Properties

Label 7605.2.a.cf.1.4
Level $7605$
Weight $2$
Character 7605.1
Self dual yes
Analytic conductor $60.726$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7605.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.7262307372\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.4752.1
Defining polynomial: \(x^{4} - 2 x^{3} - 3 x^{2} + 4 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.49551\) of defining polynomial
Character \(\chi\) \(=\) 7605.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.49551 q^{2} +0.236543 q^{4} +1.00000 q^{5} +4.82684 q^{7} -2.63726 q^{8} +O(q^{10})\) \(q+1.49551 q^{2} +0.236543 q^{4} +1.00000 q^{5} +4.82684 q^{7} -2.63726 q^{8} +1.49551 q^{10} +1.06939 q^{11} +7.21857 q^{14} -4.41713 q^{16} -3.55889 q^{17} +5.73205 q^{19} +0.236543 q^{20} +1.59928 q^{22} +7.08580 q^{23} +1.00000 q^{25} +1.14176 q^{28} -1.47309 q^{29} -1.46410 q^{31} -1.33133 q^{32} -5.32235 q^{34} +4.82684 q^{35} -0.0253983 q^{37} +8.57233 q^{38} -2.63726 q^{40} -0.267949 q^{41} +3.55889 q^{43} +0.252957 q^{44} +10.5969 q^{46} -6.51793 q^{47} +16.2984 q^{49} +1.49551 q^{50} -0.991015 q^{53} +1.06939 q^{55} -12.7296 q^{56} -2.20301 q^{58} -8.72307 q^{59} +6.33734 q^{61} -2.18958 q^{62} +6.84325 q^{64} +5.17316 q^{67} -0.841831 q^{68} +7.21857 q^{70} +7.76488 q^{71} +10.1088 q^{73} -0.0379833 q^{74} +1.35588 q^{76} +5.16177 q^{77} +8.78347 q^{79} -4.41713 q^{80} -0.400720 q^{82} +0.725474 q^{83} -3.55889 q^{85} +5.32235 q^{86} -2.82026 q^{88} -13.5065 q^{89} +1.67610 q^{92} -9.74761 q^{94} +5.73205 q^{95} -3.43870 q^{97} +24.3743 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} + 2q^{4} + 4q^{5} + 10q^{7} - 6q^{8} + O(q^{10}) \) \( 4q - 2q^{2} + 2q^{4} + 4q^{5} + 10q^{7} - 6q^{8} - 2q^{10} - 2q^{14} + 2q^{16} + 2q^{17} + 16q^{19} + 2q^{20} + 12q^{22} + 10q^{23} + 4q^{25} + 8q^{28} - 8q^{29} + 8q^{31} - 4q^{32} - 4q^{34} + 10q^{35} - 2q^{37} - 8q^{38} - 6q^{40} - 8q^{41} - 2q^{43} - 12q^{44} + 16q^{46} - 8q^{47} + 12q^{49} - 2q^{50} + 12q^{53} - 12q^{56} + 22q^{58} - 12q^{59} + 28q^{61} - 4q^{62} + 4q^{64} + 30q^{67} - 14q^{68} - 2q^{70} - 4q^{71} - 8q^{73} + 10q^{74} + 20q^{76} - 18q^{77} - 8q^{79} + 2q^{80} + 4q^{82} + 12q^{83} + 2q^{85} + 4q^{86} - 18q^{88} + 12q^{89} - 22q^{92} - 32q^{94} + 16q^{95} + 2q^{97} + 24q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.49551 1.05748 0.528742 0.848783i \(-0.322664\pi\)
0.528742 + 0.848783i \(0.322664\pi\)
\(3\) 0 0
\(4\) 0.236543 0.118272
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 4.82684 1.82437 0.912187 0.409775i \(-0.134393\pi\)
0.912187 + 0.409775i \(0.134393\pi\)
\(8\) −2.63726 −0.932413
\(9\) 0 0
\(10\) 1.49551 0.472921
\(11\) 1.06939 0.322433 0.161217 0.986919i \(-0.448458\pi\)
0.161217 + 0.986919i \(0.448458\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 7.21857 1.92924
\(15\) 0 0
\(16\) −4.41713 −1.10428
\(17\) −3.55889 −0.863157 −0.431579 0.902075i \(-0.642043\pi\)
−0.431579 + 0.902075i \(0.642043\pi\)
\(18\) 0 0
\(19\) 5.73205 1.31502 0.657511 0.753445i \(-0.271609\pi\)
0.657511 + 0.753445i \(0.271609\pi\)
\(20\) 0.236543 0.0528927
\(21\) 0 0
\(22\) 1.59928 0.340968
\(23\) 7.08580 1.47749 0.738746 0.673984i \(-0.235418\pi\)
0.738746 + 0.673984i \(0.235418\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 1.14176 0.215772
\(29\) −1.47309 −0.273545 −0.136773 0.990602i \(-0.543673\pi\)
−0.136773 + 0.990602i \(0.543673\pi\)
\(30\) 0 0
\(31\) −1.46410 −0.262960 −0.131480 0.991319i \(-0.541973\pi\)
−0.131480 + 0.991319i \(0.541973\pi\)
\(32\) −1.33133 −0.235348
\(33\) 0 0
\(34\) −5.32235 −0.912775
\(35\) 4.82684 0.815885
\(36\) 0 0
\(37\) −0.0253983 −0.00417545 −0.00208772 0.999998i \(-0.500665\pi\)
−0.00208772 + 0.999998i \(0.500665\pi\)
\(38\) 8.57233 1.39061
\(39\) 0 0
\(40\) −2.63726 −0.416988
\(41\) −0.267949 −0.0418466 −0.0209233 0.999781i \(-0.506661\pi\)
−0.0209233 + 0.999781i \(0.506661\pi\)
\(42\) 0 0
\(43\) 3.55889 0.542726 0.271363 0.962477i \(-0.412526\pi\)
0.271363 + 0.962477i \(0.412526\pi\)
\(44\) 0.252957 0.0381347
\(45\) 0 0
\(46\) 10.5969 1.56242
\(47\) −6.51793 −0.950738 −0.475369 0.879787i \(-0.657685\pi\)
−0.475369 + 0.879787i \(0.657685\pi\)
\(48\) 0 0
\(49\) 16.2984 2.32834
\(50\) 1.49551 0.211497
\(51\) 0 0
\(52\) 0 0
\(53\) −0.991015 −0.136126 −0.0680632 0.997681i \(-0.521682\pi\)
−0.0680632 + 0.997681i \(0.521682\pi\)
\(54\) 0 0
\(55\) 1.06939 0.144196
\(56\) −12.7296 −1.70107
\(57\) 0 0
\(58\) −2.20301 −0.289270
\(59\) −8.72307 −1.13565 −0.567823 0.823151i \(-0.692214\pi\)
−0.567823 + 0.823151i \(0.692214\pi\)
\(60\) 0 0
\(61\) 6.33734 0.811413 0.405707 0.914003i \(-0.367026\pi\)
0.405707 + 0.914003i \(0.367026\pi\)
\(62\) −2.18958 −0.278076
\(63\) 0 0
\(64\) 6.84325 0.855406
\(65\) 0 0
\(66\) 0 0
\(67\) 5.17316 0.632002 0.316001 0.948759i \(-0.397660\pi\)
0.316001 + 0.948759i \(0.397660\pi\)
\(68\) −0.841831 −0.102087
\(69\) 0 0
\(70\) 7.21857 0.862785
\(71\) 7.76488 0.921521 0.460761 0.887524i \(-0.347577\pi\)
0.460761 + 0.887524i \(0.347577\pi\)
\(72\) 0 0
\(73\) 10.1088 1.18314 0.591572 0.806252i \(-0.298507\pi\)
0.591572 + 0.806252i \(0.298507\pi\)
\(74\) −0.0379833 −0.00441547
\(75\) 0 0
\(76\) 1.35588 0.155530
\(77\) 5.16177 0.588238
\(78\) 0 0
\(79\) 8.78347 0.988218 0.494109 0.869400i \(-0.335494\pi\)
0.494109 + 0.869400i \(0.335494\pi\)
\(80\) −4.41713 −0.493851
\(81\) 0 0
\(82\) −0.400720 −0.0442521
\(83\) 0.725474 0.0796311 0.0398155 0.999207i \(-0.487323\pi\)
0.0398155 + 0.999207i \(0.487323\pi\)
\(84\) 0 0
\(85\) −3.55889 −0.386016
\(86\) 5.32235 0.573923
\(87\) 0 0
\(88\) −2.82026 −0.300641
\(89\) −13.5065 −1.43169 −0.715845 0.698259i \(-0.753958\pi\)
−0.715845 + 0.698259i \(0.753958\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.67610 0.174745
\(93\) 0 0
\(94\) −9.74761 −1.00539
\(95\) 5.73205 0.588096
\(96\) 0 0
\(97\) −3.43870 −0.349147 −0.174574 0.984644i \(-0.555855\pi\)
−0.174574 + 0.984644i \(0.555855\pi\)
\(98\) 24.3743 2.46218
\(99\) 0 0
\(100\) 0.236543 0.0236543
\(101\) −2.85527 −0.284110 −0.142055 0.989859i \(-0.545371\pi\)
−0.142055 + 0.989859i \(0.545371\pi\)
\(102\) 0 0
\(103\) −5.54488 −0.546354 −0.273177 0.961964i \(-0.588074\pi\)
−0.273177 + 0.961964i \(0.588074\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −1.48207 −0.143951
\(107\) 4.44111 0.429338 0.214669 0.976687i \(-0.431133\pi\)
0.214669 + 0.976687i \(0.431133\pi\)
\(108\) 0 0
\(109\) 13.7804 1.31993 0.659963 0.751298i \(-0.270572\pi\)
0.659963 + 0.751298i \(0.270572\pi\)
\(110\) 1.59928 0.152485
\(111\) 0 0
\(112\) −21.3208 −2.01463
\(113\) 8.04399 0.756715 0.378358 0.925660i \(-0.376489\pi\)
0.378358 + 0.925660i \(0.376489\pi\)
\(114\) 0 0
\(115\) 7.08580 0.660755
\(116\) −0.348448 −0.0323526
\(117\) 0 0
\(118\) −13.0454 −1.20093
\(119\) −17.1782 −1.57472
\(120\) 0 0
\(121\) −9.85641 −0.896037
\(122\) 9.47754 0.858056
\(123\) 0 0
\(124\) −0.346323 −0.0311007
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 0.706653 0.0627053 0.0313526 0.999508i \(-0.490019\pi\)
0.0313526 + 0.999508i \(0.490019\pi\)
\(128\) 12.8968 1.13993
\(129\) 0 0
\(130\) 0 0
\(131\) −6.26554 −0.547423 −0.273711 0.961812i \(-0.588251\pi\)
−0.273711 + 0.961812i \(0.588251\pi\)
\(132\) 0 0
\(133\) 27.6677 2.39909
\(134\) 7.73650 0.668332
\(135\) 0 0
\(136\) 9.38573 0.804820
\(137\) −16.3058 −1.39310 −0.696549 0.717509i \(-0.745282\pi\)
−0.696549 + 0.717509i \(0.745282\pi\)
\(138\) 0 0
\(139\) −6.82528 −0.578913 −0.289456 0.957191i \(-0.593475\pi\)
−0.289456 + 0.957191i \(0.593475\pi\)
\(140\) 1.14176 0.0964960
\(141\) 0 0
\(142\) 11.6124 0.974494
\(143\) 0 0
\(144\) 0 0
\(145\) −1.47309 −0.122333
\(146\) 15.1178 1.25116
\(147\) 0 0
\(148\) −0.00600778 −0.000493837 0
\(149\) 8.43955 0.691395 0.345698 0.938346i \(-0.387642\pi\)
0.345698 + 0.938346i \(0.387642\pi\)
\(150\) 0 0
\(151\) −1.37017 −0.111503 −0.0557513 0.998445i \(-0.517755\pi\)
−0.0557513 + 0.998445i \(0.517755\pi\)
\(152\) −15.1169 −1.22614
\(153\) 0 0
\(154\) 7.71947 0.622052
\(155\) −1.46410 −0.117599
\(156\) 0 0
\(157\) 11.9700 0.955311 0.477656 0.878547i \(-0.341487\pi\)
0.477656 + 0.878547i \(0.341487\pi\)
\(158\) 13.1357 1.04502
\(159\) 0 0
\(160\) −1.33133 −0.105251
\(161\) 34.2020 2.69550
\(162\) 0 0
\(163\) −22.5713 −1.76792 −0.883962 0.467559i \(-0.845134\pi\)
−0.883962 + 0.467559i \(0.845134\pi\)
\(164\) −0.0633815 −0.00494927
\(165\) 0 0
\(166\) 1.08495 0.0842085
\(167\) −8.19700 −0.634303 −0.317152 0.948375i \(-0.602726\pi\)
−0.317152 + 0.948375i \(0.602726\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −5.32235 −0.408205
\(171\) 0 0
\(172\) 0.841831 0.0641890
\(173\) 9.16772 0.697009 0.348505 0.937307i \(-0.386690\pi\)
0.348505 + 0.937307i \(0.386690\pi\)
\(174\) 0 0
\(175\) 4.82684 0.364875
\(176\) −4.72364 −0.356057
\(177\) 0 0
\(178\) −20.1991 −1.51399
\(179\) 10.0370 0.750200 0.375100 0.926984i \(-0.377608\pi\)
0.375100 + 0.926984i \(0.377608\pi\)
\(180\) 0 0
\(181\) 17.0238 1.26537 0.632686 0.774408i \(-0.281952\pi\)
0.632686 + 0.774408i \(0.281952\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −18.6871 −1.37763
\(185\) −0.0253983 −0.00186732
\(186\) 0 0
\(187\) −3.80584 −0.278310
\(188\) −1.54177 −0.112445
\(189\) 0 0
\(190\) 8.57233 0.621902
\(191\) 3.87741 0.280559 0.140280 0.990112i \(-0.455200\pi\)
0.140280 + 0.990112i \(0.455200\pi\)
\(192\) 0 0
\(193\) 1.25394 0.0902608 0.0451304 0.998981i \(-0.485630\pi\)
0.0451304 + 0.998981i \(0.485630\pi\)
\(194\) −5.14261 −0.369218
\(195\) 0 0
\(196\) 3.85527 0.275376
\(197\) 15.2820 1.08879 0.544397 0.838828i \(-0.316758\pi\)
0.544397 + 0.838828i \(0.316758\pi\)
\(198\) 0 0
\(199\) −13.2296 −0.937822 −0.468911 0.883246i \(-0.655353\pi\)
−0.468911 + 0.883246i \(0.655353\pi\)
\(200\) −2.63726 −0.186483
\(201\) 0 0
\(202\) −4.27007 −0.300441
\(203\) −7.11035 −0.499049
\(204\) 0 0
\(205\) −0.267949 −0.0187144
\(206\) −8.29242 −0.577760
\(207\) 0 0
\(208\) 0 0
\(209\) 6.12979 0.424007
\(210\) 0 0
\(211\) −4.81042 −0.331163 −0.165582 0.986196i \(-0.552950\pi\)
−0.165582 + 0.986196i \(0.552950\pi\)
\(212\) −0.234418 −0.0160999
\(213\) 0 0
\(214\) 6.64172 0.454018
\(215\) 3.55889 0.242714
\(216\) 0 0
\(217\) −7.06698 −0.479738
\(218\) 20.6088 1.39580
\(219\) 0 0
\(220\) 0.252957 0.0170543
\(221\) 0 0
\(222\) 0 0
\(223\) 14.7132 0.985271 0.492635 0.870236i \(-0.336034\pi\)
0.492635 + 0.870236i \(0.336034\pi\)
\(224\) −6.42612 −0.429363
\(225\) 0 0
\(226\) 12.0299 0.800214
\(227\) −14.9028 −0.989134 −0.494567 0.869140i \(-0.664673\pi\)
−0.494567 + 0.869140i \(0.664673\pi\)
\(228\) 0 0
\(229\) −19.3074 −1.27587 −0.637933 0.770092i \(-0.720210\pi\)
−0.637933 + 0.770092i \(0.720210\pi\)
\(230\) 10.5969 0.698737
\(231\) 0 0
\(232\) 3.88492 0.255057
\(233\) 21.1937 1.38845 0.694224 0.719759i \(-0.255748\pi\)
0.694224 + 0.719759i \(0.255748\pi\)
\(234\) 0 0
\(235\) −6.51793 −0.425183
\(236\) −2.06338 −0.134315
\(237\) 0 0
\(238\) −25.6901 −1.66524
\(239\) 14.8971 0.963612 0.481806 0.876278i \(-0.339981\pi\)
0.481806 + 0.876278i \(0.339981\pi\)
\(240\) 0 0
\(241\) 9.39168 0.604971 0.302486 0.953154i \(-0.402184\pi\)
0.302486 + 0.953154i \(0.402184\pi\)
\(242\) −14.7403 −0.947544
\(243\) 0 0
\(244\) 1.49905 0.0959671
\(245\) 16.2984 1.04126
\(246\) 0 0
\(247\) 0 0
\(248\) 3.86122 0.245188
\(249\) 0 0
\(250\) 1.49551 0.0945842
\(251\) −11.3163 −0.714281 −0.357140 0.934051i \(-0.616248\pi\)
−0.357140 + 0.934051i \(0.616248\pi\)
\(252\) 0 0
\(253\) 7.57748 0.476392
\(254\) 1.05680 0.0663098
\(255\) 0 0
\(256\) 5.60076 0.350047
\(257\) 26.5319 1.65502 0.827508 0.561453i \(-0.189758\pi\)
0.827508 + 0.561453i \(0.189758\pi\)
\(258\) 0 0
\(259\) −0.122593 −0.00761758
\(260\) 0 0
\(261\) 0 0
\(262\) −9.37017 −0.578891
\(263\) 14.1408 0.871957 0.435979 0.899957i \(-0.356402\pi\)
0.435979 + 0.899957i \(0.356402\pi\)
\(264\) 0 0
\(265\) −0.991015 −0.0608776
\(266\) 41.3772 2.53700
\(267\) 0 0
\(268\) 1.22368 0.0747479
\(269\) −24.7745 −1.51053 −0.755264 0.655421i \(-0.772491\pi\)
−0.755264 + 0.655421i \(0.772491\pi\)
\(270\) 0 0
\(271\) 18.7171 1.13698 0.568492 0.822689i \(-0.307527\pi\)
0.568492 + 0.822689i \(0.307527\pi\)
\(272\) 15.7201 0.953170
\(273\) 0 0
\(274\) −24.3854 −1.47318
\(275\) 1.06939 0.0644866
\(276\) 0 0
\(277\) −22.6647 −1.36179 −0.680893 0.732382i \(-0.738408\pi\)
−0.680893 + 0.732382i \(0.738408\pi\)
\(278\) −10.2073 −0.612191
\(279\) 0 0
\(280\) −12.7296 −0.760742
\(281\) 27.8384 1.66070 0.830351 0.557241i \(-0.188140\pi\)
0.830351 + 0.557241i \(0.188140\pi\)
\(282\) 0 0
\(283\) 7.92007 0.470799 0.235400 0.971899i \(-0.424360\pi\)
0.235400 + 0.971899i \(0.424360\pi\)
\(284\) 1.83673 0.108990
\(285\) 0 0
\(286\) 0 0
\(287\) −1.29335 −0.0763439
\(288\) 0 0
\(289\) −4.33431 −0.254959
\(290\) −2.20301 −0.129365
\(291\) 0 0
\(292\) 2.39117 0.139932
\(293\) 0.272971 0.0159471 0.00797356 0.999968i \(-0.497462\pi\)
0.00797356 + 0.999968i \(0.497462\pi\)
\(294\) 0 0
\(295\) −8.72307 −0.507877
\(296\) 0.0669819 0.00389324
\(297\) 0 0
\(298\) 12.6214 0.731139
\(299\) 0 0
\(300\) 0 0
\(301\) 17.1782 0.990134
\(302\) −2.04909 −0.117912
\(303\) 0 0
\(304\) −25.3192 −1.45216
\(305\) 6.33734 0.362875
\(306\) 0 0
\(307\) 6.85224 0.391078 0.195539 0.980696i \(-0.437354\pi\)
0.195539 + 0.980696i \(0.437354\pi\)
\(308\) 1.22098 0.0695719
\(309\) 0 0
\(310\) −2.18958 −0.124360
\(311\) −10.6447 −0.603605 −0.301803 0.953370i \(-0.597588\pi\)
−0.301803 + 0.953370i \(0.597588\pi\)
\(312\) 0 0
\(313\) 17.8236 1.00745 0.503724 0.863865i \(-0.331963\pi\)
0.503724 + 0.863865i \(0.331963\pi\)
\(314\) 17.9012 1.01023
\(315\) 0 0
\(316\) 2.07767 0.116878
\(317\) −8.17161 −0.458963 −0.229482 0.973313i \(-0.573703\pi\)
−0.229482 + 0.973313i \(0.573703\pi\)
\(318\) 0 0
\(319\) −1.57530 −0.0882000
\(320\) 6.84325 0.382549
\(321\) 0 0
\(322\) 51.1494 2.85044
\(323\) −20.3997 −1.13507
\(324\) 0 0
\(325\) 0 0
\(326\) −33.7556 −1.86955
\(327\) 0 0
\(328\) 0.706653 0.0390184
\(329\) −31.4610 −1.73450
\(330\) 0 0
\(331\) 24.9395 1.37080 0.685400 0.728167i \(-0.259627\pi\)
0.685400 + 0.728167i \(0.259627\pi\)
\(332\) 0.171606 0.00941809
\(333\) 0 0
\(334\) −12.2587 −0.670765
\(335\) 5.17316 0.282640
\(336\) 0 0
\(337\) −19.6057 −1.06799 −0.533996 0.845487i \(-0.679310\pi\)
−0.533996 + 0.845487i \(0.679310\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −0.841831 −0.0456547
\(341\) −1.56569 −0.0847871
\(342\) 0 0
\(343\) 44.8817 2.42339
\(344\) −9.38573 −0.506045
\(345\) 0 0
\(346\) 13.7104 0.737076
\(347\) −17.0810 −0.916955 −0.458478 0.888706i \(-0.651605\pi\)
−0.458478 + 0.888706i \(0.651605\pi\)
\(348\) 0 0
\(349\) −28.3719 −1.51871 −0.759357 0.650674i \(-0.774486\pi\)
−0.759357 + 0.650674i \(0.774486\pi\)
\(350\) 7.21857 0.385849
\(351\) 0 0
\(352\) −1.42371 −0.0758840
\(353\) −21.2520 −1.13113 −0.565564 0.824704i \(-0.691342\pi\)
−0.565564 + 0.824704i \(0.691342\pi\)
\(354\) 0 0
\(355\) 7.76488 0.412117
\(356\) −3.19488 −0.169328
\(357\) 0 0
\(358\) 15.0104 0.793325
\(359\) −32.6519 −1.72330 −0.861650 0.507502i \(-0.830569\pi\)
−0.861650 + 0.507502i \(0.830569\pi\)
\(360\) 0 0
\(361\) 13.8564 0.729285
\(362\) 25.4593 1.33811
\(363\) 0 0
\(364\) 0 0
\(365\) 10.1088 0.529118
\(366\) 0 0
\(367\) −5.91837 −0.308936 −0.154468 0.987998i \(-0.549366\pi\)
−0.154468 + 0.987998i \(0.549366\pi\)
\(368\) −31.2989 −1.63157
\(369\) 0 0
\(370\) −0.0379833 −0.00197466
\(371\) −4.78347 −0.248345
\(372\) 0 0
\(373\) −13.3185 −0.689607 −0.344803 0.938675i \(-0.612054\pi\)
−0.344803 + 0.938675i \(0.612054\pi\)
\(374\) −5.69166 −0.294309
\(375\) 0 0
\(376\) 17.1895 0.886480
\(377\) 0 0
\(378\) 0 0
\(379\) 25.4186 1.30566 0.652832 0.757503i \(-0.273581\pi\)
0.652832 + 0.757503i \(0.273581\pi\)
\(380\) 1.35588 0.0695550
\(381\) 0 0
\(382\) 5.79869 0.296687
\(383\) 10.8268 0.553226 0.276613 0.960981i \(-0.410788\pi\)
0.276613 + 0.960981i \(0.410788\pi\)
\(384\) 0 0
\(385\) 5.16177 0.263068
\(386\) 1.87528 0.0954493
\(387\) 0 0
\(388\) −0.813402 −0.0412942
\(389\) 23.0370 1.16802 0.584011 0.811746i \(-0.301482\pi\)
0.584011 + 0.811746i \(0.301482\pi\)
\(390\) 0 0
\(391\) −25.2176 −1.27531
\(392\) −42.9831 −2.17097
\(393\) 0 0
\(394\) 22.8543 1.15138
\(395\) 8.78347 0.441944
\(396\) 0 0
\(397\) 21.0864 1.05830 0.529149 0.848529i \(-0.322511\pi\)
0.529149 + 0.848529i \(0.322511\pi\)
\(398\) −19.7850 −0.991731
\(399\) 0 0
\(400\) −4.41713 −0.220857
\(401\) 19.7769 0.987611 0.493805 0.869572i \(-0.335605\pi\)
0.493805 + 0.869572i \(0.335605\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −0.675394 −0.0336021
\(405\) 0 0
\(406\) −10.6336 −0.527736
\(407\) −0.0271606 −0.00134630
\(408\) 0 0
\(409\) −31.8809 −1.57641 −0.788204 0.615414i \(-0.788989\pi\)
−0.788204 + 0.615414i \(0.788989\pi\)
\(410\) −0.400720 −0.0197902
\(411\) 0 0
\(412\) −1.31160 −0.0646181
\(413\) −42.1048 −2.07184
\(414\) 0 0
\(415\) 0.725474 0.0356121
\(416\) 0 0
\(417\) 0 0
\(418\) 9.16715 0.448380
\(419\) −30.7296 −1.50124 −0.750621 0.660733i \(-0.770245\pi\)
−0.750621 + 0.660733i \(0.770245\pi\)
\(420\) 0 0
\(421\) 17.9820 0.876391 0.438195 0.898880i \(-0.355618\pi\)
0.438195 + 0.898880i \(0.355618\pi\)
\(422\) −7.19403 −0.350200
\(423\) 0 0
\(424\) 2.61357 0.126926
\(425\) −3.55889 −0.172631
\(426\) 0 0
\(427\) 30.5893 1.48032
\(428\) 1.05051 0.0507785
\(429\) 0 0
\(430\) 5.32235 0.256666
\(431\) −4.89949 −0.236000 −0.118000 0.993014i \(-0.537648\pi\)
−0.118000 + 0.993014i \(0.537648\pi\)
\(432\) 0 0
\(433\) −19.2394 −0.924588 −0.462294 0.886727i \(-0.652974\pi\)
−0.462294 + 0.886727i \(0.652974\pi\)
\(434\) −10.5687 −0.507315
\(435\) 0 0
\(436\) 3.25967 0.156110
\(437\) 40.6162 1.94294
\(438\) 0 0
\(439\) −8.55974 −0.408534 −0.204267 0.978915i \(-0.565481\pi\)
−0.204267 + 0.978915i \(0.565481\pi\)
\(440\) −2.82026 −0.134451
\(441\) 0 0
\(442\) 0 0
\(443\) 37.9652 1.80378 0.901891 0.431965i \(-0.142179\pi\)
0.901891 + 0.431965i \(0.142179\pi\)
\(444\) 0 0
\(445\) −13.5065 −0.640271
\(446\) 22.0037 1.04191
\(447\) 0 0
\(448\) 33.0313 1.56058
\(449\) 26.7062 1.26035 0.630173 0.776455i \(-0.282984\pi\)
0.630173 + 0.776455i \(0.282984\pi\)
\(450\) 0 0
\(451\) −0.286542 −0.0134927
\(452\) 1.90275 0.0894979
\(453\) 0 0
\(454\) −22.2873 −1.04599
\(455\) 0 0
\(456\) 0 0
\(457\) −4.27127 −0.199801 −0.0999007 0.994997i \(-0.531853\pi\)
−0.0999007 + 0.994997i \(0.531853\pi\)
\(458\) −28.8743 −1.34921
\(459\) 0 0
\(460\) 1.67610 0.0781485
\(461\) −20.6423 −0.961407 −0.480704 0.876883i \(-0.659619\pi\)
−0.480704 + 0.876883i \(0.659619\pi\)
\(462\) 0 0
\(463\) −32.1040 −1.49200 −0.745999 0.665947i \(-0.768028\pi\)
−0.745999 + 0.665947i \(0.768028\pi\)
\(464\) 6.50682 0.302071
\(465\) 0 0
\(466\) 31.6954 1.46826
\(467\) −23.3774 −1.08178 −0.540888 0.841095i \(-0.681912\pi\)
−0.540888 + 0.841095i \(0.681912\pi\)
\(468\) 0 0
\(469\) 24.9700 1.15301
\(470\) −9.74761 −0.449624
\(471\) 0 0
\(472\) 23.0050 1.05889
\(473\) 3.80584 0.174993
\(474\) 0 0
\(475\) 5.73205 0.263005
\(476\) −4.06338 −0.186245
\(477\) 0 0
\(478\) 22.2787 1.01900
\(479\) 5.17534 0.236467 0.118234 0.992986i \(-0.462277\pi\)
0.118234 + 0.992986i \(0.462277\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 14.0453 0.639747
\(483\) 0 0
\(484\) −2.33147 −0.105976
\(485\) −3.43870 −0.156143
\(486\) 0 0
\(487\) −30.7729 −1.39445 −0.697227 0.716850i \(-0.745583\pi\)
−0.697227 + 0.716850i \(0.745583\pi\)
\(488\) −16.7132 −0.756572
\(489\) 0 0
\(490\) 24.3743 1.10112
\(491\) −35.7983 −1.61556 −0.807778 0.589487i \(-0.799330\pi\)
−0.807778 + 0.589487i \(0.799330\pi\)
\(492\) 0 0
\(493\) 5.24255 0.236113
\(494\) 0 0
\(495\) 0 0
\(496\) 6.46713 0.290383
\(497\) 37.4798 1.68120
\(498\) 0 0
\(499\) 28.8971 1.29361 0.646805 0.762655i \(-0.276105\pi\)
0.646805 + 0.762655i \(0.276105\pi\)
\(500\) 0.236543 0.0105785
\(501\) 0 0
\(502\) −16.9237 −0.755340
\(503\) 7.86321 0.350603 0.175302 0.984515i \(-0.443910\pi\)
0.175302 + 0.984515i \(0.443910\pi\)
\(504\) 0 0
\(505\) −2.85527 −0.127058
\(506\) 11.3322 0.503777
\(507\) 0 0
\(508\) 0.167154 0.00741625
\(509\) 28.0113 1.24158 0.620790 0.783977i \(-0.286812\pi\)
0.620790 + 0.783977i \(0.286812\pi\)
\(510\) 0 0
\(511\) 48.7935 2.15850
\(512\) −17.4176 −0.769757
\(513\) 0 0
\(514\) 39.6787 1.75015
\(515\) −5.54488 −0.244337
\(516\) 0 0
\(517\) −6.97020 −0.306549
\(518\) −0.183339 −0.00805546
\(519\) 0 0
\(520\) 0 0
\(521\) 37.5609 1.64557 0.822786 0.568351i \(-0.192419\pi\)
0.822786 + 0.568351i \(0.192419\pi\)
\(522\) 0 0
\(523\) −45.3106 −1.98129 −0.990647 0.136450i \(-0.956431\pi\)
−0.990647 + 0.136450i \(0.956431\pi\)
\(524\) −1.48207 −0.0647446
\(525\) 0 0
\(526\) 21.1476 0.922080
\(527\) 5.21058 0.226976
\(528\) 0 0
\(529\) 27.2086 1.18298
\(530\) −1.48207 −0.0643770
\(531\) 0 0
\(532\) 6.54460 0.283744
\(533\) 0 0
\(534\) 0 0
\(535\) 4.44111 0.192006
\(536\) −13.6430 −0.589287
\(537\) 0 0
\(538\) −37.0504 −1.59736
\(539\) 17.4293 0.750733
\(540\) 0 0
\(541\) 19.7445 0.848882 0.424441 0.905456i \(-0.360471\pi\)
0.424441 + 0.905456i \(0.360471\pi\)
\(542\) 27.9916 1.20234
\(543\) 0 0
\(544\) 4.73806 0.203143
\(545\) 13.7804 0.590289
\(546\) 0 0
\(547\) −11.8312 −0.505867 −0.252934 0.967484i \(-0.581395\pi\)
−0.252934 + 0.967484i \(0.581395\pi\)
\(548\) −3.85702 −0.164764
\(549\) 0 0
\(550\) 1.59928 0.0681935
\(551\) −8.44381 −0.359718
\(552\) 0 0
\(553\) 42.3964 1.80288
\(554\) −33.8952 −1.44007
\(555\) 0 0
\(556\) −1.61447 −0.0684689
\(557\) 4.04621 0.171443 0.0857217 0.996319i \(-0.472680\pi\)
0.0857217 + 0.996319i \(0.472680\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −21.3208 −0.900968
\(561\) 0 0
\(562\) 41.6326 1.75617
\(563\) −3.89926 −0.164334 −0.0821671 0.996619i \(-0.526184\pi\)
−0.0821671 + 0.996619i \(0.526184\pi\)
\(564\) 0 0
\(565\) 8.04399 0.338413
\(566\) 11.8445 0.497863
\(567\) 0 0
\(568\) −20.4780 −0.859239
\(569\) 17.3356 0.726744 0.363372 0.931644i \(-0.381625\pi\)
0.363372 + 0.931644i \(0.381625\pi\)
\(570\) 0 0
\(571\) 29.5118 1.23503 0.617515 0.786559i \(-0.288140\pi\)
0.617515 + 0.786559i \(0.288140\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −1.93421 −0.0807324
\(575\) 7.08580 0.295498
\(576\) 0 0
\(577\) −28.3684 −1.18099 −0.590496 0.807041i \(-0.701068\pi\)
−0.590496 + 0.807041i \(0.701068\pi\)
\(578\) −6.48199 −0.269615
\(579\) 0 0
\(580\) −0.348448 −0.0144685
\(581\) 3.50174 0.145277
\(582\) 0 0
\(583\) −1.05978 −0.0438917
\(584\) −26.6595 −1.10318
\(585\) 0 0
\(586\) 0.408230 0.0168638
\(587\) 34.3877 1.41933 0.709667 0.704538i \(-0.248846\pi\)
0.709667 + 0.704538i \(0.248846\pi\)
\(588\) 0 0
\(589\) −8.39230 −0.345799
\(590\) −13.0454 −0.537071
\(591\) 0 0
\(592\) 0.112187 0.00461088
\(593\) 5.47612 0.224877 0.112439 0.993659i \(-0.464134\pi\)
0.112439 + 0.993659i \(0.464134\pi\)
\(594\) 0 0
\(595\) −17.1782 −0.704237
\(596\) 1.99632 0.0817724
\(597\) 0 0
\(598\) 0 0
\(599\) −38.6039 −1.57731 −0.788657 0.614833i \(-0.789223\pi\)
−0.788657 + 0.614833i \(0.789223\pi\)
\(600\) 0 0
\(601\) 6.57896 0.268361 0.134181 0.990957i \(-0.457160\pi\)
0.134181 + 0.990957i \(0.457160\pi\)
\(602\) 25.6901 1.04705
\(603\) 0 0
\(604\) −0.324103 −0.0131876
\(605\) −9.85641 −0.400720
\(606\) 0 0
\(607\) −16.7664 −0.680525 −0.340263 0.940330i \(-0.610516\pi\)
−0.340263 + 0.940330i \(0.610516\pi\)
\(608\) −7.63126 −0.309488
\(609\) 0 0
\(610\) 9.47754 0.383734
\(611\) 0 0
\(612\) 0 0
\(613\) −28.7789 −1.16237 −0.581184 0.813772i \(-0.697410\pi\)
−0.581184 + 0.813772i \(0.697410\pi\)
\(614\) 10.2476 0.413558
\(615\) 0 0
\(616\) −13.6129 −0.548481
\(617\) −37.3291 −1.50281 −0.751406 0.659840i \(-0.770624\pi\)
−0.751406 + 0.659840i \(0.770624\pi\)
\(618\) 0 0
\(619\) −12.7535 −0.512606 −0.256303 0.966597i \(-0.582504\pi\)
−0.256303 + 0.966597i \(0.582504\pi\)
\(620\) −0.346323 −0.0139087
\(621\) 0 0
\(622\) −15.9192 −0.638303
\(623\) −65.1939 −2.61194
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 26.6553 1.06536
\(627\) 0 0
\(628\) 2.83143 0.112986
\(629\) 0.0903896 0.00360407
\(630\) 0 0
\(631\) −28.7242 −1.14349 −0.571746 0.820430i \(-0.693734\pi\)
−0.571746 + 0.820430i \(0.693734\pi\)
\(632\) −23.1643 −0.921427
\(633\) 0 0
\(634\) −12.2207 −0.485346
\(635\) 0.706653 0.0280427
\(636\) 0 0
\(637\) 0 0
\(638\) −2.35588 −0.0932701
\(639\) 0 0
\(640\) 12.8968 0.509791
\(641\) 22.3970 0.884630 0.442315 0.896860i \(-0.354157\pi\)
0.442315 + 0.896860i \(0.354157\pi\)
\(642\) 0 0
\(643\) 14.1642 0.558581 0.279290 0.960207i \(-0.409901\pi\)
0.279290 + 0.960207i \(0.409901\pi\)
\(644\) 8.09025 0.318801
\(645\) 0 0
\(646\) −30.5080 −1.20032
\(647\) 23.6097 0.928193 0.464096 0.885785i \(-0.346379\pi\)
0.464096 + 0.885785i \(0.346379\pi\)
\(648\) 0 0
\(649\) −9.32835 −0.366170
\(650\) 0 0
\(651\) 0 0
\(652\) −5.33910 −0.209095
\(653\) −33.2765 −1.30221 −0.651105 0.758987i \(-0.725694\pi\)
−0.651105 + 0.758987i \(0.725694\pi\)
\(654\) 0 0
\(655\) −6.26554 −0.244815
\(656\) 1.18357 0.0462105
\(657\) 0 0
\(658\) −47.0502 −1.83421
\(659\) −23.0908 −0.899491 −0.449745 0.893157i \(-0.648485\pi\)
−0.449745 + 0.893157i \(0.648485\pi\)
\(660\) 0 0
\(661\) 13.4365 0.522620 0.261310 0.965255i \(-0.415845\pi\)
0.261310 + 0.965255i \(0.415845\pi\)
\(662\) 37.2972 1.44960
\(663\) 0 0
\(664\) −1.91326 −0.0742491
\(665\) 27.6677 1.07291
\(666\) 0 0
\(667\) −10.4380 −0.404161
\(668\) −1.93895 −0.0750200
\(669\) 0 0
\(670\) 7.73650 0.298887
\(671\) 6.77708 0.261626
\(672\) 0 0
\(673\) −1.94524 −0.0749835 −0.0374918 0.999297i \(-0.511937\pi\)
−0.0374918 + 0.999297i \(0.511937\pi\)
\(674\) −29.3205 −1.12938
\(675\) 0 0
\(676\) 0 0
\(677\) −24.8683 −0.955768 −0.477884 0.878423i \(-0.658596\pi\)
−0.477884 + 0.878423i \(0.658596\pi\)
\(678\) 0 0
\(679\) −16.5981 −0.636975
\(680\) 9.38573 0.359926
\(681\) 0 0
\(682\) −2.34151 −0.0896610
\(683\) −14.6221 −0.559500 −0.279750 0.960073i \(-0.590252\pi\)
−0.279750 + 0.960073i \(0.590252\pi\)
\(684\) 0 0
\(685\) −16.3058 −0.623013
\(686\) 67.1210 2.56269
\(687\) 0 0
\(688\) −15.7201 −0.599323
\(689\) 0 0
\(690\) 0 0
\(691\) 3.52451 0.134079 0.0670393 0.997750i \(-0.478645\pi\)
0.0670393 + 0.997750i \(0.478645\pi\)
\(692\) 2.16856 0.0824364
\(693\) 0 0
\(694\) −25.5447 −0.969665
\(695\) −6.82528 −0.258898
\(696\) 0 0
\(697\) 0.953601 0.0361202
\(698\) −42.4304 −1.60602
\(699\) 0 0
\(700\) 1.14176 0.0431543
\(701\) 1.53457 0.0579599 0.0289800 0.999580i \(-0.490774\pi\)
0.0289800 + 0.999580i \(0.490774\pi\)
\(702\) 0 0
\(703\) −0.145584 −0.00549081
\(704\) 7.31810 0.275811
\(705\) 0 0
\(706\) −31.7825 −1.19615
\(707\) −13.7819 −0.518322
\(708\) 0 0
\(709\) −14.0052 −0.525978 −0.262989 0.964799i \(-0.584708\pi\)
−0.262989 + 0.964799i \(0.584708\pi\)
\(710\) 11.6124 0.435807
\(711\) 0 0
\(712\) 35.6203 1.33493
\(713\) −10.3743 −0.388522
\(714\) 0 0
\(715\) 0 0
\(716\) 2.37418 0.0887274
\(717\) 0 0
\(718\) −48.8312 −1.82236
\(719\) −22.4761 −0.838218 −0.419109 0.907936i \(-0.637657\pi\)
−0.419109 + 0.907936i \(0.637657\pi\)
\(720\) 0 0
\(721\) −26.7643 −0.996753
\(722\) 20.7224 0.771206
\(723\) 0 0
\(724\) 4.02687 0.149658
\(725\) −1.47309 −0.0547091
\(726\) 0 0
\(727\) −10.3421 −0.383566 −0.191783 0.981437i \(-0.561427\pi\)
−0.191783 + 0.981437i \(0.561427\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 15.1178 0.559534
\(731\) −12.6657 −0.468458
\(732\) 0 0
\(733\) −27.3533 −1.01032 −0.505159 0.863026i \(-0.668566\pi\)
−0.505159 + 0.863026i \(0.668566\pi\)
\(734\) −8.85096 −0.326695
\(735\) 0 0
\(736\) −9.43355 −0.347725
\(737\) 5.53212 0.203778
\(738\) 0 0
\(739\) 13.4517 0.494827 0.247413 0.968910i \(-0.420419\pi\)
0.247413 + 0.968910i \(0.420419\pi\)
\(740\) −0.00600778 −0.000220851 0
\(741\) 0 0
\(742\) −7.15372 −0.262621
\(743\) 16.3926 0.601388 0.300694 0.953721i \(-0.402782\pi\)
0.300694 + 0.953721i \(0.402782\pi\)
\(744\) 0 0
\(745\) 8.43955 0.309201
\(746\) −19.9179 −0.729248
\(747\) 0 0
\(748\) −0.900245 −0.0329162
\(749\) 21.4365 0.783274
\(750\) 0 0
\(751\) −27.6655 −1.00953 −0.504764 0.863257i \(-0.668421\pi\)
−0.504764 + 0.863257i \(0.668421\pi\)
\(752\) 28.7906 1.04988
\(753\) 0 0
\(754\) 0 0
\(755\) −1.37017 −0.0498654
\(756\) 0 0
\(757\) 22.9978 0.835870 0.417935 0.908477i \(-0.362754\pi\)
0.417935 + 0.908477i \(0.362754\pi\)
\(758\) 38.0136 1.38072
\(759\) 0 0
\(760\) −15.1169 −0.548349
\(761\) 7.66442 0.277835 0.138918 0.990304i \(-0.455638\pi\)
0.138918 + 0.990304i \(0.455638\pi\)
\(762\) 0 0
\(763\) 66.5160 2.40804
\(764\) 0.917174 0.0331822
\(765\) 0 0
\(766\) 16.1916 0.585027
\(767\) 0 0
\(768\) 0 0
\(769\) −7.23095 −0.260755 −0.130377 0.991464i \(-0.541619\pi\)
−0.130377 + 0.991464i \(0.541619\pi\)
\(770\) 7.71947 0.278190
\(771\) 0 0
\(772\) 0.296612 0.0106753
\(773\) 33.5995 1.20849 0.604246 0.796798i \(-0.293475\pi\)
0.604246 + 0.796798i \(0.293475\pi\)
\(774\) 0 0
\(775\) −1.46410 −0.0525921
\(776\) 9.06877 0.325550
\(777\) 0 0
\(778\) 34.4520 1.23516
\(779\) −1.53590 −0.0550293
\(780\) 0 0
\(781\) 8.30368 0.297129
\(782\) −37.7131 −1.34862
\(783\) 0 0
\(784\) −71.9921 −2.57115
\(785\) 11.9700 0.427228
\(786\) 0 0
\(787\) −2.97168 −0.105929 −0.0529645 0.998596i \(-0.516867\pi\)
−0.0529645 + 0.998596i \(0.516867\pi\)
\(788\) 3.61484 0.128773
\(789\) 0 0
\(790\) 13.1357 0.467349
\(791\) 38.8270 1.38053
\(792\) 0 0
\(793\) 0 0
\(794\) 31.5349 1.11913
\(795\) 0 0
\(796\) −3.12937 −0.110918
\(797\) 22.5751 0.799650 0.399825 0.916591i \(-0.369071\pi\)
0.399825 + 0.916591i \(0.369071\pi\)
\(798\) 0 0
\(799\) 23.1966 0.820636
\(800\) −1.33133 −0.0470696
\(801\) 0 0
\(802\) 29.5765 1.04438
\(803\) 10.8102 0.381485
\(804\) 0 0
\(805\) 34.2020 1.20546
\(806\) 0 0
\(807\) 0 0
\(808\) 7.53009 0.264908
\(809\) −13.6584 −0.480204 −0.240102 0.970748i \(-0.577181\pi\)
−0.240102 + 0.970748i \(0.577181\pi\)
\(810\) 0 0
\(811\) 14.1147 0.495636 0.247818 0.968807i \(-0.420287\pi\)
0.247818 + 0.968807i \(0.420287\pi\)
\(812\) −1.68190 −0.0590233
\(813\) 0 0
\(814\) −0.0406189 −0.00142369
\(815\) −22.5713 −0.790640
\(816\) 0 0
\(817\) 20.3997 0.713696
\(818\) −47.6781 −1.66703
\(819\) 0 0
\(820\) −0.0633815 −0.00221338
\(821\) −1.58562 −0.0553383 −0.0276692 0.999617i \(-0.508808\pi\)
−0.0276692 + 0.999617i \(0.508808\pi\)
\(822\) 0 0
\(823\) 18.5648 0.647127 0.323563 0.946206i \(-0.395119\pi\)
0.323563 + 0.946206i \(0.395119\pi\)
\(824\) 14.6233 0.509427
\(825\) 0 0
\(826\) −62.9681 −2.19094
\(827\) −9.01023 −0.313316 −0.156658 0.987653i \(-0.550072\pi\)
−0.156658 + 0.987653i \(0.550072\pi\)
\(828\) 0 0
\(829\) 47.1177 1.63647 0.818233 0.574887i \(-0.194954\pi\)
0.818233 + 0.574887i \(0.194954\pi\)
\(830\) 1.08495 0.0376592
\(831\) 0 0
\(832\) 0 0
\(833\) −58.0041 −2.00972
\(834\) 0 0
\(835\) −8.19700 −0.283669
\(836\) 1.44996 0.0501479
\(837\) 0 0
\(838\) −45.9564 −1.58754
\(839\) 53.4766 1.84622 0.923108 0.384541i \(-0.125640\pi\)
0.923108 + 0.384541i \(0.125640\pi\)
\(840\) 0 0
\(841\) −26.8300 −0.925173
\(842\) 26.8923 0.926769
\(843\) 0 0
\(844\) −1.13787 −0.0391672
\(845\) 0 0
\(846\) 0 0
\(847\) −47.5753 −1.63471
\(848\) 4.37745 0.150322
\(849\) 0 0
\(850\) −5.32235 −0.182555
\(851\) −0.179967 −0.00616919
\(852\) 0 0
\(853\) 27.7756 0.951019 0.475510 0.879711i \(-0.342264\pi\)
0.475510 + 0.879711i \(0.342264\pi\)
\(854\) 45.7465 1.56541
\(855\) 0 0
\(856\) −11.7124 −0.400321
\(857\) −53.6917 −1.83407 −0.917037 0.398801i \(-0.869426\pi\)
−0.917037 + 0.398801i \(0.869426\pi\)
\(858\) 0 0
\(859\) 2.08958 0.0712955 0.0356477 0.999364i \(-0.488651\pi\)
0.0356477 + 0.999364i \(0.488651\pi\)
\(860\) 0.841831 0.0287062
\(861\) 0 0
\(862\) −7.32722 −0.249566
\(863\) −1.75413 −0.0597113 −0.0298557 0.999554i \(-0.509505\pi\)
−0.0298557 + 0.999554i \(0.509505\pi\)
\(864\) 0 0
\(865\) 9.16772 0.311712
\(866\) −28.7727 −0.977737
\(867\) 0 0
\(868\) −1.67165 −0.0567394
\(869\) 9.39295 0.318634
\(870\) 0 0
\(871\) 0 0
\(872\) −36.3426 −1.23072
\(873\) 0 0
\(874\) 60.7418 2.05462
\(875\) 4.82684 0.163177
\(876\) 0 0
\(877\) −21.5672 −0.728272 −0.364136 0.931346i \(-0.618636\pi\)
−0.364136 + 0.931346i \(0.618636\pi\)
\(878\) −12.8012 −0.432018
\(879\) 0 0
\(880\) −4.72364 −0.159234
\(881\) −25.0263 −0.843158 −0.421579 0.906792i \(-0.638524\pi\)
−0.421579 + 0.906792i \(0.638524\pi\)
\(882\) 0 0
\(883\) −48.7832 −1.64169 −0.820843 0.571154i \(-0.806496\pi\)
−0.820843 + 0.571154i \(0.806496\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 56.7772 1.90747
\(887\) 33.7933 1.13467 0.567334 0.823488i \(-0.307975\pi\)
0.567334 + 0.823488i \(0.307975\pi\)
\(888\) 0 0
\(889\) 3.41090 0.114398
\(890\) −20.1991 −0.677076
\(891\) 0 0
\(892\) 3.48031 0.116530
\(893\) −37.3611 −1.25024
\(894\) 0 0
\(895\) 10.0370 0.335500
\(896\) 62.2508 2.07965
\(897\) 0 0
\(898\) 39.9394 1.33279
\(899\) 2.15675 0.0719316
\(900\) 0 0
\(901\) 3.52691 0.117499
\(902\) −0.428526 −0.0142683
\(903\) 0 0
\(904\) −21.2141 −0.705571
\(905\) 17.0238 0.565892
\(906\) 0 0
\(907\) −34.6270 −1.14977 −0.574885 0.818234i \(-0.694953\pi\)
−0.574885 + 0.818234i \(0.694953\pi\)
\(908\) −3.52516 −0.116986
\(909\) 0 0
\(910\) 0 0
\(911\) −31.1865 −1.03326 −0.516628 0.856210i \(-0.672813\pi\)
−0.516628 + 0.856210i \(0.672813\pi\)
\(912\) 0 0
\(913\) 0.775814 0.0256757
\(914\) −6.38771 −0.211287
\(915\) 0 0
\(916\) −4.56702 −0.150899
\(917\) −30.2428 −0.998704
\(918\) 0 0
\(919\) −51.9220 −1.71275 −0.856374 0.516356i \(-0.827288\pi\)
−0.856374 + 0.516356i \(0.827288\pi\)
\(920\) −18.6871 −0.616096
\(921\) 0 0
\(922\) −30.8707 −1.01667
\(923\) 0 0
\(924\) 0 0
\(925\) −0.0253983 −0.000835090 0
\(926\) −48.0117 −1.57776
\(927\) 0 0
\(928\) 1.96117 0.0643784
\(929\) 20.4915 0.672304 0.336152 0.941808i \(-0.390874\pi\)
0.336152 + 0.941808i \(0.390874\pi\)
\(930\) 0 0
\(931\) 93.4231 3.06182
\(932\) 5.01324 0.164214
\(933\) 0 0
\(934\) −34.9610 −1.14396
\(935\) −3.80584 −0.124464
\(936\) 0 0
\(937\) −39.6806 −1.29631 −0.648154 0.761510i \(-0.724459\pi\)
−0.648154 + 0.761510i \(0.724459\pi\)
\(938\) 37.3428 1.21929
\(939\) 0 0
\(940\) −1.54177 −0.0502870
\(941\) −19.6189 −0.639557 −0.319779 0.947492i \(-0.603609\pi\)
−0.319779 + 0.947492i \(0.603609\pi\)
\(942\) 0 0
\(943\) −1.89864 −0.0618281
\(944\) 38.5309 1.25408
\(945\) 0 0
\(946\) 5.69166 0.185052
\(947\) 57.2124 1.85915 0.929576 0.368631i \(-0.120173\pi\)
0.929576 + 0.368631i \(0.120173\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 8.57233 0.278123
\(951\) 0 0
\(952\) 45.3034 1.46829
\(953\) −27.4770 −0.890066 −0.445033 0.895514i \(-0.646808\pi\)
−0.445033 + 0.895514i \(0.646808\pi\)
\(954\) 0 0
\(955\) 3.87741 0.125470
\(956\) 3.52380 0.113968
\(957\) 0 0
\(958\) 7.73976 0.250060
\(959\) −78.7055 −2.54153
\(960\) 0 0
\(961\) −28.8564 −0.930852
\(962\) 0 0
\(963\) 0 0
\(964\) 2.22154 0.0715509
\(965\) 1.25394 0.0403659
\(966\) 0 0
\(967\) 10.3643 0.333293 0.166647 0.986017i \(-0.446706\pi\)
0.166647 + 0.986017i \(0.446706\pi\)
\(968\) 25.9939 0.835477
\(969\) 0 0
\(970\) −5.14261 −0.165119
\(971\) −41.7515 −1.33987 −0.669935 0.742420i \(-0.733678\pi\)
−0.669935 + 0.742420i \(0.733678\pi\)
\(972\) 0 0
\(973\) −32.9445 −1.05615
\(974\) −46.0212 −1.47461
\(975\) 0 0
\(976\) −27.9929 −0.896030
\(977\) −13.7938 −0.441303 −0.220652 0.975353i \(-0.570818\pi\)
−0.220652 + 0.975353i \(0.570818\pi\)
\(978\) 0 0
\(979\) −14.4437 −0.461624
\(980\) 3.85527 0.123152
\(981\) 0 0
\(982\) −53.5367 −1.70842
\(983\) 37.9997 1.21200 0.606002 0.795463i \(-0.292773\pi\)
0.606002 + 0.795463i \(0.292773\pi\)
\(984\) 0 0
\(985\) 15.2820 0.486924
\(986\) 7.84028 0.249685
\(987\) 0 0
\(988\) 0 0
\(989\) 25.2176 0.801873
\(990\) 0 0
\(991\) −52.5530 −1.66940 −0.834700 0.550705i \(-0.814359\pi\)
−0.834700 + 0.550705i \(0.814359\pi\)
\(992\) 1.94920 0.0618873
\(993\) 0 0
\(994\) 56.0513 1.77784
\(995\) −13.2296 −0.419407
\(996\) 0 0
\(997\) 18.5899 0.588749 0.294375 0.955690i \(-0.404889\pi\)
0.294375 + 0.955690i \(0.404889\pi\)
\(998\) 43.2158 1.36797
\(999\) 0 0
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 7605.2.a.cf.1.4 4
3.2 odd 2 845.2.a.m.1.1 4
13.6 odd 12 585.2.bu.c.361.1 8
13.11 odd 12 585.2.bu.c.316.1 8
13.12 even 2 7605.2.a.cj.1.1 4
15.14 odd 2 4225.2.a.bi.1.4 4
39.2 even 12 845.2.m.g.316.1 8
39.5 even 4 845.2.c.g.506.7 8
39.8 even 4 845.2.c.g.506.2 8
39.11 even 12 65.2.m.a.56.4 yes 8
39.17 odd 6 845.2.e.n.146.1 8
39.20 even 12 845.2.m.g.361.1 8
39.23 odd 6 845.2.e.n.191.1 8
39.29 odd 6 845.2.e.m.191.4 8
39.32 even 12 65.2.m.a.36.4 8
39.35 odd 6 845.2.e.m.146.4 8
39.38 odd 2 845.2.a.l.1.4 4
156.11 odd 12 1040.2.da.b.641.3 8
156.71 odd 12 1040.2.da.b.881.3 8
195.32 odd 12 325.2.m.b.49.4 8
195.89 even 12 325.2.n.d.251.1 8
195.128 odd 12 325.2.m.b.199.4 8
195.149 even 12 325.2.n.d.101.1 8
195.167 odd 12 325.2.m.c.199.1 8
195.188 odd 12 325.2.m.c.49.1 8
195.194 odd 2 4225.2.a.bl.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.m.a.36.4 8 39.32 even 12
65.2.m.a.56.4 yes 8 39.11 even 12
325.2.m.b.49.4 8 195.32 odd 12
325.2.m.b.199.4 8 195.128 odd 12
325.2.m.c.49.1 8 195.188 odd 12
325.2.m.c.199.1 8 195.167 odd 12
325.2.n.d.101.1 8 195.149 even 12
325.2.n.d.251.1 8 195.89 even 12
585.2.bu.c.316.1 8 13.11 odd 12
585.2.bu.c.361.1 8 13.6 odd 12
845.2.a.l.1.4 4 39.38 odd 2
845.2.a.m.1.1 4 3.2 odd 2
845.2.c.g.506.2 8 39.8 even 4
845.2.c.g.506.7 8 39.5 even 4
845.2.e.m.146.4 8 39.35 odd 6
845.2.e.m.191.4 8 39.29 odd 6
845.2.e.n.146.1 8 39.17 odd 6
845.2.e.n.191.1 8 39.23 odd 6
845.2.m.g.316.1 8 39.2 even 12
845.2.m.g.361.1 8 39.20 even 12
1040.2.da.b.641.3 8 156.11 odd 12
1040.2.da.b.881.3 8 156.71 odd 12
4225.2.a.bi.1.4 4 15.14 odd 2
4225.2.a.bl.1.1 4 195.194 odd 2
7605.2.a.cf.1.4 4 1.1 even 1 trivial
7605.2.a.cj.1.1 4 13.12 even 2