Properties

Label 3465.2.a.bh.1.3
Level $3465$
Weight $2$
Character 3465.1
Self dual yes
Analytic conductor $27.668$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 3465 = 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3465.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.6681643004\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 385)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.48119\) of defining polynomial
Character \(\chi\) \(=\) 3465.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.67513 q^{2} +5.15633 q^{4} +1.00000 q^{5} -1.00000 q^{7} +8.44358 q^{8} +O(q^{10})\) \(q+2.67513 q^{2} +5.15633 q^{4} +1.00000 q^{5} -1.00000 q^{7} +8.44358 q^{8} +2.67513 q^{10} +1.00000 q^{11} +5.83146 q^{13} -2.67513 q^{14} +12.2750 q^{16} -5.44358 q^{17} -1.35026 q^{19} +5.15633 q^{20} +2.67513 q^{22} +3.19394 q^{23} +1.00000 q^{25} +15.5999 q^{26} -5.15633 q^{28} +3.61213 q^{29} -5.28726 q^{31} +15.9502 q^{32} -14.5623 q^{34} -1.00000 q^{35} -8.54420 q^{37} -3.61213 q^{38} +8.44358 q^{40} +5.02539 q^{41} +5.89446 q^{43} +5.15633 q^{44} +8.54420 q^{46} +11.8315 q^{47} +1.00000 q^{49} +2.67513 q^{50} +30.0689 q^{52} +0.231548 q^{53} +1.00000 q^{55} -8.44358 q^{56} +9.66291 q^{58} -13.5999 q^{59} -1.41327 q^{61} -14.1441 q^{62} +18.1187 q^{64} +5.83146 q^{65} -10.8568 q^{67} -28.0689 q^{68} -2.67513 q^{70} +15.5369 q^{71} -11.3684 q^{73} -22.8568 q^{74} -6.96239 q^{76} -1.00000 q^{77} +1.96968 q^{79} +12.2750 q^{80} +13.4436 q^{82} -10.6253 q^{83} -5.44358 q^{85} +15.7685 q^{86} +8.44358 q^{88} -7.22425 q^{89} -5.83146 q^{91} +16.4690 q^{92} +31.6507 q^{94} -1.35026 q^{95} -0.836381 q^{97} +2.67513 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 5 q^{4} + 3 q^{5} - 3 q^{7} + 9 q^{8} + O(q^{10}) \) \( 3 q + 3 q^{2} + 5 q^{4} + 3 q^{5} - 3 q^{7} + 9 q^{8} + 3 q^{10} + 3 q^{11} + 2 q^{13} - 3 q^{14} + 5 q^{16} + 6 q^{19} + 5 q^{20} + 3 q^{22} + 10 q^{23} + 3 q^{25} + 20 q^{26} - 5 q^{28} + 10 q^{29} - 10 q^{31} + 11 q^{32} - 6 q^{34} - 3 q^{35} - 16 q^{37} - 10 q^{38} + 9 q^{40} - 2 q^{43} + 5 q^{44} + 16 q^{46} + 20 q^{47} + 3 q^{49} + 3 q^{50} + 32 q^{52} + 12 q^{53} + 3 q^{55} - 9 q^{56} - 2 q^{58} - 14 q^{59} + 10 q^{61} - 6 q^{62} + 33 q^{64} + 2 q^{65} - 2 q^{67} - 26 q^{68} - 3 q^{70} + 24 q^{71} + 4 q^{73} - 38 q^{74} - 10 q^{76} - 3 q^{77} + 8 q^{79} + 5 q^{80} + 24 q^{82} + 10 q^{83} + 36 q^{86} + 9 q^{88} - 20 q^{89} - 2 q^{91} + 18 q^{92} + 38 q^{94} + 6 q^{95} + 3 q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.67513 1.89160 0.945802 0.324745i \(-0.105279\pi\)
0.945802 + 0.324745i \(0.105279\pi\)
\(3\) 0 0
\(4\) 5.15633 2.57816
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 8.44358 2.98526
\(9\) 0 0
\(10\) 2.67513 0.845951
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 5.83146 1.61735 0.808677 0.588252i \(-0.200184\pi\)
0.808677 + 0.588252i \(0.200184\pi\)
\(14\) −2.67513 −0.714959
\(15\) 0 0
\(16\) 12.2750 3.06876
\(17\) −5.44358 −1.32026 −0.660131 0.751150i \(-0.729499\pi\)
−0.660131 + 0.751150i \(0.729499\pi\)
\(18\) 0 0
\(19\) −1.35026 −0.309771 −0.154886 0.987932i \(-0.549501\pi\)
−0.154886 + 0.987932i \(0.549501\pi\)
\(20\) 5.15633 1.15299
\(21\) 0 0
\(22\) 2.67513 0.570340
\(23\) 3.19394 0.665982 0.332991 0.942930i \(-0.391942\pi\)
0.332991 + 0.942930i \(0.391942\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 15.5999 3.05939
\(27\) 0 0
\(28\) −5.15633 −0.974454
\(29\) 3.61213 0.670755 0.335378 0.942084i \(-0.391136\pi\)
0.335378 + 0.942084i \(0.391136\pi\)
\(30\) 0 0
\(31\) −5.28726 −0.949620 −0.474810 0.880088i \(-0.657483\pi\)
−0.474810 + 0.880088i \(0.657483\pi\)
\(32\) 15.9502 2.81962
\(33\) 0 0
\(34\) −14.5623 −2.49741
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) −8.54420 −1.40466 −0.702329 0.711853i \(-0.747856\pi\)
−0.702329 + 0.711853i \(0.747856\pi\)
\(38\) −3.61213 −0.585964
\(39\) 0 0
\(40\) 8.44358 1.33505
\(41\) 5.02539 0.784834 0.392417 0.919787i \(-0.371639\pi\)
0.392417 + 0.919787i \(0.371639\pi\)
\(42\) 0 0
\(43\) 5.89446 0.898897 0.449448 0.893306i \(-0.351621\pi\)
0.449448 + 0.893306i \(0.351621\pi\)
\(44\) 5.15633 0.777345
\(45\) 0 0
\(46\) 8.54420 1.25977
\(47\) 11.8315 1.72580 0.862898 0.505379i \(-0.168647\pi\)
0.862898 + 0.505379i \(0.168647\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 2.67513 0.378321
\(51\) 0 0
\(52\) 30.0689 4.16980
\(53\) 0.231548 0.0318056 0.0159028 0.999874i \(-0.494938\pi\)
0.0159028 + 0.999874i \(0.494938\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) −8.44358 −1.12832
\(57\) 0 0
\(58\) 9.66291 1.26880
\(59\) −13.5999 −1.77056 −0.885279 0.465061i \(-0.846032\pi\)
−0.885279 + 0.465061i \(0.846032\pi\)
\(60\) 0 0
\(61\) −1.41327 −0.180950 −0.0904751 0.995899i \(-0.528839\pi\)
−0.0904751 + 0.995899i \(0.528839\pi\)
\(62\) −14.1441 −1.79630
\(63\) 0 0
\(64\) 18.1187 2.26484
\(65\) 5.83146 0.723303
\(66\) 0 0
\(67\) −10.8568 −1.32638 −0.663188 0.748453i \(-0.730797\pi\)
−0.663188 + 0.748453i \(0.730797\pi\)
\(68\) −28.0689 −3.40385
\(69\) 0 0
\(70\) −2.67513 −0.319739
\(71\) 15.5369 1.84389 0.921946 0.387319i \(-0.126599\pi\)
0.921946 + 0.387319i \(0.126599\pi\)
\(72\) 0 0
\(73\) −11.3684 −1.33057 −0.665283 0.746591i \(-0.731689\pi\)
−0.665283 + 0.746591i \(0.731689\pi\)
\(74\) −22.8568 −2.65705
\(75\) 0 0
\(76\) −6.96239 −0.798641
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) 1.96968 0.221607 0.110803 0.993842i \(-0.464658\pi\)
0.110803 + 0.993842i \(0.464658\pi\)
\(80\) 12.2750 1.37239
\(81\) 0 0
\(82\) 13.4436 1.48460
\(83\) −10.6253 −1.16628 −0.583139 0.812372i \(-0.698176\pi\)
−0.583139 + 0.812372i \(0.698176\pi\)
\(84\) 0 0
\(85\) −5.44358 −0.590439
\(86\) 15.7685 1.70036
\(87\) 0 0
\(88\) 8.44358 0.900089
\(89\) −7.22425 −0.765769 −0.382885 0.923796i \(-0.625069\pi\)
−0.382885 + 0.923796i \(0.625069\pi\)
\(90\) 0 0
\(91\) −5.83146 −0.611303
\(92\) 16.4690 1.71701
\(93\) 0 0
\(94\) 31.6507 3.26452
\(95\) −1.35026 −0.138534
\(96\) 0 0
\(97\) −0.836381 −0.0849216 −0.0424608 0.999098i \(-0.513520\pi\)
−0.0424608 + 0.999098i \(0.513520\pi\)
\(98\) 2.67513 0.270229
\(99\) 0 0
\(100\) 5.15633 0.515633
\(101\) −7.41327 −0.737648 −0.368824 0.929499i \(-0.620239\pi\)
−0.368824 + 0.929499i \(0.620239\pi\)
\(102\) 0 0
\(103\) 4.21933 0.415743 0.207871 0.978156i \(-0.433346\pi\)
0.207871 + 0.978156i \(0.433346\pi\)
\(104\) 49.2384 4.82822
\(105\) 0 0
\(106\) 0.619421 0.0601635
\(107\) 11.5369 1.11531 0.557657 0.830071i \(-0.311700\pi\)
0.557657 + 0.830071i \(0.311700\pi\)
\(108\) 0 0
\(109\) −2.18664 −0.209442 −0.104721 0.994502i \(-0.533395\pi\)
−0.104721 + 0.994502i \(0.533395\pi\)
\(110\) 2.67513 0.255064
\(111\) 0 0
\(112\) −12.2750 −1.15988
\(113\) 9.35026 0.879599 0.439799 0.898096i \(-0.355050\pi\)
0.439799 + 0.898096i \(0.355050\pi\)
\(114\) 0 0
\(115\) 3.19394 0.297836
\(116\) 18.6253 1.72932
\(117\) 0 0
\(118\) −36.3815 −3.34919
\(119\) 5.44358 0.499012
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −3.78067 −0.342286
\(123\) 0 0
\(124\) −27.2628 −2.44827
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −16.9624 −1.50517 −0.752584 0.658496i \(-0.771193\pi\)
−0.752584 + 0.658496i \(0.771193\pi\)
\(128\) 16.5696 1.46456
\(129\) 0 0
\(130\) 15.5999 1.36820
\(131\) 9.92478 0.867132 0.433566 0.901122i \(-0.357255\pi\)
0.433566 + 0.901122i \(0.357255\pi\)
\(132\) 0 0
\(133\) 1.35026 0.117083
\(134\) −29.0435 −2.50898
\(135\) 0 0
\(136\) −45.9633 −3.94132
\(137\) 10.9927 0.939170 0.469585 0.882887i \(-0.344404\pi\)
0.469585 + 0.882887i \(0.344404\pi\)
\(138\) 0 0
\(139\) 6.88717 0.584162 0.292081 0.956394i \(-0.405652\pi\)
0.292081 + 0.956394i \(0.405652\pi\)
\(140\) −5.15633 −0.435789
\(141\) 0 0
\(142\) 41.5633 3.48791
\(143\) 5.83146 0.487651
\(144\) 0 0
\(145\) 3.61213 0.299971
\(146\) −30.4119 −2.51690
\(147\) 0 0
\(148\) −44.0567 −3.62144
\(149\) −22.8119 −1.86883 −0.934414 0.356190i \(-0.884076\pi\)
−0.934414 + 0.356190i \(0.884076\pi\)
\(150\) 0 0
\(151\) −3.24472 −0.264052 −0.132026 0.991246i \(-0.542148\pi\)
−0.132026 + 0.991246i \(0.542148\pi\)
\(152\) −11.4010 −0.924747
\(153\) 0 0
\(154\) −2.67513 −0.215568
\(155\) −5.28726 −0.424683
\(156\) 0 0
\(157\) −5.42548 −0.433001 −0.216500 0.976283i \(-0.569464\pi\)
−0.216500 + 0.976283i \(0.569464\pi\)
\(158\) 5.26916 0.419192
\(159\) 0 0
\(160\) 15.9502 1.26097
\(161\) −3.19394 −0.251717
\(162\) 0 0
\(163\) 3.38058 0.264787 0.132394 0.991197i \(-0.457734\pi\)
0.132394 + 0.991197i \(0.457734\pi\)
\(164\) 25.9126 2.02343
\(165\) 0 0
\(166\) −28.4241 −2.20614
\(167\) −11.2750 −0.872489 −0.436244 0.899828i \(-0.643692\pi\)
−0.436244 + 0.899828i \(0.643692\pi\)
\(168\) 0 0
\(169\) 21.0059 1.61584
\(170\) −14.5623 −1.11688
\(171\) 0 0
\(172\) 30.3938 2.31750
\(173\) 8.98049 0.682774 0.341387 0.939923i \(-0.389103\pi\)
0.341387 + 0.939923i \(0.389103\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 12.2750 0.925266
\(177\) 0 0
\(178\) −19.3258 −1.44853
\(179\) 26.2374 1.96108 0.980539 0.196326i \(-0.0629010\pi\)
0.980539 + 0.196326i \(0.0629010\pi\)
\(180\) 0 0
\(181\) −11.1998 −0.832476 −0.416238 0.909256i \(-0.636652\pi\)
−0.416238 + 0.909256i \(0.636652\pi\)
\(182\) −15.5999 −1.15634
\(183\) 0 0
\(184\) 26.9683 1.98813
\(185\) −8.54420 −0.628182
\(186\) 0 0
\(187\) −5.44358 −0.398074
\(188\) 61.0068 4.44938
\(189\) 0 0
\(190\) −3.61213 −0.262051
\(191\) −11.1998 −0.810390 −0.405195 0.914230i \(-0.632796\pi\)
−0.405195 + 0.914230i \(0.632796\pi\)
\(192\) 0 0
\(193\) 0.604833 0.0435368 0.0217684 0.999763i \(-0.493070\pi\)
0.0217684 + 0.999763i \(0.493070\pi\)
\(194\) −2.23743 −0.160638
\(195\) 0 0
\(196\) 5.15633 0.368309
\(197\) −15.3054 −1.09046 −0.545231 0.838286i \(-0.683558\pi\)
−0.545231 + 0.838286i \(0.683558\pi\)
\(198\) 0 0
\(199\) −12.5623 −0.890518 −0.445259 0.895402i \(-0.646888\pi\)
−0.445259 + 0.895402i \(0.646888\pi\)
\(200\) 8.44358 0.597051
\(201\) 0 0
\(202\) −19.8315 −1.39534
\(203\) −3.61213 −0.253522
\(204\) 0 0
\(205\) 5.02539 0.350989
\(206\) 11.2873 0.786421
\(207\) 0 0
\(208\) 71.5814 4.96327
\(209\) −1.35026 −0.0933996
\(210\) 0 0
\(211\) −4.43866 −0.305570 −0.152785 0.988259i \(-0.548824\pi\)
−0.152785 + 0.988259i \(0.548824\pi\)
\(212\) 1.19394 0.0819999
\(213\) 0 0
\(214\) 30.8627 2.10973
\(215\) 5.89446 0.401999
\(216\) 0 0
\(217\) 5.28726 0.358922
\(218\) −5.84955 −0.396182
\(219\) 0 0
\(220\) 5.15633 0.347639
\(221\) −31.7440 −2.13533
\(222\) 0 0
\(223\) −7.78067 −0.521032 −0.260516 0.965469i \(-0.583893\pi\)
−0.260516 + 0.965469i \(0.583893\pi\)
\(224\) −15.9502 −1.06572
\(225\) 0 0
\(226\) 25.0132 1.66385
\(227\) 10.4485 0.693492 0.346746 0.937959i \(-0.387287\pi\)
0.346746 + 0.937959i \(0.387287\pi\)
\(228\) 0 0
\(229\) −29.4518 −1.94623 −0.973116 0.230316i \(-0.926024\pi\)
−0.973116 + 0.230316i \(0.926024\pi\)
\(230\) 8.54420 0.563388
\(231\) 0 0
\(232\) 30.4993 2.00238
\(233\) 8.73084 0.571976 0.285988 0.958233i \(-0.407678\pi\)
0.285988 + 0.958233i \(0.407678\pi\)
\(234\) 0 0
\(235\) 11.8315 0.771799
\(236\) −70.1255 −4.56478
\(237\) 0 0
\(238\) 14.5623 0.943933
\(239\) 21.2144 1.37225 0.686123 0.727486i \(-0.259311\pi\)
0.686123 + 0.727486i \(0.259311\pi\)
\(240\) 0 0
\(241\) −9.33804 −0.601516 −0.300758 0.953700i \(-0.597240\pi\)
−0.300758 + 0.953700i \(0.597240\pi\)
\(242\) 2.67513 0.171964
\(243\) 0 0
\(244\) −7.28726 −0.466519
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) −7.87399 −0.501010
\(248\) −44.6434 −2.83486
\(249\) 0 0
\(250\) 2.67513 0.169190
\(251\) −1.87636 −0.118435 −0.0592174 0.998245i \(-0.518861\pi\)
−0.0592174 + 0.998245i \(0.518861\pi\)
\(252\) 0 0
\(253\) 3.19394 0.200801
\(254\) −45.3766 −2.84718
\(255\) 0 0
\(256\) 8.08840 0.505525
\(257\) −27.1392 −1.69290 −0.846448 0.532472i \(-0.821263\pi\)
−0.846448 + 0.532472i \(0.821263\pi\)
\(258\) 0 0
\(259\) 8.54420 0.530911
\(260\) 30.0689 1.86479
\(261\) 0 0
\(262\) 26.5501 1.64027
\(263\) −12.8119 −0.790018 −0.395009 0.918677i \(-0.629259\pi\)
−0.395009 + 0.918677i \(0.629259\pi\)
\(264\) 0 0
\(265\) 0.231548 0.0142239
\(266\) 3.61213 0.221474
\(267\) 0 0
\(268\) −55.9814 −3.41961
\(269\) 6.26187 0.381793 0.190896 0.981610i \(-0.438861\pi\)
0.190896 + 0.981610i \(0.438861\pi\)
\(270\) 0 0
\(271\) −5.73813 −0.348567 −0.174283 0.984696i \(-0.555761\pi\)
−0.174283 + 0.984696i \(0.555761\pi\)
\(272\) −66.8202 −4.05157
\(273\) 0 0
\(274\) 29.4069 1.77654
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) −8.35756 −0.502157 −0.251078 0.967967i \(-0.580785\pi\)
−0.251078 + 0.967967i \(0.580785\pi\)
\(278\) 18.4241 1.10500
\(279\) 0 0
\(280\) −8.44358 −0.504601
\(281\) 8.44851 0.503996 0.251998 0.967728i \(-0.418912\pi\)
0.251998 + 0.967728i \(0.418912\pi\)
\(282\) 0 0
\(283\) 0.836381 0.0497177 0.0248588 0.999691i \(-0.492086\pi\)
0.0248588 + 0.999691i \(0.492086\pi\)
\(284\) 80.1133 4.75385
\(285\) 0 0
\(286\) 15.5999 0.922442
\(287\) −5.02539 −0.296640
\(288\) 0 0
\(289\) 12.6326 0.743094
\(290\) 9.66291 0.567426
\(291\) 0 0
\(292\) −58.6190 −3.43042
\(293\) 2.71862 0.158824 0.0794118 0.996842i \(-0.474696\pi\)
0.0794118 + 0.996842i \(0.474696\pi\)
\(294\) 0 0
\(295\) −13.5999 −0.791817
\(296\) −72.1436 −4.19326
\(297\) 0 0
\(298\) −61.0249 −3.53508
\(299\) 18.6253 1.07713
\(300\) 0 0
\(301\) −5.89446 −0.339751
\(302\) −8.68006 −0.499481
\(303\) 0 0
\(304\) −16.5745 −0.950614
\(305\) −1.41327 −0.0809234
\(306\) 0 0
\(307\) −8.36344 −0.477326 −0.238663 0.971102i \(-0.576709\pi\)
−0.238663 + 0.971102i \(0.576709\pi\)
\(308\) −5.15633 −0.293809
\(309\) 0 0
\(310\) −14.1441 −0.803331
\(311\) −4.43629 −0.251559 −0.125779 0.992058i \(-0.540143\pi\)
−0.125779 + 0.992058i \(0.540143\pi\)
\(312\) 0 0
\(313\) 29.7889 1.68377 0.841885 0.539658i \(-0.181446\pi\)
0.841885 + 0.539658i \(0.181446\pi\)
\(314\) −14.5139 −0.819066
\(315\) 0 0
\(316\) 10.1563 0.571338
\(317\) −15.4010 −0.865009 −0.432504 0.901632i \(-0.642370\pi\)
−0.432504 + 0.901632i \(0.642370\pi\)
\(318\) 0 0
\(319\) 3.61213 0.202240
\(320\) 18.1187 1.01287
\(321\) 0 0
\(322\) −8.54420 −0.476150
\(323\) 7.35026 0.408980
\(324\) 0 0
\(325\) 5.83146 0.323471
\(326\) 9.04349 0.500873
\(327\) 0 0
\(328\) 42.4323 2.34293
\(329\) −11.8315 −0.652289
\(330\) 0 0
\(331\) 6.26187 0.344183 0.172092 0.985081i \(-0.444947\pi\)
0.172092 + 0.985081i \(0.444947\pi\)
\(332\) −54.7875 −3.00685
\(333\) 0 0
\(334\) −30.1622 −1.65040
\(335\) −10.8568 −0.593173
\(336\) 0 0
\(337\) −15.8700 −0.864495 −0.432248 0.901755i \(-0.642279\pi\)
−0.432248 + 0.901755i \(0.642279\pi\)
\(338\) 56.1935 3.05652
\(339\) 0 0
\(340\) −28.0689 −1.52225
\(341\) −5.28726 −0.286321
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 49.7704 2.68344
\(345\) 0 0
\(346\) 24.0240 1.29154
\(347\) −6.79147 −0.364585 −0.182293 0.983244i \(-0.558352\pi\)
−0.182293 + 0.983244i \(0.558352\pi\)
\(348\) 0 0
\(349\) 26.7489 1.43184 0.715919 0.698183i \(-0.246008\pi\)
0.715919 + 0.698183i \(0.246008\pi\)
\(350\) −2.67513 −0.142992
\(351\) 0 0
\(352\) 15.9502 0.850147
\(353\) 16.8627 0.897512 0.448756 0.893654i \(-0.351867\pi\)
0.448756 + 0.893654i \(0.351867\pi\)
\(354\) 0 0
\(355\) 15.5369 0.824613
\(356\) −37.2506 −1.97428
\(357\) 0 0
\(358\) 70.1886 3.70958
\(359\) 3.79289 0.200181 0.100091 0.994978i \(-0.468087\pi\)
0.100091 + 0.994978i \(0.468087\pi\)
\(360\) 0 0
\(361\) −17.1768 −0.904042
\(362\) −29.9610 −1.57471
\(363\) 0 0
\(364\) −30.0689 −1.57604
\(365\) −11.3684 −0.595047
\(366\) 0 0
\(367\) −6.36977 −0.332500 −0.166250 0.986084i \(-0.553166\pi\)
−0.166250 + 0.986084i \(0.553166\pi\)
\(368\) 39.2057 2.04374
\(369\) 0 0
\(370\) −22.8568 −1.18827
\(371\) −0.231548 −0.0120214
\(372\) 0 0
\(373\) −21.3317 −1.10451 −0.552257 0.833674i \(-0.686233\pi\)
−0.552257 + 0.833674i \(0.686233\pi\)
\(374\) −14.5623 −0.752998
\(375\) 0 0
\(376\) 99.8999 5.15194
\(377\) 21.0640 1.08485
\(378\) 0 0
\(379\) 24.7875 1.27325 0.636624 0.771174i \(-0.280330\pi\)
0.636624 + 0.771174i \(0.280330\pi\)
\(380\) −6.96239 −0.357163
\(381\) 0 0
\(382\) −29.9610 −1.53294
\(383\) 5.45817 0.278900 0.139450 0.990229i \(-0.455467\pi\)
0.139450 + 0.990229i \(0.455467\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) 1.61801 0.0823544
\(387\) 0 0
\(388\) −4.31265 −0.218942
\(389\) 13.7235 0.695811 0.347906 0.937530i \(-0.386893\pi\)
0.347906 + 0.937530i \(0.386893\pi\)
\(390\) 0 0
\(391\) −17.3865 −0.879271
\(392\) 8.44358 0.426465
\(393\) 0 0
\(394\) −40.9438 −2.06272
\(395\) 1.96968 0.0991055
\(396\) 0 0
\(397\) 2.11142 0.105969 0.0529846 0.998595i \(-0.483127\pi\)
0.0529846 + 0.998595i \(0.483127\pi\)
\(398\) −33.6058 −1.68451
\(399\) 0 0
\(400\) 12.2750 0.613752
\(401\) −19.1490 −0.956257 −0.478128 0.878290i \(-0.658685\pi\)
−0.478128 + 0.878290i \(0.658685\pi\)
\(402\) 0 0
\(403\) −30.8324 −1.53587
\(404\) −38.2252 −1.90178
\(405\) 0 0
\(406\) −9.66291 −0.479562
\(407\) −8.54420 −0.423520
\(408\) 0 0
\(409\) −18.6883 −0.924077 −0.462039 0.886860i \(-0.652882\pi\)
−0.462039 + 0.886860i \(0.652882\pi\)
\(410\) 13.4436 0.663931
\(411\) 0 0
\(412\) 21.7562 1.07185
\(413\) 13.5999 0.669208
\(414\) 0 0
\(415\) −10.6253 −0.521575
\(416\) 93.0127 4.56032
\(417\) 0 0
\(418\) −3.61213 −0.176675
\(419\) 0.773377 0.0377819 0.0188910 0.999822i \(-0.493986\pi\)
0.0188910 + 0.999822i \(0.493986\pi\)
\(420\) 0 0
\(421\) −10.5198 −0.512702 −0.256351 0.966584i \(-0.582520\pi\)
−0.256351 + 0.966584i \(0.582520\pi\)
\(422\) −11.8740 −0.578017
\(423\) 0 0
\(424\) 1.95509 0.0949478
\(425\) −5.44358 −0.264053
\(426\) 0 0
\(427\) 1.41327 0.0683927
\(428\) 59.4880 2.87546
\(429\) 0 0
\(430\) 15.7685 0.760422
\(431\) 24.7308 1.19124 0.595621 0.803265i \(-0.296906\pi\)
0.595621 + 0.803265i \(0.296906\pi\)
\(432\) 0 0
\(433\) −18.5599 −0.891933 −0.445967 0.895050i \(-0.647140\pi\)
−0.445967 + 0.895050i \(0.647140\pi\)
\(434\) 14.1441 0.678939
\(435\) 0 0
\(436\) −11.2750 −0.539976
\(437\) −4.31265 −0.206302
\(438\) 0 0
\(439\) −1.42548 −0.0680347 −0.0340173 0.999421i \(-0.510830\pi\)
−0.0340173 + 0.999421i \(0.510830\pi\)
\(440\) 8.44358 0.402532
\(441\) 0 0
\(442\) −84.9194 −4.03920
\(443\) −40.1925 −1.90960 −0.954802 0.297242i \(-0.903933\pi\)
−0.954802 + 0.297242i \(0.903933\pi\)
\(444\) 0 0
\(445\) −7.22425 −0.342462
\(446\) −20.8143 −0.985586
\(447\) 0 0
\(448\) −18.1187 −0.856029
\(449\) −12.6556 −0.597256 −0.298628 0.954370i \(-0.596529\pi\)
−0.298628 + 0.954370i \(0.596529\pi\)
\(450\) 0 0
\(451\) 5.02539 0.236636
\(452\) 48.2130 2.26775
\(453\) 0 0
\(454\) 27.9511 1.31181
\(455\) −5.83146 −0.273383
\(456\) 0 0
\(457\) −0.544198 −0.0254565 −0.0127283 0.999919i \(-0.504052\pi\)
−0.0127283 + 0.999919i \(0.504052\pi\)
\(458\) −78.7875 −3.68150
\(459\) 0 0
\(460\) 16.4690 0.767870
\(461\) 11.5755 0.539123 0.269562 0.962983i \(-0.413121\pi\)
0.269562 + 0.962983i \(0.413121\pi\)
\(462\) 0 0
\(463\) 23.7948 1.10584 0.552919 0.833235i \(-0.313514\pi\)
0.552919 + 0.833235i \(0.313514\pi\)
\(464\) 44.3390 2.05839
\(465\) 0 0
\(466\) 23.3561 1.08195
\(467\) 2.66784 0.123453 0.0617264 0.998093i \(-0.480339\pi\)
0.0617264 + 0.998093i \(0.480339\pi\)
\(468\) 0 0
\(469\) 10.8568 0.501323
\(470\) 31.6507 1.45994
\(471\) 0 0
\(472\) −114.832 −5.28557
\(473\) 5.89446 0.271028
\(474\) 0 0
\(475\) −1.35026 −0.0619543
\(476\) 28.0689 1.28654
\(477\) 0 0
\(478\) 56.7513 2.59574
\(479\) 10.7104 0.489369 0.244685 0.969603i \(-0.421316\pi\)
0.244685 + 0.969603i \(0.421316\pi\)
\(480\) 0 0
\(481\) −49.8251 −2.27183
\(482\) −24.9805 −1.13783
\(483\) 0 0
\(484\) 5.15633 0.234378
\(485\) −0.836381 −0.0379781
\(486\) 0 0
\(487\) 17.4314 0.789891 0.394945 0.918705i \(-0.370764\pi\)
0.394945 + 0.918705i \(0.370764\pi\)
\(488\) −11.9330 −0.540183
\(489\) 0 0
\(490\) 2.67513 0.120850
\(491\) 28.3693 1.28029 0.640145 0.768254i \(-0.278874\pi\)
0.640145 + 0.768254i \(0.278874\pi\)
\(492\) 0 0
\(493\) −19.6629 −0.885573
\(494\) −21.0640 −0.947712
\(495\) 0 0
\(496\) −64.9013 −2.91415
\(497\) −15.5369 −0.696925
\(498\) 0 0
\(499\) 27.4763 1.23001 0.615003 0.788524i \(-0.289155\pi\)
0.615003 + 0.788524i \(0.289155\pi\)
\(500\) 5.15633 0.230598
\(501\) 0 0
\(502\) −5.01951 −0.224032
\(503\) −20.2981 −0.905046 −0.452523 0.891753i \(-0.649476\pi\)
−0.452523 + 0.891753i \(0.649476\pi\)
\(504\) 0 0
\(505\) −7.41327 −0.329886
\(506\) 8.54420 0.379836
\(507\) 0 0
\(508\) −87.4636 −3.88057
\(509\) 24.2619 1.07539 0.537694 0.843140i \(-0.319296\pi\)
0.537694 + 0.843140i \(0.319296\pi\)
\(510\) 0 0
\(511\) 11.3684 0.502907
\(512\) −11.5017 −0.508306
\(513\) 0 0
\(514\) −72.6009 −3.20229
\(515\) 4.21933 0.185926
\(516\) 0 0
\(517\) 11.8315 0.520347
\(518\) 22.8568 1.00427
\(519\) 0 0
\(520\) 49.2384 2.15925
\(521\) −2.20123 −0.0964377 −0.0482188 0.998837i \(-0.515354\pi\)
−0.0482188 + 0.998837i \(0.515354\pi\)
\(522\) 0 0
\(523\) −22.1378 −0.968017 −0.484008 0.875063i \(-0.660820\pi\)
−0.484008 + 0.875063i \(0.660820\pi\)
\(524\) 51.1754 2.23561
\(525\) 0 0
\(526\) −34.2736 −1.49440
\(527\) 28.7816 1.25375
\(528\) 0 0
\(529\) −12.7988 −0.556468
\(530\) 0.619421 0.0269059
\(531\) 0 0
\(532\) 6.96239 0.301858
\(533\) 29.3054 1.26936
\(534\) 0 0
\(535\) 11.5369 0.498784
\(536\) −91.6707 −3.95957
\(537\) 0 0
\(538\) 16.7513 0.722200
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 23.0640 0.991597 0.495799 0.868438i \(-0.334875\pi\)
0.495799 + 0.868438i \(0.334875\pi\)
\(542\) −15.3503 −0.659350
\(543\) 0 0
\(544\) −86.8261 −3.72264
\(545\) −2.18664 −0.0936655
\(546\) 0 0
\(547\) 21.3766 0.913998 0.456999 0.889467i \(-0.348924\pi\)
0.456999 + 0.889467i \(0.348924\pi\)
\(548\) 56.6820 2.42133
\(549\) 0 0
\(550\) 2.67513 0.114068
\(551\) −4.87732 −0.207781
\(552\) 0 0
\(553\) −1.96968 −0.0837594
\(554\) −22.3576 −0.949882
\(555\) 0 0
\(556\) 35.5125 1.50606
\(557\) 9.19394 0.389560 0.194780 0.980847i \(-0.437601\pi\)
0.194780 + 0.980847i \(0.437601\pi\)
\(558\) 0 0
\(559\) 34.3733 1.45384
\(560\) −12.2750 −0.518715
\(561\) 0 0
\(562\) 22.6009 0.953360
\(563\) 9.79877 0.412969 0.206484 0.978450i \(-0.433798\pi\)
0.206484 + 0.978450i \(0.433798\pi\)
\(564\) 0 0
\(565\) 9.35026 0.393368
\(566\) 2.23743 0.0940461
\(567\) 0 0
\(568\) 131.187 5.50449
\(569\) 33.5125 1.40492 0.702458 0.711725i \(-0.252086\pi\)
0.702458 + 0.711725i \(0.252086\pi\)
\(570\) 0 0
\(571\) 43.1392 1.80532 0.902659 0.430356i \(-0.141612\pi\)
0.902659 + 0.430356i \(0.141612\pi\)
\(572\) 30.0689 1.25724
\(573\) 0 0
\(574\) −13.4436 −0.561124
\(575\) 3.19394 0.133196
\(576\) 0 0
\(577\) −14.8510 −0.618254 −0.309127 0.951021i \(-0.600037\pi\)
−0.309127 + 0.951021i \(0.600037\pi\)
\(578\) 33.7938 1.40564
\(579\) 0 0
\(580\) 18.6253 0.773374
\(581\) 10.6253 0.440812
\(582\) 0 0
\(583\) 0.231548 0.00958974
\(584\) −95.9897 −3.97208
\(585\) 0 0
\(586\) 7.27267 0.300431
\(587\) −14.7938 −0.610607 −0.305304 0.952255i \(-0.598758\pi\)
−0.305304 + 0.952255i \(0.598758\pi\)
\(588\) 0 0
\(589\) 7.13918 0.294165
\(590\) −36.3815 −1.49780
\(591\) 0 0
\(592\) −104.880 −4.31056
\(593\) 27.4191 1.12597 0.562985 0.826467i \(-0.309653\pi\)
0.562985 + 0.826467i \(0.309653\pi\)
\(594\) 0 0
\(595\) 5.44358 0.223165
\(596\) −117.626 −4.81814
\(597\) 0 0
\(598\) 49.8251 2.03750
\(599\) −11.3258 −0.462761 −0.231380 0.972863i \(-0.574324\pi\)
−0.231380 + 0.972863i \(0.574324\pi\)
\(600\) 0 0
\(601\) 15.5393 0.633860 0.316930 0.948449i \(-0.397348\pi\)
0.316930 + 0.948449i \(0.397348\pi\)
\(602\) −15.7685 −0.642674
\(603\) 0 0
\(604\) −16.7308 −0.680768
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) 17.7235 0.719377 0.359688 0.933073i \(-0.382883\pi\)
0.359688 + 0.933073i \(0.382883\pi\)
\(608\) −21.5369 −0.873437
\(609\) 0 0
\(610\) −3.78067 −0.153075
\(611\) 68.9946 2.79122
\(612\) 0 0
\(613\) 22.2941 0.900450 0.450225 0.892915i \(-0.351344\pi\)
0.450225 + 0.892915i \(0.351344\pi\)
\(614\) −22.3733 −0.902912
\(615\) 0 0
\(616\) −8.44358 −0.340202
\(617\) 30.9438 1.24575 0.622876 0.782321i \(-0.285964\pi\)
0.622876 + 0.782321i \(0.285964\pi\)
\(618\) 0 0
\(619\) 32.4119 1.30274 0.651371 0.758759i \(-0.274194\pi\)
0.651371 + 0.758759i \(0.274194\pi\)
\(620\) −27.2628 −1.09490
\(621\) 0 0
\(622\) −11.8677 −0.475850
\(623\) 7.22425 0.289434
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 79.6893 3.18502
\(627\) 0 0
\(628\) −27.9756 −1.11635
\(629\) 46.5111 1.85452
\(630\) 0 0
\(631\) −27.3258 −1.08782 −0.543912 0.839142i \(-0.683057\pi\)
−0.543912 + 0.839142i \(0.683057\pi\)
\(632\) 16.6312 0.661553
\(633\) 0 0
\(634\) −41.1998 −1.63625
\(635\) −16.9624 −0.673132
\(636\) 0 0
\(637\) 5.83146 0.231051
\(638\) 9.66291 0.382558
\(639\) 0 0
\(640\) 16.5696 0.654971
\(641\) −19.4460 −0.768069 −0.384034 0.923319i \(-0.625466\pi\)
−0.384034 + 0.923319i \(0.625466\pi\)
\(642\) 0 0
\(643\) 5.29314 0.208741 0.104370 0.994538i \(-0.466717\pi\)
0.104370 + 0.994538i \(0.466717\pi\)
\(644\) −16.4690 −0.648969
\(645\) 0 0
\(646\) 19.6629 0.773627
\(647\) −35.0966 −1.37979 −0.689896 0.723909i \(-0.742344\pi\)
−0.689896 + 0.723909i \(0.742344\pi\)
\(648\) 0 0
\(649\) −13.5999 −0.533843
\(650\) 15.5999 0.611879
\(651\) 0 0
\(652\) 17.4314 0.682665
\(653\) −27.7988 −1.08785 −0.543925 0.839134i \(-0.683062\pi\)
−0.543925 + 0.839134i \(0.683062\pi\)
\(654\) 0 0
\(655\) 9.92478 0.387793
\(656\) 61.6869 2.40847
\(657\) 0 0
\(658\) −31.6507 −1.23387
\(659\) −19.6180 −0.764209 −0.382105 0.924119i \(-0.624801\pi\)
−0.382105 + 0.924119i \(0.624801\pi\)
\(660\) 0 0
\(661\) 21.5633 0.838713 0.419357 0.907822i \(-0.362256\pi\)
0.419357 + 0.907822i \(0.362256\pi\)
\(662\) 16.7513 0.651058
\(663\) 0 0
\(664\) −89.7156 −3.48164
\(665\) 1.35026 0.0523609
\(666\) 0 0
\(667\) 11.5369 0.446711
\(668\) −58.1378 −2.24942
\(669\) 0 0
\(670\) −29.0435 −1.12205
\(671\) −1.41327 −0.0545585
\(672\) 0 0
\(673\) −21.0679 −0.812109 −0.406054 0.913849i \(-0.633096\pi\)
−0.406054 + 0.913849i \(0.633096\pi\)
\(674\) −42.4544 −1.63528
\(675\) 0 0
\(676\) 108.313 4.16589
\(677\) −34.5174 −1.32661 −0.663306 0.748349i \(-0.730847\pi\)
−0.663306 + 0.748349i \(0.730847\pi\)
\(678\) 0 0
\(679\) 0.836381 0.0320973
\(680\) −45.9633 −1.76261
\(681\) 0 0
\(682\) −14.1441 −0.541606
\(683\) −33.7802 −1.29256 −0.646282 0.763099i \(-0.723677\pi\)
−0.646282 + 0.763099i \(0.723677\pi\)
\(684\) 0 0
\(685\) 10.9927 0.420010
\(686\) −2.67513 −0.102137
\(687\) 0 0
\(688\) 72.3547 2.75850
\(689\) 1.35026 0.0514409
\(690\) 0 0
\(691\) −13.8618 −0.527327 −0.263663 0.964615i \(-0.584931\pi\)
−0.263663 + 0.964615i \(0.584931\pi\)
\(692\) 46.3063 1.76030
\(693\) 0 0
\(694\) −18.1681 −0.689651
\(695\) 6.88717 0.261245
\(696\) 0 0
\(697\) −27.3561 −1.03619
\(698\) 71.5569 2.70847
\(699\) 0 0
\(700\) −5.15633 −0.194891
\(701\) −40.5256 −1.53063 −0.765316 0.643655i \(-0.777417\pi\)
−0.765316 + 0.643655i \(0.777417\pi\)
\(702\) 0 0
\(703\) 11.5369 0.435123
\(704\) 18.1187 0.682875
\(705\) 0 0
\(706\) 45.1100 1.69774
\(707\) 7.41327 0.278805
\(708\) 0 0
\(709\) −0.850969 −0.0319588 −0.0159794 0.999872i \(-0.505087\pi\)
−0.0159794 + 0.999872i \(0.505087\pi\)
\(710\) 41.5633 1.55984
\(711\) 0 0
\(712\) −60.9986 −2.28602
\(713\) −16.8872 −0.632429
\(714\) 0 0
\(715\) 5.83146 0.218084
\(716\) 135.289 5.05598
\(717\) 0 0
\(718\) 10.1465 0.378663
\(719\) −22.5769 −0.841976 −0.420988 0.907066i \(-0.638317\pi\)
−0.420988 + 0.907066i \(0.638317\pi\)
\(720\) 0 0
\(721\) −4.21933 −0.157136
\(722\) −45.9502 −1.71009
\(723\) 0 0
\(724\) −57.7499 −2.14626
\(725\) 3.61213 0.134151
\(726\) 0 0
\(727\) 12.5174 0.464244 0.232122 0.972687i \(-0.425433\pi\)
0.232122 + 0.972687i \(0.425433\pi\)
\(728\) −49.2384 −1.82490
\(729\) 0 0
\(730\) −30.4119 −1.12559
\(731\) −32.0870 −1.18678
\(732\) 0 0
\(733\) 16.6678 0.615641 0.307820 0.951445i \(-0.400400\pi\)
0.307820 + 0.951445i \(0.400400\pi\)
\(734\) −17.0400 −0.628957
\(735\) 0 0
\(736\) 50.9438 1.87781
\(737\) −10.8568 −0.399917
\(738\) 0 0
\(739\) 42.7005 1.57076 0.785382 0.619011i \(-0.212467\pi\)
0.785382 + 0.619011i \(0.212467\pi\)
\(740\) −44.0567 −1.61956
\(741\) 0 0
\(742\) −0.619421 −0.0227397
\(743\) 19.6873 0.722259 0.361129 0.932516i \(-0.382391\pi\)
0.361129 + 0.932516i \(0.382391\pi\)
\(744\) 0 0
\(745\) −22.8119 −0.835765
\(746\) −57.0651 −2.08930
\(747\) 0 0
\(748\) −28.0689 −1.02630
\(749\) −11.5369 −0.421549
\(750\) 0 0
\(751\) −5.85940 −0.213813 −0.106906 0.994269i \(-0.534094\pi\)
−0.106906 + 0.994269i \(0.534094\pi\)
\(752\) 145.232 5.29605
\(753\) 0 0
\(754\) 56.3488 2.05210
\(755\) −3.24472 −0.118088
\(756\) 0 0
\(757\) −40.5863 −1.47513 −0.737567 0.675274i \(-0.764025\pi\)
−0.737567 + 0.675274i \(0.764025\pi\)
\(758\) 66.3098 2.40848
\(759\) 0 0
\(760\) −11.4010 −0.413559
\(761\) −21.8472 −0.791960 −0.395980 0.918259i \(-0.629595\pi\)
−0.395980 + 0.918259i \(0.629595\pi\)
\(762\) 0 0
\(763\) 2.18664 0.0791618
\(764\) −57.7499 −2.08932
\(765\) 0 0
\(766\) 14.6013 0.527567
\(767\) −79.3073 −2.86362
\(768\) 0 0
\(769\) 45.2892 1.63317 0.816585 0.577226i \(-0.195865\pi\)
0.816585 + 0.577226i \(0.195865\pi\)
\(770\) −2.67513 −0.0964050
\(771\) 0 0
\(772\) 3.11871 0.112245
\(773\) 33.8153 1.21625 0.608125 0.793841i \(-0.291922\pi\)
0.608125 + 0.793841i \(0.291922\pi\)
\(774\) 0 0
\(775\) −5.28726 −0.189924
\(776\) −7.06205 −0.253513
\(777\) 0 0
\(778\) 36.7123 1.31620
\(779\) −6.78560 −0.243119
\(780\) 0 0
\(781\) 15.5369 0.555954
\(782\) −46.5111 −1.66323
\(783\) 0 0
\(784\) 12.2750 0.438394
\(785\) −5.42548 −0.193644
\(786\) 0 0
\(787\) 1.27504 0.0454502 0.0227251 0.999742i \(-0.492766\pi\)
0.0227251 + 0.999742i \(0.492766\pi\)
\(788\) −78.9194 −2.81139
\(789\) 0 0
\(790\) 5.26916 0.187468
\(791\) −9.35026 −0.332457
\(792\) 0 0
\(793\) −8.24140 −0.292661
\(794\) 5.64832 0.200452
\(795\) 0 0
\(796\) −64.7753 −2.29590
\(797\) 42.5256 1.50634 0.753168 0.657829i \(-0.228525\pi\)
0.753168 + 0.657829i \(0.228525\pi\)
\(798\) 0 0
\(799\) −64.4055 −2.27850
\(800\) 15.9502 0.563924
\(801\) 0 0
\(802\) −51.2262 −1.80886
\(803\) −11.3684 −0.401181
\(804\) 0 0
\(805\) −3.19394 −0.112571
\(806\) −82.4807 −2.90526
\(807\) 0 0
\(808\) −62.5945 −2.20207
\(809\) 14.7151 0.517356 0.258678 0.965964i \(-0.416713\pi\)
0.258678 + 0.965964i \(0.416713\pi\)
\(810\) 0 0
\(811\) 51.7743 1.81804 0.909021 0.416750i \(-0.136831\pi\)
0.909021 + 0.416750i \(0.136831\pi\)
\(812\) −18.6253 −0.653620
\(813\) 0 0
\(814\) −22.8568 −0.801132
\(815\) 3.38058 0.118417
\(816\) 0 0
\(817\) −7.95906 −0.278452
\(818\) −49.9937 −1.74799
\(819\) 0 0
\(820\) 25.9126 0.904906
\(821\) −2.64974 −0.0924765 −0.0462383 0.998930i \(-0.514723\pi\)
−0.0462383 + 0.998930i \(0.514723\pi\)
\(822\) 0 0
\(823\) 5.76845 0.201076 0.100538 0.994933i \(-0.467944\pi\)
0.100538 + 0.994933i \(0.467944\pi\)
\(824\) 35.6263 1.24110
\(825\) 0 0
\(826\) 36.3815 1.26588
\(827\) 13.4920 0.469163 0.234581 0.972096i \(-0.424628\pi\)
0.234581 + 0.972096i \(0.424628\pi\)
\(828\) 0 0
\(829\) −4.70052 −0.163256 −0.0816280 0.996663i \(-0.526012\pi\)
−0.0816280 + 0.996663i \(0.526012\pi\)
\(830\) −28.4241 −0.986614
\(831\) 0 0
\(832\) 105.658 3.66305
\(833\) −5.44358 −0.188609
\(834\) 0 0
\(835\) −11.2750 −0.390189
\(836\) −6.96239 −0.240799
\(837\) 0 0
\(838\) 2.06888 0.0714684
\(839\) 38.8045 1.33968 0.669839 0.742506i \(-0.266363\pi\)
0.669839 + 0.742506i \(0.266363\pi\)
\(840\) 0 0
\(841\) −15.9525 −0.550088
\(842\) −28.1417 −0.969828
\(843\) 0 0
\(844\) −22.8872 −0.787809
\(845\) 21.0059 0.722624
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) 2.84226 0.0976036
\(849\) 0 0
\(850\) −14.5623 −0.499483
\(851\) −27.2896 −0.935476
\(852\) 0 0
\(853\) 20.6824 0.708153 0.354076 0.935217i \(-0.384795\pi\)
0.354076 + 0.935217i \(0.384795\pi\)
\(854\) 3.78067 0.129372
\(855\) 0 0
\(856\) 97.4128 3.32950
\(857\) −26.3453 −0.899940 −0.449970 0.893044i \(-0.648565\pi\)
−0.449970 + 0.893044i \(0.648565\pi\)
\(858\) 0 0
\(859\) −8.51151 −0.290409 −0.145205 0.989402i \(-0.546384\pi\)
−0.145205 + 0.989402i \(0.546384\pi\)
\(860\) 30.3938 1.03642
\(861\) 0 0
\(862\) 66.1582 2.25336
\(863\) −7.56722 −0.257591 −0.128796 0.991671i \(-0.541111\pi\)
−0.128796 + 0.991671i \(0.541111\pi\)
\(864\) 0 0
\(865\) 8.98049 0.305346
\(866\) −49.6502 −1.68718
\(867\) 0 0
\(868\) 27.2628 0.925360
\(869\) 1.96968 0.0668169
\(870\) 0 0
\(871\) −63.3112 −2.14522
\(872\) −18.4631 −0.625239
\(873\) 0 0
\(874\) −11.5369 −0.390242
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) 17.2955 0.584028 0.292014 0.956414i \(-0.405675\pi\)
0.292014 + 0.956414i \(0.405675\pi\)
\(878\) −3.81336 −0.128695
\(879\) 0 0
\(880\) 12.2750 0.413791
\(881\) −20.4504 −0.688992 −0.344496 0.938788i \(-0.611950\pi\)
−0.344496 + 0.938788i \(0.611950\pi\)
\(882\) 0 0
\(883\) −49.6589 −1.67116 −0.835578 0.549371i \(-0.814867\pi\)
−0.835578 + 0.549371i \(0.814867\pi\)
\(884\) −163.682 −5.50524
\(885\) 0 0
\(886\) −107.520 −3.61221
\(887\) 47.1100 1.58180 0.790900 0.611946i \(-0.209613\pi\)
0.790900 + 0.611946i \(0.209613\pi\)
\(888\) 0 0
\(889\) 16.9624 0.568900
\(890\) −19.3258 −0.647803
\(891\) 0 0
\(892\) −40.1197 −1.34331
\(893\) −15.9756 −0.534602
\(894\) 0 0
\(895\) 26.2374 0.877020
\(896\) −16.5696 −0.553551
\(897\) 0 0
\(898\) −33.8554 −1.12977
\(899\) −19.0982 −0.636962
\(900\) 0 0
\(901\) −1.26045 −0.0419917
\(902\) 13.4436 0.447622
\(903\) 0 0
\(904\) 78.9497 2.62583
\(905\) −11.1998 −0.372294
\(906\) 0 0
\(907\) 14.4591 0.480107 0.240054 0.970760i \(-0.422835\pi\)
0.240054 + 0.970760i \(0.422835\pi\)
\(908\) 53.8759 1.78793
\(909\) 0 0
\(910\) −15.5999 −0.517132
\(911\) 31.5369 1.04486 0.522432 0.852681i \(-0.325025\pi\)
0.522432 + 0.852681i \(0.325025\pi\)
\(912\) 0 0
\(913\) −10.6253 −0.351646
\(914\) −1.45580 −0.0481536
\(915\) 0 0
\(916\) −151.863 −5.01770
\(917\) −9.92478 −0.327745
\(918\) 0 0
\(919\) 5.26328 0.173620 0.0868098 0.996225i \(-0.472333\pi\)
0.0868098 + 0.996225i \(0.472333\pi\)
\(920\) 26.9683 0.889117
\(921\) 0 0
\(922\) 30.9659 1.01981
\(923\) 90.6028 2.98223
\(924\) 0 0
\(925\) −8.54420 −0.280932
\(926\) 63.6542 2.09181
\(927\) 0 0
\(928\) 57.6140 1.89127
\(929\) −26.0508 −0.854699 −0.427349 0.904087i \(-0.640553\pi\)
−0.427349 + 0.904087i \(0.640553\pi\)
\(930\) 0 0
\(931\) −1.35026 −0.0442530
\(932\) 45.0191 1.47465
\(933\) 0 0
\(934\) 7.13681 0.233524
\(935\) −5.44358 −0.178024
\(936\) 0 0
\(937\) −29.3439 −0.958624 −0.479312 0.877645i \(-0.659114\pi\)
−0.479312 + 0.877645i \(0.659114\pi\)
\(938\) 29.0435 0.948304
\(939\) 0 0
\(940\) 61.0068 1.98982
\(941\) 28.6375 0.933556 0.466778 0.884374i \(-0.345415\pi\)
0.466778 + 0.884374i \(0.345415\pi\)
\(942\) 0 0
\(943\) 16.0508 0.522685
\(944\) −166.939 −5.43341
\(945\) 0 0
\(946\) 15.7685 0.512677
\(947\) 52.8178 1.71635 0.858174 0.513358i \(-0.171599\pi\)
0.858174 + 0.513358i \(0.171599\pi\)
\(948\) 0 0
\(949\) −66.2941 −2.15200
\(950\) −3.61213 −0.117193
\(951\) 0 0
\(952\) 45.9633 1.48968
\(953\) −37.1939 −1.20483 −0.602415 0.798183i \(-0.705795\pi\)
−0.602415 + 0.798183i \(0.705795\pi\)
\(954\) 0 0
\(955\) −11.1998 −0.362418
\(956\) 109.388 3.53787
\(957\) 0 0
\(958\) 28.6516 0.925693
\(959\) −10.9927 −0.354973
\(960\) 0 0
\(961\) −3.04491 −0.0982228
\(962\) −133.289 −4.29740
\(963\) 0 0
\(964\) −48.1500 −1.55081
\(965\) 0.604833 0.0194703
\(966\) 0 0
\(967\) 4.07125 0.130923 0.0654613 0.997855i \(-0.479148\pi\)
0.0654613 + 0.997855i \(0.479148\pi\)
\(968\) 8.44358 0.271387
\(969\) 0 0
\(970\) −2.23743 −0.0718395
\(971\) −0.773377 −0.0248188 −0.0124094 0.999923i \(-0.503950\pi\)
−0.0124094 + 0.999923i \(0.503950\pi\)
\(972\) 0 0
\(973\) −6.88717 −0.220792
\(974\) 46.6312 1.49416
\(975\) 0 0
\(976\) −17.3479 −0.555292
\(977\) 37.8740 1.21170 0.605848 0.795580i \(-0.292834\pi\)
0.605848 + 0.795580i \(0.292834\pi\)
\(978\) 0 0
\(979\) −7.22425 −0.230888
\(980\) 5.15633 0.164713
\(981\) 0 0
\(982\) 75.8916 2.42180
\(983\) 15.5794 0.496907 0.248453 0.968644i \(-0.420078\pi\)
0.248453 + 0.968644i \(0.420078\pi\)
\(984\) 0 0
\(985\) −15.3054 −0.487669
\(986\) −52.6009 −1.67515
\(987\) 0 0
\(988\) −40.6009 −1.29169
\(989\) 18.8265 0.598649
\(990\) 0 0
\(991\) 27.0982 0.860804 0.430402 0.902637i \(-0.358372\pi\)
0.430402 + 0.902637i \(0.358372\pi\)
\(992\) −84.3327 −2.67756
\(993\) 0 0
\(994\) −41.5633 −1.31831
\(995\) −12.5623 −0.398252
\(996\) 0 0
\(997\) 50.4060 1.59637 0.798187 0.602410i \(-0.205793\pi\)
0.798187 + 0.602410i \(0.205793\pi\)
\(998\) 73.5026 2.32668
\(999\) 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 3465.2.a.bh.1.3 3
3.2 odd 2 385.2.a.f.1.1 3
12.11 even 2 6160.2.a.bn.1.3 3
15.2 even 4 1925.2.b.n.1849.1 6
15.8 even 4 1925.2.b.n.1849.6 6
15.14 odd 2 1925.2.a.v.1.3 3
21.20 even 2 2695.2.a.g.1.1 3
33.32 even 2 4235.2.a.q.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
385.2.a.f.1.1 3 3.2 odd 2
1925.2.a.v.1.3 3 15.14 odd 2
1925.2.b.n.1849.1 6 15.2 even 4
1925.2.b.n.1849.6 6 15.8 even 4
2695.2.a.g.1.1 3 21.20 even 2
3465.2.a.bh.1.3 3 1.1 even 1 trivial
4235.2.a.q.1.3 3 33.32 even 2
6160.2.a.bn.1.3 3 12.11 even 2