0000 −0.240506
\(159\) 152.053i 0.956306i
\(160\) 73.7902i 0.461189i
\(161\) 182.000 1.13043
\(162\) 59.0000 0.364198
\(163\) −34.0000 −0.208589 −0.104294 0.994546i \(-0.533258\pi\)
−0.104294 + 0.994546i \(0.533258\pi\)
\(164\) − 80.4984i − 0.490844i
\(165\) −20.0000 −0.121212
\(166\) 40.2492i 0.242465i
\(167\) 107.331i 0.642702i 0.946960 + 0.321351i \(0.104137\pi\)
−0.946960 + 0.321351i \(0.895863\pi\)
\(168\) − 219.135i − 1.30437i
\(169\) −11.0000 −0.0650888
\(170\) 60.0000 0.352941
\(171\) − 147.580i − 0.863044i
\(172\) 102.000 0.593023
\(173\) − 147.580i − 0.853066i −0.904472 0.426533i \(-0.859735\pi\)
0.904472 0.426533i \(-0.140265\pi\)
\(174\) − 98.3870i − 0.565442i
\(175\) −35.0000 −0.200000
\(176\) 10.0000 0.0568182
\(177\) 180.000 1.01695
\(178\) − 26.8328i − 0.150746i
\(179\) 218.000 1.21788 0.608939 0.793217i \(-0.291596\pi\)
0.608939 + 0.793217i \(0.291596\pi\)
\(180\) − 73.7902i − 0.409946i
\(181\) 254.912i 1.40835i 0.710025 + 0.704176i \(0.248683\pi\)
−0.710025 + 0.704176i \(0.751317\pi\)
\(182\) − 93.9149i − 0.516016i
\(183\) −420.000 −2.29508
\(184\) 182.000 0.989130
\(185\) − 31.3050i − 0.169216i
\(186\) −240.000 −1.29032
\(187\) − 53.6656i − 0.286982i
\(188\) − 80.4984i − 0.428183i
\(189\) 62.6099i 0.331269i
\(190\) −30.0000 −0.157895
\(191\) −58.0000 −0.303665 −0.151832 0.988406i \(-0.548517\pi\)
−0.151832 + 0.988406i \(0.548517\pi\)
\(192\) − 58.1378i − 0.302801i
\(193\) 206.000 1.06736 0.533679 0.845687i \(-0.320809\pi\)
0.533679 + 0.845687i \(0.320809\pi\)
\(194\) − 26.8328i − 0.138313i
\(195\) − 134.164i − 0.688021i
\(196\) −147.000 −0.750000
\(197\) −226.000 −1.14721 −0.573604 0.819133i \(-0.694455\pi\)
−0.573604 + 0.819133i \(0.694455\pi\)
\(198\) 22.0000 0.111111
\(199\) − 134.164i − 0.674191i −0.941470 0.337096i \(-0.890555\pi\)
0.941470 0.337096i \(-0.109445\pi\)
\(200\) −35.0000 −0.175000
\(201\) − 62.6099i − 0.311492i
\(202\) 67.0820i 0.332089i
\(203\) −154.000 −0.758621
\(204\) 360.000 1.76471
\(205\) 60.0000 0.292683
\(206\) 160.997i 0.781538i
\(207\) −286.000 −1.38164
\(208\) 67.0820i 0.322510i
\(209\) 26.8328i 0.128387i
\(210\) 70.0000 0.333333
\(211\) −118.000 −0.559242 −0.279621 0.960111i \(-0.590209\pi\)
−0.279621 + 0.960111i \(0.590209\pi\)
\(212\) 102.000 0.481132
\(213\) − 277.272i − 1.30175i
\(214\) 106.000 0.495327
\(215\) 76.0263i 0.353611i
\(216\) 62.6099i 0.289861i
\(217\) 375.659i 1.73115i
\(218\) 142.000 0.651376
\(219\) 240.000 1.09589
\(220\) 13.4164i 0.0609837i
\(221\) 360.000 1.62896
\(222\) 62.6099i 0.282027i
\(223\) − 80.4984i − 0.360980i −0.983577 0.180490i \(-0.942232\pi\)
0.983577 0.180490i \(-0.0577683\pi\)
\(224\) −231.000 −1.03125
\(225\) 55.0000 0.244444
\(226\) 34.0000 0.150442
\(227\) − 254.912i − 1.12296i −0.827491 0.561480i \(-0.810232\pi\)
0.827491 0.561480i \(-0.189768\pi\)
\(228\) −180.000 −0.789474
\(229\) − 13.4164i − 0.0585869i −0.999571 0.0292935i \(-0.990674\pi\)
0.999571 0.0292935i \(-0.00932573\pi\)
\(230\) 58.1378i 0.252773i
\(231\) − 62.6099i − 0.271039i
\(232\) −154.000 −0.663793
\(233\) −214.000 −0.918455 −0.459227 0.888319i \(-0.651874\pi\)
−0.459227 + 0.888319i \(0.651874\pi\)
\(234\) 147.580i 0.630686i
\(235\) 60.0000 0.255319
\(236\) − 120.748i − 0.511643i
\(237\) − 169.941i − 0.717051i
\(238\) 187.830i 0.789200i
\(239\) 98.0000 0.410042 0.205021 0.978758i \(-0.434274\pi\)
0.205021 + 0.978758i \(0.434274\pi\)
\(240\) −50.0000 −0.208333
\(241\) 160.997i 0.668037i 0.942567 + 0.334018i \(0.108405\pi\)
−0.942567 + 0.334018i \(0.891595\pi\)
\(242\) 117.000 0.483471
\(243\) 344.354i 1.41710i
\(244\) 281.745i 1.15469i
\(245\) − 109.567i − 0.447214i
\(246\) −120.000 −0.487805
\(247\) −180.000 −0.728745
\(248\) 375.659i 1.51476i
\(249\) −180.000 −0.722892
\(250\) − 11.1803i − 0.0447214i
\(251\) − 335.410i − 1.33630i −0.744029 0.668148i \(-0.767087\pi\)
0.744029 0.668148i \(-0.232913\pi\)
\(252\) 231.000 0.916667
\(253\) 52.0000 0.205534
\(254\) −194.000 −0.763780
\(255\) 268.328i 1.05227i
\(256\) −171.000 −0.667969
\(257\) 134.164i 0.522039i 0.965333 + 0.261020i \(0.0840587\pi\)
−0.965333 + 0.261020i \(0.915941\pi\)
\(258\) − 152.053i − 0.589351i
\(259\) 98.0000 0.378378
\(260\) −90.0000 −0.346154
\(261\) 242.000 0.927203
\(262\) − 120.748i − 0.460869i
\(263\) −34.0000 −0.129278 −0.0646388 0.997909i \(-0.520590\pi\)
−0.0646388 + 0.997909i \(0.520590\pi\)
\(264\) − 62.6099i − 0.237159i
\(265\) 76.0263i 0.286892i
\(266\) − 93.9149i − 0.353063i
\(267\) 120.000 0.449438
\(268\) −42.0000 −0.156716
\(269\) − 254.912i − 0.947627i −0.880625 0.473814i \(-0.842877\pi\)
0.880625 0.473814i \(-0.157123\pi\)
\(270\) −20.0000 −0.0740741
\(271\) − 321.994i − 1.18817i −0.804403 0.594084i \(-0.797514\pi\)
0.804403 0.594084i \(-0.202486\pi\)
\(272\) − 134.164i − 0.493250i
\(273\) 420.000 1.53846
\(274\) 166.000 0.605839
\(275\) −10.0000 −0.0363636
\(276\) 348.827i 1.26386i
\(277\) 14.0000 0.0505415 0.0252708 0.999681i \(-0.491955\pi\)
0.0252708 + 0.999681i \(0.491955\pi\)
\(278\) − 93.9149i − 0.337823i
\(279\) − 590.322i − 2.11585i
\(280\) − 109.567i − 0.391312i
\(281\) 2.00000 0.00711744 0.00355872 0.999994i \(-0.498867\pi\)
0.00355872 + 0.999994i \(0.498867\pi\)
\(282\) −120.000 −0.425532
\(283\) 93.9149i 0.331855i 0.986138 + 0.165927i \(0.0530617\pi\)
−0.986138 + 0.165927i \(0.946938\pi\)
\(284\) −186.000 −0.654930
\(285\) − 134.164i − 0.470751i
\(286\) − 26.8328i − 0.0938210i
\(287\) 187.830i 0.654459i
\(288\) 363.000 1.26042
\(289\) −431.000 −1.49135
\(290\) − 49.1935i − 0.169633i
\(291\) 120.000 0.412371
\(292\) − 160.997i − 0.551359i
\(293\) − 335.410i − 1.14474i −0.819994 0.572372i \(-0.806023\pi\)
0.819994 0.572372i \(-0.193977\pi\)
\(294\) 219.135i 0.745356i
\(295\) 90.0000 0.305085
\(296\) 98.0000 0.331081
\(297\) 17.8885i 0.0602308i
\(298\) 142.000 0.476510
\(299\) 348.827i 1.16664i
\(300\) − 67.0820i − 0.223607i
\(301\) −238.000 −0.790698
\(302\) −2.00000 −0.00662252
\(303\) −300.000 −0.990099
\(304\) 67.0820i 0.220665i
\(305\) −210.000 −0.688525
\(306\) − 295.161i − 0.964578i
\(307\) − 201.246i − 0.655525i −0.944760 0.327762i \(-0.893705\pi\)
0.944760 0.327762i \(-0.106295\pi\)
\(308\) −42.0000 −0.136364
\(309\) −720.000 −2.33010
\(310\) −120.000 −0.387097
\(311\) 509.823i 1.63930i 0.572862 + 0.819652i \(0.305833\pi\)
−0.572862 + 0.819652i \(0.694167\pi\)
\(312\) 420.000 1.34615
\(313\) − 321.994i − 1.02873i −0.857570 0.514367i \(-0.828027\pi\)
0.857570 0.514367i \(-0.171973\pi\)
\(314\) − 67.0820i − 0.213637i
\(315\) 172.177i 0.546594i
\(316\) −114.000 −0.360759
\(317\) 374.000 1.17981 0.589905 0.807472i \(-0.299165\pi\)
0.589905 + 0.807472i \(0.299165\pi\)
\(318\) − 152.053i − 0.478153i
\(319\) −44.0000 −0.137931
\(320\) − 29.0689i − 0.0908403i
\(321\) 474.046i 1.47678i
\(322\) −182.000 −0.565217
\(323\) 360.000 1.11455
\(324\) 177.000 0.546296
\(325\) − 67.0820i − 0.206406i
\(326\) 34.0000 0.104294
\(327\) 635.043i 1.94203i
\(328\) 187.830i 0.572652i
\(329\) 187.830i 0.570911i
\(330\) 20.0000 0.0606061
\(331\) 482.000 1.45619 0.728097 0.685474i \(-0.240405\pi\)
0.728097 + 0.685474i \(0.240405\pi\)
\(332\) 120.748i 0.363698i
\(333\) −154.000 −0.462462
\(334\) − 107.331i − 0.321351i
\(335\) − 31.3050i − 0.0934476i
\(336\) − 156.525i − 0.465847i
\(337\) 494.000 1.46588 0.732938 0.680296i \(-0.238149\pi\)
0.732938 + 0.680296i \(0.238149\pi\)
\(338\) 11.0000 0.0325444
\(339\) 152.053i 0.448533i
\(340\) 180.000 0.529412
\(341\) 107.331i 0.314754i
\(342\) 147.580i 0.431522i
\(343\) 343.000 1.00000
\(344\) −238.000 −0.691860
\(345\) −260.000 −0.753623
\(346\) 147.580i 0.426533i
\(347\) −346.000 −0.997118 −0.498559 0.866856i \(-0.666137\pi\)
−0.498559 + 0.866856i \(0.666137\pi\)
\(348\) − 295.161i − 0.848164i
\(349\) 335.410i 0.961061i 0.876978 + 0.480530i \(0.159556\pi\)
−0.876978 + 0.480530i \(0.840444\pi\)
\(350\) 35.0000 0.100000
\(351\) −120.000 −0.341880
\(352\) −66.0000 −0.187500
\(353\) − 26.8328i − 0.0760136i −0.999277 0.0380068i \(-0.987899\pi\)
0.999277 0.0380068i \(-0.0121009\pi\)
\(354\) −180.000 −0.508475
\(355\) − 138.636i − 0.390525i
\(356\) − 80.4984i − 0.226119i
\(357\) −840.000 −2.35294
\(358\) −218.000 −0.608939
\(359\) 338.000 0.941504 0.470752 0.882266i \(-0.343983\pi\)
0.470752 + 0.882266i \(0.343983\pi\)
\(360\) 172.177i 0.478270i
\(361\) 181.000 0.501385
\(362\) − 254.912i − 0.704176i
\(363\) 523.240i 1.44143i
\(364\) − 281.745i − 0.774024i
\(365\) 120.000 0.328767
\(366\) 420.000 1.14754
\(367\) − 295.161i − 0.804253i −0.915584 0.402127i \(-0.868271\pi\)
0.915584 0.402127i \(-0.131729\pi\)
\(368\) 130.000 0.353261
\(369\) − 295.161i − 0.799894i
\(370\) 31.3050i 0.0846080i
\(371\) −238.000 −0.641509
\(372\) −720.000 −1.93548
\(373\) 86.0000 0.230563 0.115282 0.993333i \(-0.463223\pi\)
0.115282 + 0.993333i \(0.463223\pi\)
\(374\) 53.6656i 0.143491i
\(375\) 50.0000 0.133333
\(376\) 187.830i 0.499547i
\(377\) − 295.161i − 0.782920i
\(378\) − 62.6099i − 0.165635i
\(379\) −262.000 −0.691293 −0.345646 0.938365i \(-0.612340\pi\)
−0.345646 + 0.938365i \(0.612340\pi\)
\(380\) −90.0000 −0.236842
\(381\) − 867.594i − 2.27715i
\(382\) 58.0000 0.151832
\(383\) 563.489i 1.47125i 0.677388 + 0.735625i \(0.263112\pi\)
−0.677388 + 0.735625i \(0.736888\pi\)
\(384\) − 532.184i − 1.38590i
\(385\) − 31.3050i − 0.0813116i
\(386\) −206.000 −0.533679
\(387\) 374.000 0.966408
\(388\) − 80.4984i − 0.207470i
\(389\) 698.000 1.79434 0.897172 0.441681i \(-0.145618\pi\)
0.897172 + 0.441681i \(0.145618\pi\)
\(390\) 134.164i 0.344010i
\(391\) − 697.653i − 1.78428i
\(392\) 343.000 0.875000
\(393\) 540.000 1.37405
\(394\) 226.000 0.573604
\(395\) − 84.9706i − 0.215115i
\(396\) 66.0000 0.166667
\(397\) − 308.577i − 0.777273i −0.921391 0.388636i \(-0.872946\pi\)
0.921391 0.388636i \(-0.127054\pi\)
\(398\) 134.164i 0.337096i
\(399\) 420.000 1.05263
\(400\) −25.0000 −0.0625000
\(401\) −538.000 −1.34165 −0.670823 0.741618i \(-0.734059\pi\)
−0.670823 + 0.741618i \(0.734059\pi\)
\(402\) 62.6099i 0.155746i
\(403\) −720.000 −1.78660
\(404\) 201.246i 0.498134i
\(405\) 131.928i 0.325748i
\(406\) 154.000 0.379310
\(407\) 28.0000 0.0687961
\(408\) −840.000 −2.05882
\(409\) − 295.161i − 0.721665i −0.932631 0.360832i \(-0.882493\pi\)
0.932631 0.360832i \(-0.117507\pi\)
\(410\) −60.0000 −0.146341
\(411\) 742.375i 1.80626i
\(412\) 482.991i 1.17231i
\(413\) 281.745i 0.682190i
\(414\) 286.000 0.690821
\(415\) −90.0000 −0.216867
\(416\) − 442.741i − 1.06428i
\(417\) 420.000 1.00719
\(418\) − 26.8328i − 0.0641933i
\(419\) 818.401i 1.95322i 0.215009 + 0.976612i \(0.431022\pi\)
−0.215009 + 0.976612i \(0.568978\pi\)
\(420\) 210.000 0.500000
\(421\) −118.000 −0.280285 −0.140143 0.990131i \(-0.544756\pi\)
−0.140143 + 0.990131i \(0.544756\pi\)
\(422\) 118.000 0.279621
\(423\) − 295.161i − 0.697780i
\(424\) −238.000 −0.561321
\(425\) 134.164i 0.315680i
\(426\) 277.272i 0.650874i
\(427\) − 657.404i − 1.53959i
\(428\) 318.000 0.742991
\(429\) 120.000 0.279720
\(430\) − 76.0263i − 0.176805i
\(431\) −718.000 −1.66589 −0.832947 0.553353i \(-0.813348\pi\)
−0.832947 + 0.553353i \(0.813348\pi\)
\(432\) 44.7214i 0.103522i
\(433\) − 509.823i − 1.17742i −0.808344 0.588711i \(-0.799636\pi\)
0.808344 0.588711i \(-0.200364\pi\)
\(434\) − 375.659i − 0.865575i
\(435\) 220.000 0.505747
\(436\) 426.000 0.977064
\(437\) 348.827i 0.798230i
\(438\) −240.000 −0.547945
\(439\) 26.8328i 0.0611226i 0.999533 + 0.0305613i \(0.00972948\pi\)
−0.999533 + 0.0305613i \(0.990271\pi\)
\(440\) − 31.3050i − 0.0711476i
\(441\) −539.000 −1.22222
\(442\) −360.000 −0.814480
\(443\) −634.000 −1.43115 −0.715576 0.698535i \(-0.753836\pi\)
−0.715576 + 0.698535i \(0.753836\pi\)
\(444\) 187.830i 0.423040i
\(445\) 60.0000 0.134831
\(446\) 80.4984i 0.180490i
\(447\) 635.043i 1.42068i
\(448\) 91.0000 0.203125
\(449\) 338.000 0.752784 0.376392 0.926461i \(-0.377165\pi\)
0.376392 + 0.926461i \(0.377165\pi\)
\(450\) −55.0000 −0.122222
\(451\) 53.6656i 0.118993i
\(452\) 102.000 0.225664
\(453\) − 8.94427i − 0.0197445i
\(454\) 254.912i 0.561480i
\(455\) 210.000 0.461538
\(456\) 420.000 0.921053
\(457\) −466.000 −1.01969 −0.509847 0.860265i \(-0.670298\pi\)
−0.509847 + 0.860265i \(0.670298\pi\)
\(458\) 13.4164i 0.0292935i
\(459\) 240.000 0.522876
\(460\) 174.413i 0.379159i
\(461\) 442.741i 0.960394i 0.877161 + 0.480197i \(0.159435\pi\)
−0.877161 + 0.480197i \(0.840565\pi\)
\(462\) 62.6099i 0.135519i
\(463\) 206.000 0.444924 0.222462 0.974941i \(-0.428591\pi\)
0.222462 + 0.974941i \(0.428591\pi\)
\(464\) −110.000 −0.237069
\(465\) − 536.656i − 1.15410i
\(466\) 214.000 0.459227
\(467\) − 362.243i − 0.775681i −0.921727 0.387840i \(-0.873221\pi\)
0.921727 0.387840i \(-0.126779\pi\)
\(468\) 442.741i 0.946029i
\(469\) 98.0000 0.208955
\(470\) −60.0000 −0.127660
\(471\) 300.000 0.636943
\(472\) 281.745i 0.596916i
\(473\) −68.0000 −0.143763
\(474\) 169.941i 0.358526i
\(475\) − 67.0820i − 0.141225i
\(476\) 563.489i 1.18380i
\(477\) 374.000 0.784067
\(478\) −98.0000 −0.205021
\(479\) − 214.663i − 0.448147i −0.974572 0.224074i \(-0.928064\pi\)
0.974572 0.224074i \(-0.0719356\pi\)
\(480\) 330.000 0.687500
\(481\) 187.830i 0.390498i
\(482\) − 160.997i − 0.334018i
\(483\) − 813.929i − 1.68515i
\(484\) 351.000 0.725207
\(485\) 60.0000 0.123711
\(486\) − 344.354i − 0.708548i
\(487\) −166.000 −0.340862 −0.170431 0.985370i \(-0.554516\pi\)
−0.170431 + 0.985370i \(0.554516\pi\)
\(488\) − 657.404i − 1.34714i
\(489\) 152.053i 0.310946i
\(490\) 109.567i 0.223607i
\(491\) −838.000 −1.70672 −0.853360 0.521321i \(-0.825439\pi\)
−0.853360 + 0.521321i \(0.825439\pi\)
\(492\) −360.000 −0.731707
\(493\) 590.322i 1.19741i
\(494\) 180.000 0.364372
\(495\) 49.1935i 0.0993808i
\(496\) 268.328i 0.540984i
\(497\) 434.000 0.873239
\(498\) 180.000 0.361446
\(499\) −262.000 −0.525050 −0.262525 0.964925i \(-0.584555\pi\)
−0.262525 + 0.964925i \(0.584555\pi\)
\(500\) − 33.5410i − 0.0670820i
\(501\) 480.000 0.958084
\(502\) 335.410i 0.668148i
\(503\) − 429.325i − 0.853529i −0.904363 0.426764i \(-0.859653\pi\)
0.904363 0.426764i \(-0.140347\pi\)
\(504\) −539.000 −1.06944
\(505\) −150.000 −0.297030
\(506\) −52.0000 −0.102767
\(507\) 49.1935i 0.0970286i
\(508\) −582.000 −1.14567
\(509\) − 898.899i − 1.76601i −0.469363 0.883005i \(-0.655516\pi\)
0.469363 0.883005i \(-0.344484\pi\)
\(510\) − 268.328i − 0.526134i
\(511\) 375.659i 0.735146i
\(512\) −305.000 −0.595703
\(513\) −120.000 −0.233918
\(514\) − 134.164i − 0.261020i
\(515\) −360.000 −0.699029
\(516\) − 456.158i − 0.884027i
\(517\) 53.6656i 0.103802i
\(518\) −98.0000 −0.189189
\(519\) −660.000 −1.27168
\(520\) 210.000 0.403846
\(521\) − 724.486i − 1.39057i −0.718735 0.695284i \(-0.755279\pi\)
0.718735 0.695284i \(-0.244721\pi\)
\(522\) −242.000 −0.463602
\(523\) 523.240i 1.00046i 0.865893 + 0.500229i \(0.166751\pi\)
−0.865893 + 0.500229i \(0.833249\pi\)
\(524\) − 362.243i − 0.691303i
\(525\) 156.525i 0.298142i
\(526\) 34.0000 0.0646388
\(527\) 1440.00 2.73245
\(528\) − 44.7214i − 0.0846995i
\(529\) 147.000 0.277883
\(530\) − 76.0263i − 0.143446i
\(531\) − 442.741i − 0.833788i
\(532\) − 281.745i − 0.529595i
\(533\) −360.000 −0.675422
\(534\) −120.000 −0.224719
\(535\) 237.023i 0.443034i
\(536\) 98.0000 0.182836
\(537\) − 974.926i − 1.81550i
\(538\) 254.912i 0.473814i
\(539\) 98.0000 0.181818
\(540\) −60.0000 −0.111111
\(541\) 842.000 1.55638 0.778189 0.628031i \(-0.216139\pi\)
0.778189 + 0.628031i \(0.216139\pi\)
\(542\) 321.994i 0.594084i
\(543\) 1140.00 2.09945
\(544\) 885.483i 1.62773i
\(545\) 317.522i 0.582609i
\(546\) −420.000 −0.769231
\(547\) 134.000 0.244973 0.122486 0.992470i \(-0.460913\pi\)
0.122486 + 0.992470i \(0.460913\pi\)
\(548\) 498.000 0.908759
\(549\) 1033.06i 1.88172i
\(550\) 10.0000 0.0181818
\(551\) − 295.161i − 0.535682i
\(552\) − 813.929i − 1.47451i
\(553\) 266.000 0.481013
\(554\) −14.0000 −0.0252708
\(555\) −140.000 −0.252252
\(556\) − 281.745i − 0.506735i
\(557\) −706.000 −1.26750 −0.633752 0.773536i \(-0.718486\pi\)
−0.633752 + 0.773536i \(0.718486\pi\)
\(558\) 590.322i 1.05792i
\(559\) − 456.158i − 0.816025i
\(560\) − 78.2624i − 0.139754i
\(561\) −240.000 −0.427807
\(562\) −2.00000 −0.00355872
\(563\) 13.4164i 0.0238302i 0.999929 + 0.0119151i \(0.00379279\pi\)
−0.999929 + 0.0119151i \(0.996207\pi\)
\(564\) −360.000 −0.638298
\(565\) 76.0263i 0.134560i
\(566\) − 93.9149i − 0.165927i
\(567\) −413.000 −0.728395
\(568\) 434.000 0.764085
\(569\) −82.0000 −0.144112 −0.0720562 0.997401i \(-0.522956\pi\)
−0.0720562 + 0.997401i \(0.522956\pi\)
\(570\) 134.164i 0.235376i
\(571\) −118.000 −0.206655 −0.103327 0.994647i \(-0.532949\pi\)
−0.103327 + 0.994647i \(0.532949\pi\)
\(572\) − 80.4984i − 0.140732i
\(573\) 259.384i 0.452677i
\(574\) − 187.830i − 0.327229i
\(575\) −130.000 −0.226087
\(576\) −143.000 −0.248264
\(577\) 885.483i 1.53463i 0.641269 + 0.767316i \(0.278408\pi\)
−0.641269 + 0.767316i \(0.721592\pi\)
\(578\) 431.000 0.745675
\(579\) − 921.260i − 1.59112i
\(580\) − 147.580i − 0.254449i
\(581\) − 281.745i − 0.484930i
\(582\) −120.000 −0.206186
\(583\) −68.0000 −0.116638
\(584\) 375.659i 0.643252i
\(585\) −330.000 −0.564103
\(586\) 335.410i 0.572372i
\(587\) 791.568i 1.34850i 0.738504 + 0.674249i \(0.235532\pi\)
−0.738504 + 0.674249i \(0.764468\pi\)
\(588\) 657.404i 1.11803i
\(589\) −720.000 −1.22241
\(590\) −90.0000 −0.152542
\(591\) 1010.70i 1.71016i
\(592\) 70.0000 0.118243
\(593\) 134.164i 0.226246i 0.993581 + 0.113123i \(0.0360855\pi\)
−0.993581 + 0.113123i \(0.963915\pi\)
\(594\) − 17.8885i − 0.0301154i
\(595\) −420.000 −0.705882
\(596\) 426.000 0.714765
\(597\) −600.000 −1.00503
\(598\) − 348.827i − 0.583322i
\(599\) 398.000 0.664441 0.332220 0.943202i \(-0.392202\pi\)
0.332220 + 0.943202i \(0.392202\pi\)
\(600\) 156.525i 0.260875i
\(601\) − 134.164i − 0.223235i −0.993751 0.111617i \(-0.964397\pi\)
0.993751 0.111617i \(-0.0356031\pi\)
\(602\) 238.000 0.395349
\(603\) −154.000 −0.255390
\(604\) −6.00000 −0.00993377
\(605\) 261.620i 0.432430i
\(606\) 300.000 0.495050
\(607\) 939.149i 1.54720i 0.633676 + 0.773598i \(0.281545\pi\)
−0.633676 + 0.773598i \(0.718455\pi\)
\(608\) − 442.741i − 0.728193i
\(609\) 688.709i 1.13088i
\(610\) 210.000 0.344262
\(611\) −360.000 −0.589198
\(612\) − 885.483i − 1.44687i
\(613\) 206.000 0.336052 0.168026 0.985783i \(-0.446261\pi\)
0.168026 + 0.985783i \(0.446261\pi\)
\(614\) 201.246i 0.327762i
\(615\) − 268.328i − 0.436306i
\(616\) 98.0000 0.159091
\(617\) 494.000 0.800648 0.400324 0.916374i \(-0.368898\pi\)
0.400324 + 0.916374i \(0.368898\pi\)
\(618\) 720.000 1.16505
\(619\) − 120.748i − 0.195069i −0.995232 0.0975345i \(-0.968904\pi\)
0.995232 0.0975345i \(-0.0310956\pi\)
\(620\) −360.000 −0.580645
\(621\) 232.551i 0.374478i
\(622\) − 509.823i − 0.819652i
\(623\) 187.830i 0.301492i
\(624\) 300.000 0.480769
\(625\) 25.0000 0.0400000
\(626\) 321.994i 0.514367i
\(627\) 120.000 0.191388
\(628\) − 201.246i − 0.320456i
\(629\) − 375.659i − 0.597233i
\(630\) − 172.177i − 0.273297i
\(631\) 542.000 0.858954 0.429477 0.903078i \(-0.358698\pi\)
0.429477 + 0.903078i \(0.358698\pi\)
\(632\) 266.000 0.420886
\(633\) 527.712i 0.833668i
\(634\) −374.000 −0.589905
\(635\) − 433.797i − 0.683145i
\(636\) − 456.158i − 0.717229i
\(637\) 657.404i 1.03203i
\(638\) 44.0000 0.0689655
\(639\) −682.000 −1.06729
\(640\) − 266.092i − 0.415769i
\(641\) −298.000 −0.464899 −0.232449 0.972609i \(-0.574674\pi\)
−0.232449 + 0.972609i \(0.574674\pi\)
\(642\) − 474.046i − 0.738390i
\(643\) − 1006.23i − 1.56490i −0.622714 0.782450i \(-0.713970\pi\)
0.622714 0.782450i \(-0.286030\pi\)
\(644\) −546.000 −0.847826
\(645\) 340.000 0.527132
\(646\) −360.000 −0.557276
\(647\) − 643.988i − 0.995344i −0.867365 0.497672i \(-0.834188\pi\)
0.867365 0.497672i \(-0.165812\pi\)
\(648\) −413.000 −0.637346
\(649\) 80.4984i 0.124035i
\(650\) 67.0820i 0.103203i
\(651\) 1680.00 2.58065
\(652\) 102.000 0.156442
\(653\) −154.000 −0.235835 −0.117917 0.993023i \(-0.537622\pi\)
−0.117917 + 0.993023i \(0.537622\pi\)
\(654\) − 635.043i − 0.971014i
\(655\) 270.000 0.412214
\(656\) 134.164i 0.204518i
\(657\) − 590.322i − 0.898511i
\(658\) − 187.830i − 0.285455i
\(659\) 338.000 0.512898 0.256449 0.966558i \(-0.417447\pi\)
0.256449 + 0.966558i \(0.417447\pi\)
\(660\) 60.0000 0.0909091
\(661\) 576.906i 0.872777i 0.899758 + 0.436388i \(0.143743\pi\)
−0.899758 + 0.436388i \(0.856257\pi\)
\(662\) −482.000 −0.728097
\(663\) − 1609.97i − 2.42831i
\(664\) − 281.745i − 0.424314i
\(665\) 210.000 0.315789
\(666\) 154.000 0.231231
\(667\) −572.000 −0.857571
\(668\) − 321.994i − 0.482027i
\(669\) −360.000 −0.538117
\(670\) 31.3050i 0.0467238i
\(671\) − 187.830i − 0.279925i
\(672\) 1033.06i 1.53730i
\(673\) −814.000 −1.20951 −0.604755 0.796412i \(-0.706729\pi\)
−0.604755 + 0.796412i \(0.706729\pi\)
\(674\) −494.000 −0.732938
\(675\) − 44.7214i − 0.0662539i
\(676\) 33.0000 0.0488166
\(677\) 684.237i 1.01069i 0.862918 + 0.505345i \(0.168635\pi\)
−0.862918 + 0.505345i \(0.831365\pi\)
\(678\) − 152.053i − 0.224266i
\(679\) 187.830i 0.276627i
\(680\) −420.000 −0.617647
\(681\) −1140.00 −1.67401
\(682\) − 107.331i − 0.157377i
\(683\) 926.000 1.35578 0.677892 0.735162i \(-0.262894\pi\)
0.677892 + 0.735162i \(0.262894\pi\)
\(684\) 442.741i 0.647283i
\(685\) 371.187i 0.541879i
\(686\) −343.000 −0.500000
\(687\) −60.0000 −0.0873362
\(688\) −170.000 −0.247093
\(689\) − 456.158i − 0.662058i
\(690\) 260.000 0.376812
\(691\) − 576.906i − 0.834885i −0.908703 0.417443i \(-0.862927\pi\)
0.908703 0.417443i \(-0.137073\pi\)
\(692\) 442.741i 0.639800i
\(693\) −154.000 −0.222222
\(694\) 346.000 0.498559
\(695\) 210.000 0.302158
\(696\) 688.709i 0.989524i
\(697\) 720.000 1.03300
\(698\) − 335.410i − 0.480530i
\(699\) 957.037i 1.36915i
\(700\) 105.000 0.150000
\(701\) 362.000 0.516405 0.258203 0.966091i \(-0.416870\pi\)
0.258203 + 0.966091i \(0.416870\pi\)
\(702\) 120.000 0.170940
\(703\) 187.830i 0.267183i
\(704\) 26.0000 0.0369318
\(705\) − 268.328i − 0.380607i
\(706\) 26.8328i 0.0380068i
\(707\) − 469.574i − 0.664179i
\(708\) −540.000 −0.762712
\(709\) 1058.00 1.49224 0.746121 0.665810i \(-0.231914\pi\)
0.746121 + 0.665810i \(0.231914\pi\)
\(710\) 138.636i 0.195262i
\(711\) −418.000 −0.587904
\(712\) 187.830i 0.263806i
\(713\) 1395.31i 1.95695i
\(714\) 840.000 1.17647
\(715\) 60.0000 0.0839161
\(716\) −654.000 −0.913408
\(717\) − 438.269i − 0.611254i
\(718\) −338.000 −0.470752
\(719\) 482.991i 0.671753i 0.941906 + 0.335877i \(0.109033\pi\)
−0.941906 + 0.335877i \(0.890967\pi\)
\(720\) 122.984i 0.170811i
\(721\) − 1126.98i − 1.56308i
\(722\) −181.000 −0.250693
\(723\) 720.000 0.995851
\(724\) − 764.735i − 1.05626i
\(725\) 110.000 0.151724
\(726\) − 523.240i − 0.720716i
\(727\) − 1126.98i − 1.55018i −0.631853 0.775088i \(-0.717705\pi\)
0.631853 0.775088i \(-0.282295\pi\)
\(728\) 657.404i 0.903027i
\(729\) 1009.00 1.38409
\(730\) −120.000 −0.164384
\(731\) 912.316i 1.24804i
\(732\) 1260.00 1.72131
\(733\) 1301.39i 1.77543i 0.460392 + 0.887716i \(0.347709\pi\)
−0.460392 + 0.887716i \(0.652291\pi\)
\(734\) 295.161i 0.402127i
\(735\) −490.000 −0.666667
\(736\) −858.000 −1.16576
\(737\) 28.0000 0.0379919
\(738\) 295.161i 0.399947i
\(739\) −982.000 −1.32882 −0.664411 0.747367i \(-0.731318\pi\)
−0.664411 + 0.747367i \(0.731318\pi\)
\(740\) 93.9149i 0.126912i
\(741\) 804.984i 1.08635i
\(742\) 238.000 0.320755
\(743\) −694.000 −0.934051 −0.467026 0.884244i \(-0.654674\pi\)
−0.467026 + 0.884244i \(0.654674\pi\)
\(744\) 1680.00 2.25806
\(745\) 317.522i 0.426204i
\(746\) −86.0000 −0.115282
\(747\) 442.741i 0.592693i
\(748\) 160.997i 0.215236i
\(749\) −742.000 −0.990654
\(750\) −50.0000 −0.0666667
\(751\) 242.000 0.322237 0.161119 0.986935i \(-0.448490\pi\)
0.161119 + 0.986935i \(0.448490\pi\)
\(752\) 134.164i 0.178410i
\(753\) −1500.00 −1.99203
\(754\) 295.161i 0.391460i
\(755\) − 4.47214i − 0.00592336i
\(756\) − 187.830i − 0.248452i
\(757\) −106.000 −0.140026 −0.0700132 0.997546i \(-0.522304\pi\)
−0.0700132 + 0.997546i \(0.522304\pi\)
\(758\) 262.000 0.345646
\(759\) − 232.551i − 0.306391i
\(760\) 210.000 0.276316
\(761\) − 1100.15i − 1.44566i −0.691027 0.722829i \(-0.742842\pi\)
0.691027 0.722829i \(-0.257158\pi\)
\(762\) 867.594i 1.13858i
\(763\) −994.000 −1.30275
\(764\) 174.000 0.227749
\(765\) 660.000 0.862745
\(766\) − 563.489i − 0.735625i
\(767\) −540.000 −0.704042
\(768\) 764.735i 0.995749i
\(769\) − 1126.98i − 1.46551i −0.680492 0.732756i \(-0.738234\pi\)
0.680492 0.732756i \(-0.261766\pi\)
\(770\) 31.3050i 0.0406558i
\(771\) 600.000 0.778210
\(772\) −618.000 −0.800518
\(773\) − 818.401i − 1.05873i −0.848393 0.529367i \(-0.822430\pi\)
0.848393 0.529367i \(-0.177570\pi\)
\(774\) −374.000 −0.483204
\(775\) − 268.328i − 0.346230i
\(776\) 187.830i 0.242049i
\(777\) − 438.269i − 0.564053i
\(778\) −698.000 −0.897172
\(779\) −360.000 −0.462131
\(780\) 402.492i 0.516016i
\(781\) 124.000 0.158771
\(782\) 697.653i 0.892140i
\(783\) − 196.774i − 0.251308i
\(784\) 245.000 0.312500
\(785\) 150.000 0.191083
\(786\) −540.000 −0.687023
\(787\) − 684.237i − 0.869424i −0.900569 0.434712i \(-0.856850\pi\)
0.900569 0.434712i \(-0.143150\pi\)
\(788\) 678.000 0.860406
\(789\) 152.053i 0.192716i
\(790\) 84.9706i 0.107558i
\(791\) −238.000 −0.300885
\(792\) −154.000 −0.194444
\(793\) 1260.00 1.58890
\(794\) 308.577i 0.388636i
\(795\) 340.000 0.427673
\(796\) 402.492i 0.505644i
\(797\) − 308.577i − 0.387174i −0.981083 0.193587i \(-0.937988\pi\)
0.981083 0.193587i \(-0.0620121\pi\)
\(798\) −420.000 −0.526316
\(799\) 720.000 0.901126
\(800\) 165.000 0.206250
\(801\) − 295.161i − 0.368491i
\(802\) 538.000 0.670823
\(803\) 107.331i 0.133663i
\(804\) 187.830i 0.233619i
\(805\) − 406.964i − 0.505546i
\(806\) 720.000 0.893300
\(807\) −1140.00 −1.41264
\(808\) − 469.574i − 0.581156i
\(809\) 1358.00 1.67862 0.839308 0.543657i \(-0.182961\pi\)
0.839308 + 0.543657i \(0.182961\pi\)
\(810\) − 131.928i − 0.162874i
\(811\) 308.577i 0.380490i 0.981737 + 0.190245i \(0.0609282\pi\)
−0.981737 + 0.190245i \(0.939072\pi\)
\(812\) 462.000 0.568966
\(813\) −1440.00 −1.77122
\(814\) −28.0000 −0.0343980
\(815\) 76.0263i 0.0932838i
\(816\) −600.000 −0.735294
\(817\) − 456.158i − 0.558333i
\(818\) 295.161i 0.360832i
\(819\) − 1033.06i − 1.26137i
\(820\) −180.000 −0.219512
\(821\) 482.000 0.587089 0.293544 0.955945i \(-0.405165\pi\)
0.293544 + 0.955945i \(0.405165\pi\)
\(822\) − 742.375i − 0.903132i
\(823\) 926.000 1.12515 0.562576 0.826746i \(-0.309810\pi\)
0.562576 + 0.826746i \(0.309810\pi\)
\(824\) − 1126.98i − 1.36769i
\(825\) 44.7214i 0.0542077i
\(826\) − 281.745i − 0.341095i
\(827\) −226.000 −0.273277 −0.136638 0.990621i \(-0.543630\pi\)
−0.136638 + 0.990621i \(0.543630\pi\)
\(828\) 858.000 1.03623
\(829\) 1462.39i 1.76404i 0.471213 + 0.882020i \(0.343816\pi\)
−0.471213 + 0.882020i \(0.656184\pi\)
\(830\) 90.0000 0.108434
\(831\) − 62.6099i − 0.0753428i
\(832\) 174.413i 0.209631i
\(833\) − 1314.81i − 1.57840i
\(834\) −420.000 −0.503597
\(835\) 240.000 0.287425
\(836\) − 80.4984i − 0.0962900i
\(837\) −480.000 −0.573477
\(838\) − 818.401i − 0.976612i
\(839\) 831.817i 0.991439i 0.868483 + 0.495719i \(0.165096\pi\)
−0.868483 + 0.495719i \(0.834904\pi\)
\(840\) −490.000 −0.583333
\(841\) −357.000 −0.424495
\(842\) 118.000 0.140143
\(843\) − 8.94427i − 0.0106100i
\(844\) 354.000 0.419431
\(845\) 24.5967i 0.0291086i
\(846\) 295.161i 0.348890i
\(847\) −819.000 −0.966942
\(848\) −170.000 −0.200472
\(849\) 420.000 0.494700
\(850\) − 134.164i − 0.157840i
\(851\) 364.000 0.427732
\(852\) 831.817i 0.976311i
\(853\) 40.2492i 0.0471855i 0.999722 + 0.0235927i \(0.00751050\pi\)
−0.999722 + 0.0235927i \(0.992489\pi\)
\(854\) 657.404i 0.769794i
\(855\) −330.000 −0.385965
\(856\) −742.000 −0.866822
\(857\) − 268.328i − 0.313102i −0.987670 0.156551i \(-0.949962\pi\)
0.987670 0.156551i \(-0.0500375\pi\)
\(858\) −120.000 −0.139860
\(859\) 308.577i 0.359229i 0.983737 + 0.179614i \(0.0574850\pi\)
−0.983737 + 0.179614i \(0.942515\pi\)
\(860\) − 228.079i − 0.265208i
\(861\) 840.000 0.975610
\(862\) 718.000 0.832947
\(863\) −514.000 −0.595597 −0.297798 0.954629i \(-0.596252\pi\)
−0.297798 + 0.954629i \(0.596252\pi\)
\(864\) − 295.161i − 0.341621i
\(865\) −330.000 −0.381503
\(866\) 509.823i 0.588711i
\(867\) 1927.49i 2.22317i
\(868\) − 1126.98i − 1.29836i
\(869\) 76.0000 0.0874568
\(870\) −220.000 −0.252874
\(871\) 187.830i 0.215648i
\(872\) −994.000 −1.13991
\(873\) − 295.161i − 0.338100i
\(874\) − 348.827i − 0.399115i
\(875\) 78.2624i 0.0894427i
\(876\) −720.000 −0.821918
\(877\) −1306.00 −1.48917 −0.744584 0.667529i \(-0.767352\pi\)
−0.744584 + 0.667529i \(0.767352\pi\)
\(878\) − 26.8328i − 0.0305613i
\(879\) −1500.00 −1.70648
\(880\) − 22.3607i − 0.0254099i
\(881\) 1126.98i 1.27920i 0.768706 + 0.639602i \(0.220901\pi\)
−0.768706 + 0.639602i \(0.779099\pi\)
\(882\) 539.000 0.611111
\(883\) 1526.00 1.72820 0.864100 0.503321i \(-0.167889\pi\)
0.864100 + 0.503321i \(0.167889\pi\)
\(884\) −1080.00 −1.22172
\(885\) − 402.492i − 0.454793i
\(886\) 634.000 0.715576
\(887\) − 1556.30i − 1.75457i −0.479970 0.877285i \(-0.659352\pi\)
0.479970 0.877285i \(-0.340648\pi\)
\(888\) − 438.269i − 0.493547i
\(889\) 1358.00 1.52756
\(890\) −60.0000 −0.0674157
\(891\) −118.000 −0.132435
\(892\) 241.495i 0.270735i
\(893\) −360.000 −0.403135
\(894\) − 635.043i − 0.710339i
\(895\) − 487.463i − 0.544651i
\(896\) 833.000 0.929688
\(897\) 1560.00 1.73913
\(898\) −338.000 −0.376392
\(899\) − 1180.64i − 1.31329i
\(900\) −165.000 −0.183333
\(901\) 912.316i 1.01256i
\(902\) − 53.6656i − 0.0594963i
\(903\) 1064.37i 1.17870i
\(904\) −238.000 −0.263274
\(905\) 570.000 0.629834
\(906\) 8.94427i 0.00987226i
\(907\) 734.000 0.809261 0.404631 0.914480i \(-0.367400\pi\)
0.404631 + 0.914480i \(0.367400\pi\)
\(908\) 764.735i 0.842219i
\(909\) 737.902i 0.811774i
\(910\) −210.000 −0.230769
\(911\) 1202.00 1.31943 0.659715 0.751516i \(-0.270677\pi\)
0.659715 + 0.751516i \(0.270677\pi\)
\(912\) 300.000 0.328947
\(913\) − 80.4984i − 0.0881692i
\(914\) 466.000 0.509847
\(915\) 939.149i 1.02639i
\(916\) 40.2492i 0.0439402i
\(917\) 845.234i 0.921738i
\(918\) −240.000 −0.261438
\(919\) −1282.00 −1.39499 −0.697497 0.716587i \(-0.745703\pi\)
−0.697497 + 0.716587i \(0.745703\pi\)
\(920\) − 406.964i − 0.442353i
\(921\) −900.000 −0.977199
\(922\) − 442.741i − 0.480197i
\(923\) 831.817i 0.901210i
\(924\) 187.830i 0.203279i
\(925\) −70.0000 −0.0756757
\(926\) −206.000 −0.222462
\(927\) 1770.97i 1.91043i
\(928\) 726.000 0.782328
\(929\) 1126.98i 1.21311i 0.795042 + 0.606554i \(0.207449\pi\)
−0.795042 + 0.606554i \(0.792551\pi\)
\(930\) 536.656i 0.577050i
\(931\) 657.404i 0.706127i
\(932\) 642.000 0.688841
\(933\) 2280.00 2.44373
\(934\) 362.243i 0.387840i
\(935\) −120.000 −0.128342
\(936\) − 1033.06i − 1.10370i
\(937\) − 214.663i − 0.229096i −0.993418 0.114548i \(-0.963458\pi\)
0.993418 0.114548i \(-0.0365419\pi\)
\(938\) −98.0000 −0.104478
\(939\) −1440.00 −1.53355
\(940\) −180.000 −0.191489
\(941\) − 845.234i − 0.898229i −0.893474 0.449115i \(-0.851739\pi\)
0.893474 0.449115i \(-0.148261\pi\)
\(942\) −300.000 −0.318471
\(943\) 697.653i 0.739823i
\(944\) 201.246i 0.213184i
\(945\) 140.000 0.148148
\(946\) 68.0000 0.0718816
\(947\) 734.000 0.775079 0.387540 0.921853i \(-0.373325\pi\)
0.387540 + 0.921853i \(0.373325\pi\)
\(948\) 509.823i 0.537789i
\(949\) −720.000 −0.758693
\(950\) 67.0820i 0.0706127i
\(951\) − 1672.58i − 1.75876i
\(952\) − 1314.81i − 1.38110i
\(953\) −934.000 −0.980063 −0.490031 0.871705i \(-0.663015\pi\)
−0.490031 + 0.871705i \(0.663015\pi\)
\(954\) −374.000 −0.392034
\(955\) 129.692i 0.135803i
\(956\) −294.000 −0.307531
\(957\) 196.774i 0.205615i
\(958\) 214.663i 0.224074i
\(959\) −1162.00 −1.21168
\(960\) −130.000 −0.135417
\(961\) −1919.00 −1.99688
\(962\) − 187.830i − 0.195249i
\(963\) 1166.00 1.21080
\(964\) − 482.991i − 0.501028i
\(965\) − 460.630i − 0.477337i
\(966\) 813.929i 0.842576i
\(967\) 314.000 0.324716 0.162358 0.986732i \(-0.448090\pi\)
0.162358 + 0.986732i \(0.448090\pi\)
\(968\) −819.000 −0.846074
\(969\) − 1609.97i − 1.66147i
\(970\) −60.0000 −0.0618557
\(971\) 147.580i 0.151988i 0.997108 + 0.0759941i \(0.0242130\pi\)
−0.997108 + 0.0759941i \(0.975787\pi\)
\(972\) − 1033.06i − 1.06282i
\(973\) 657.404i 0.675646i
\(974\) 166.000 0.170431
\(975\) −300.000 −0.307692
\(976\) − 469.574i − 0.481121i
\(977\) −1486.00 −1.52098 −0.760491 0.649348i \(-0.775042\pi\)
−0.760491 + 0.649348i \(0.775042\pi\)
\(978\) − 152.053i − 0.155473i
\(979\) 53.6656i 0.0548168i
\(980\) 328.702i 0.335410i
\(981\) 1562.00 1.59225
\(982\) 838.000 0.853360
\(983\) 965.981i 0.982687i 0.870966 + 0.491344i \(0.163494\pi\)
−0.870966 + 0.491344i \(0.836506\pi\)
\(984\) 840.000 0.853659
\(985\) 505.351i 0.513047i
\(986\) − 590.322i − 0.598704i
\(987\) 840.000 0.851064
\(988\) 540.000 0.546559
\(989\) −884.000 −0.893832
\(990\) − 49.1935i − 0.0496904i
\(991\) −58.0000 −0.0585267 −0.0292634 0.999572i \(-0.509316\pi\)
−0.0292634 + 0.999572i \(0.509316\pi\)
\(992\) − 1770.97i − 1.78525i
\(993\) − 2155.57i − 2.17076i
\(994\) −434.000 −0.436620
\(995\) −300.000 −0.301508
\(996\) 540.000 0.542169
\(997\) 630.571i 0.632469i 0.948681 + 0.316234i \(0.102419\pi\)
−0.948681 + 0.316234i \(0.897581\pi\)
\(998\) 262.000 0.262525
\(999\) 125.220i 0.125345i
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 35.3.d.a.6.1 2
3.2 odd 2 315.3.h.b.181.2 2
4.3 odd 2 560.3.f.a.321.2 2
5.2 odd 4 175.3.c.d.174.1 4
5.3 odd 4 175.3.c.d.174.4 4
5.4 even 2 175.3.d.f.76.2 2
7.2 even 3 245.3.h.b.31.2 4
7.3 odd 6 245.3.h.b.166.2 4
7.4 even 3 245.3.h.b.166.1 4
7.5 odd 6 245.3.h.b.31.1 4
7.6 odd 2 inner 35.3.d.a.6.2 yes 2
21.20 even 2 315.3.h.b.181.1 2
28.27 even 2 560.3.f.a.321.1 2
35.13 even 4 175.3.c.d.174.3 4
35.27 even 4 175.3.c.d.174.2 4
35.34 odd 2 175.3.d.f.76.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.3.d.a.6.1 2 1.1 even 1 trivial
35.3.d.a.6.2 yes 2 7.6 odd 2 inner
175.3.c.d.174.1 4 5.2 odd 4
175.3.c.d.174.2 4 35.27 even 4
175.3.c.d.174.3 4 35.13 even 4
175.3.c.d.174.4 4 5.3 odd 4
175.3.d.f.76.1 2 35.34 odd 2
175.3.d.f.76.2 2 5.4 even 2
245.3.h.b.31.1 4 7.5 odd 6
245.3.h.b.31.2 4 7.2 even 3
245.3.h.b.166.1 4 7.4 even 3
245.3.h.b.166.2 4 7.3 odd 6
315.3.h.b.181.1 2 21.20 even 2
315.3.h.b.181.2 2 3.2 odd 2
560.3.f.a.321.1 2 28.27 even 2
560.3.f.a.321.2 2 4.3 odd 2