Properties

Label 2475.2.a.bj.1.1
Level $2475$
Weight $2$
Character 2475.1
Self dual yes
Analytic conductor $19.763$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2475,2,Mod(1,2475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2475, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2475.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(19.7629745003\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.48704.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 6x^{2} + 4x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 495)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.28632\) of defining polynomial
Character \(\chi\) \(=\) 2475.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.28632 q^{2} +3.22727 q^{4} -2.51962 q^{7} -2.80595 q^{8} +O(q^{10})\) \(q-2.28632 q^{2} +3.22727 q^{4} -2.51962 q^{7} -2.80595 q^{8} +1.00000 q^{11} -6.05302 q^{13} +5.76067 q^{14} -0.0392472 q^{16} +4.97417 q^{17} -7.02720 q^{19} -2.28632 q^{22} -4.45455 q^{23} +13.8392 q^{26} -8.13152 q^{28} +0.921150 q^{29} +3.03925 q^{31} +5.70163 q^{32} -11.3726 q^{34} +3.49380 q^{37} +16.0664 q^{38} -10.0664 q^{41} -1.48038 q^{43} +3.22727 q^{44} +10.1845 q^{46} +8.10605 q^{47} -0.651497 q^{49} -19.5348 q^{52} +1.54545 q^{53} +7.06993 q^{56} -2.10605 q^{58} -7.59984 q^{59} +10.1060 q^{61} -6.94870 q^{62} -12.9573 q^{64} +8.69074 q^{67} +16.0530 q^{68} +1.54545 q^{71} -6.05302 q^{73} -7.98795 q^{74} -22.6787 q^{76} -2.51962 q^{77} +11.0272 q^{79} +23.0151 q^{82} +8.50757 q^{83} +3.38462 q^{86} -2.80595 q^{88} +15.1453 q^{89} +15.2513 q^{91} -14.3761 q^{92} -18.5330 q^{94} +11.5998 q^{97} +1.48953 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 8 q^{4} - 4 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 8 q^{4} - 4 q^{7} + 6 q^{8} + 4 q^{11} - 8 q^{13} + 8 q^{14} + 12 q^{16} + 4 q^{17} + 4 q^{19} + 2 q^{22} - 8 q^{23} + 16 q^{26} + 8 q^{28} + 4 q^{29} + 14 q^{32} + 4 q^{34} - 8 q^{37} + 20 q^{38} + 4 q^{41} - 12 q^{43} + 8 q^{44} - 16 q^{46} + 20 q^{49} - 20 q^{52} + 16 q^{53} + 48 q^{56} + 24 q^{58} + 24 q^{59} + 8 q^{61} - 20 q^{62} + 48 q^{68} + 16 q^{71} - 8 q^{73} - 12 q^{74} - 36 q^{76} - 4 q^{77} + 12 q^{79} + 40 q^{82} + 8 q^{83} - 16 q^{86} + 6 q^{88} + 16 q^{89} - 16 q^{91} - 72 q^{92} - 40 q^{94} - 8 q^{97} + 62 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.28632 −1.61667 −0.808337 0.588720i \(-0.799632\pi\)
−0.808337 + 0.588720i \(0.799632\pi\)
\(3\) 0 0
\(4\) 3.22727 1.61364
\(5\) 0 0
\(6\) 0 0
\(7\) −2.51962 −0.952328 −0.476164 0.879356i \(-0.657973\pi\)
−0.476164 + 0.879356i \(0.657973\pi\)
\(8\) −2.80595 −0.992052
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −6.05302 −1.67881 −0.839403 0.543509i \(-0.817095\pi\)
−0.839403 + 0.543509i \(0.817095\pi\)
\(14\) 5.76067 1.53960
\(15\) 0 0
\(16\) −0.0392472 −0.00981179
\(17\) 4.97417 1.20641 0.603207 0.797585i \(-0.293889\pi\)
0.603207 + 0.797585i \(0.293889\pi\)
\(18\) 0 0
\(19\) −7.02720 −1.61215 −0.806075 0.591814i \(-0.798412\pi\)
−0.806075 + 0.591814i \(0.798412\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −2.28632 −0.487446
\(23\) −4.45455 −0.928838 −0.464419 0.885616i \(-0.653737\pi\)
−0.464419 + 0.885616i \(0.653737\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 13.8392 2.71408
\(27\) 0 0
\(28\) −8.13152 −1.53671
\(29\) 0.921150 0.171053 0.0855266 0.996336i \(-0.472743\pi\)
0.0855266 + 0.996336i \(0.472743\pi\)
\(30\) 0 0
\(31\) 3.03925 0.545865 0.272932 0.962033i \(-0.412006\pi\)
0.272932 + 0.962033i \(0.412006\pi\)
\(32\) 5.70163 1.00791
\(33\) 0 0
\(34\) −11.3726 −1.95038
\(35\) 0 0
\(36\) 0 0
\(37\) 3.49380 0.574377 0.287188 0.957874i \(-0.407279\pi\)
0.287188 + 0.957874i \(0.407279\pi\)
\(38\) 16.0664 2.60632
\(39\) 0 0
\(40\) 0 0
\(41\) −10.0664 −1.57211 −0.786057 0.618154i \(-0.787881\pi\)
−0.786057 + 0.618154i \(0.787881\pi\)
\(42\) 0 0
\(43\) −1.48038 −0.225755 −0.112878 0.993609i \(-0.536007\pi\)
−0.112878 + 0.993609i \(0.536007\pi\)
\(44\) 3.22727 0.486530
\(45\) 0 0
\(46\) 10.1845 1.50163
\(47\) 8.10605 1.18239 0.591194 0.806529i \(-0.298657\pi\)
0.591194 + 0.806529i \(0.298657\pi\)
\(48\) 0 0
\(49\) −0.651497 −0.0930710
\(50\) 0 0
\(51\) 0 0
\(52\) −19.5348 −2.70899
\(53\) 1.54545 0.212284 0.106142 0.994351i \(-0.466150\pi\)
0.106142 + 0.994351i \(0.466150\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 7.06993 0.944759
\(57\) 0 0
\(58\) −2.10605 −0.276537
\(59\) −7.59984 −0.989415 −0.494708 0.869059i \(-0.664725\pi\)
−0.494708 + 0.869059i \(0.664725\pi\)
\(60\) 0 0
\(61\) 10.1060 1.29395 0.646973 0.762513i \(-0.276034\pi\)
0.646973 + 0.762513i \(0.276034\pi\)
\(62\) −6.94870 −0.882486
\(63\) 0 0
\(64\) −12.9573 −1.61966
\(65\) 0 0
\(66\) 0 0
\(67\) 8.69074 1.06174 0.530872 0.847452i \(-0.321865\pi\)
0.530872 + 0.847452i \(0.321865\pi\)
\(68\) 16.0530 1.94672
\(69\) 0 0
\(70\) 0 0
\(71\) 1.54545 0.183411 0.0917056 0.995786i \(-0.470768\pi\)
0.0917056 + 0.995786i \(0.470768\pi\)
\(72\) 0 0
\(73\) −6.05302 −0.708453 −0.354226 0.935160i \(-0.615256\pi\)
−0.354226 + 0.935160i \(0.615256\pi\)
\(74\) −7.98795 −0.928580
\(75\) 0 0
\(76\) −22.6787 −2.60142
\(77\) −2.51962 −0.287138
\(78\) 0 0
\(79\) 11.0272 1.24066 0.620328 0.784342i \(-0.286999\pi\)
0.620328 + 0.784342i \(0.286999\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 23.0151 2.54160
\(83\) 8.50757 0.933827 0.466914 0.884303i \(-0.345366\pi\)
0.466914 + 0.884303i \(0.345366\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 3.38462 0.364973
\(87\) 0 0
\(88\) −2.80595 −0.299115
\(89\) 15.1453 1.60540 0.802699 0.596384i \(-0.203397\pi\)
0.802699 + 0.596384i \(0.203397\pi\)
\(90\) 0 0
\(91\) 15.2513 1.59877
\(92\) −14.3761 −1.49881
\(93\) 0 0
\(94\) −18.5330 −1.91154
\(95\) 0 0
\(96\) 0 0
\(97\) 11.5998 1.17779 0.588893 0.808211i \(-0.299564\pi\)
0.588893 + 0.808211i \(0.299564\pi\)
\(98\) 1.48953 0.150466
\(99\) 0 0
\(100\) 0 0
\(101\) −0.118097 −0.0117511 −0.00587556 0.999983i \(-0.501870\pi\)
−0.00587556 + 0.999983i \(0.501870\pi\)
\(102\) 0 0
\(103\) −10.6391 −1.04830 −0.524150 0.851626i \(-0.675617\pi\)
−0.524150 + 0.851626i \(0.675617\pi\)
\(104\) 16.9845 1.66546
\(105\) 0 0
\(106\) −3.53340 −0.343194
\(107\) −18.6921 −1.80703 −0.903517 0.428551i \(-0.859024\pi\)
−0.903517 + 0.428551i \(0.859024\pi\)
\(108\) 0 0
\(109\) −7.14529 −0.684395 −0.342198 0.939628i \(-0.611171\pi\)
−0.342198 + 0.939628i \(0.611171\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0.0988881 0.00934405
\(113\) 2.58470 0.243148 0.121574 0.992582i \(-0.461206\pi\)
0.121574 + 0.992582i \(0.461206\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 2.97280 0.276018
\(117\) 0 0
\(118\) 17.3757 1.59956
\(119\) −12.5330 −1.14890
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −23.1057 −2.09189
\(123\) 0 0
\(124\) 9.80849 0.880828
\(125\) 0 0
\(126\) 0 0
\(127\) −11.4287 −1.01414 −0.507068 0.861906i \(-0.669271\pi\)
−0.507068 + 0.861906i \(0.669271\pi\)
\(128\) 18.2212 1.61055
\(129\) 0 0
\(130\) 0 0
\(131\) 8.10605 0.708229 0.354114 0.935202i \(-0.384782\pi\)
0.354114 + 0.935202i \(0.384782\pi\)
\(132\) 0 0
\(133\) 17.7059 1.53530
\(134\) −19.8699 −1.71649
\(135\) 0 0
\(136\) −13.9573 −1.19683
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) −20.0664 −1.70201 −0.851007 0.525155i \(-0.824007\pi\)
−0.851007 + 0.525155i \(0.824007\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −3.53340 −0.296516
\(143\) −6.05302 −0.506179
\(144\) 0 0
\(145\) 0 0
\(146\) 13.8392 1.14534
\(147\) 0 0
\(148\) 11.2754 0.926836
\(149\) 21.2634 1.74196 0.870982 0.491314i \(-0.163483\pi\)
0.870982 + 0.491314i \(0.163483\pi\)
\(150\) 0 0
\(151\) −7.02720 −0.571865 −0.285933 0.958250i \(-0.592303\pi\)
−0.285933 + 0.958250i \(0.592303\pi\)
\(152\) 19.7179 1.59934
\(153\) 0 0
\(154\) 5.76067 0.464208
\(155\) 0 0
\(156\) 0 0
\(157\) −21.0936 −1.68346 −0.841728 0.539902i \(-0.818461\pi\)
−0.841728 + 0.539902i \(0.818461\pi\)
\(158\) −25.2117 −2.00574
\(159\) 0 0
\(160\) 0 0
\(161\) 11.2238 0.884558
\(162\) 0 0
\(163\) 7.65150 0.599311 0.299656 0.954047i \(-0.403128\pi\)
0.299656 + 0.954047i \(0.403128\pi\)
\(164\) −32.4872 −2.53682
\(165\) 0 0
\(166\) −19.4511 −1.50970
\(167\) 3.49243 0.270252 0.135126 0.990828i \(-0.456856\pi\)
0.135126 + 0.990828i \(0.456856\pi\)
\(168\) 0 0
\(169\) 23.6391 1.81839
\(170\) 0 0
\(171\) 0 0
\(172\) −4.77758 −0.364287
\(173\) 16.9742 1.29052 0.645261 0.763962i \(-0.276749\pi\)
0.645261 + 0.763962i \(0.276749\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −0.0392472 −0.00295837
\(177\) 0 0
\(178\) −34.6270 −2.59541
\(179\) 9.14529 0.683551 0.341776 0.939782i \(-0.388972\pi\)
0.341776 + 0.939782i \(0.388972\pi\)
\(180\) 0 0
\(181\) 22.6391 1.68275 0.841375 0.540452i \(-0.181747\pi\)
0.841375 + 0.540452i \(0.181747\pi\)
\(182\) −34.8695 −2.58470
\(183\) 0 0
\(184\) 12.4992 0.921455
\(185\) 0 0
\(186\) 0 0
\(187\) 4.97417 0.363748
\(188\) 26.1604 1.90795
\(189\) 0 0
\(190\) 0 0
\(191\) 12.2362 0.885380 0.442690 0.896675i \(-0.354024\pi\)
0.442690 + 0.896675i \(0.354024\pi\)
\(192\) 0 0
\(193\) −1.03788 −0.0747081 −0.0373540 0.999302i \(-0.511893\pi\)
−0.0373540 + 0.999302i \(0.511893\pi\)
\(194\) −26.5210 −1.90410
\(195\) 0 0
\(196\) −2.10256 −0.150183
\(197\) 8.06507 0.574613 0.287306 0.957839i \(-0.407240\pi\)
0.287306 + 0.957839i \(0.407240\pi\)
\(198\) 0 0
\(199\) −13.1453 −0.931845 −0.465923 0.884825i \(-0.654277\pi\)
−0.465923 + 0.884825i \(0.654277\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0.270009 0.0189977
\(203\) −2.32095 −0.162899
\(204\) 0 0
\(205\) 0 0
\(206\) 24.3244 1.69476
\(207\) 0 0
\(208\) 0.237564 0.0164721
\(209\) −7.02720 −0.486081
\(210\) 0 0
\(211\) −3.05130 −0.210060 −0.105030 0.994469i \(-0.533494\pi\)
−0.105030 + 0.994469i \(0.533494\pi\)
\(212\) 4.98759 0.342549
\(213\) 0 0
\(214\) 42.7362 2.92139
\(215\) 0 0
\(216\) 0 0
\(217\) −7.65776 −0.519843
\(218\) 16.3365 1.10644
\(219\) 0 0
\(220\) 0 0
\(221\) −30.1088 −2.02534
\(222\) 0 0
\(223\) 13.9483 0.934050 0.467025 0.884244i \(-0.345326\pi\)
0.467025 + 0.884244i \(0.345326\pi\)
\(224\) −14.3660 −0.959865
\(225\) 0 0
\(226\) −5.90945 −0.393091
\(227\) 29.6529 1.96813 0.984065 0.177809i \(-0.0569011\pi\)
0.984065 + 0.177809i \(0.0569011\pi\)
\(228\) 0 0
\(229\) 18.2121 1.20349 0.601744 0.798689i \(-0.294473\pi\)
0.601744 + 0.798689i \(0.294473\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −2.58470 −0.169694
\(233\) −13.0802 −0.856914 −0.428457 0.903562i \(-0.640943\pi\)
−0.428457 + 0.903562i \(0.640943\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −24.5268 −1.59656
\(237\) 0 0
\(238\) 28.6546 1.85740
\(239\) 0.803053 0.0519452 0.0259726 0.999663i \(-0.491732\pi\)
0.0259726 + 0.999663i \(0.491732\pi\)
\(240\) 0 0
\(241\) 3.03925 0.195775 0.0978876 0.995197i \(-0.468791\pi\)
0.0978876 + 0.995197i \(0.468791\pi\)
\(242\) −2.28632 −0.146970
\(243\) 0 0
\(244\) 32.6150 2.08796
\(245\) 0 0
\(246\) 0 0
\(247\) 42.5358 2.70649
\(248\) −8.52797 −0.541526
\(249\) 0 0
\(250\) 0 0
\(251\) 7.36365 0.464789 0.232395 0.972622i \(-0.425344\pi\)
0.232395 + 0.972622i \(0.425344\pi\)
\(252\) 0 0
\(253\) −4.45455 −0.280055
\(254\) 26.1298 1.63953
\(255\) 0 0
\(256\) −15.7451 −0.984071
\(257\) 3.94835 0.246291 0.123146 0.992389i \(-0.460702\pi\)
0.123146 + 0.992389i \(0.460702\pi\)
\(258\) 0 0
\(259\) −8.80305 −0.546995
\(260\) 0 0
\(261\) 0 0
\(262\) −18.5330 −1.14498
\(263\) −28.6405 −1.76605 −0.883023 0.469329i \(-0.844496\pi\)
−0.883023 + 0.469329i \(0.844496\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −40.4814 −2.48207
\(267\) 0 0
\(268\) 28.0474 1.71327
\(269\) 28.2121 1.72012 0.860061 0.510191i \(-0.170425\pi\)
0.860061 + 0.510191i \(0.170425\pi\)
\(270\) 0 0
\(271\) 11.2634 0.684202 0.342101 0.939663i \(-0.388861\pi\)
0.342101 + 0.939663i \(0.388861\pi\)
\(272\) −0.195222 −0.0118371
\(273\) 0 0
\(274\) 13.7179 0.828731
\(275\) 0 0
\(276\) 0 0
\(277\) 29.0165 1.74343 0.871717 0.490010i \(-0.163007\pi\)
0.871717 + 0.490010i \(0.163007\pi\)
\(278\) 45.8784 2.75160
\(279\) 0 0
\(280\) 0 0
\(281\) 30.9755 1.84785 0.923923 0.382579i \(-0.124964\pi\)
0.923923 + 0.382579i \(0.124964\pi\)
\(282\) 0 0
\(283\) 23.7951 1.41447 0.707235 0.706979i \(-0.249942\pi\)
0.707235 + 0.706979i \(0.249942\pi\)
\(284\) 4.98759 0.295959
\(285\) 0 0
\(286\) 13.8392 0.818327
\(287\) 25.3636 1.49717
\(288\) 0 0
\(289\) 7.74240 0.455435
\(290\) 0 0
\(291\) 0 0
\(292\) −19.5348 −1.14319
\(293\) 0.998275 0.0583198 0.0291599 0.999575i \(-0.490717\pi\)
0.0291599 + 0.999575i \(0.490717\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −9.80341 −0.569812
\(297\) 0 0
\(298\) −48.6150 −2.81619
\(299\) 26.9635 1.55934
\(300\) 0 0
\(301\) 3.72999 0.214993
\(302\) 16.0664 0.924520
\(303\) 0 0
\(304\) 0.275798 0.0158181
\(305\) 0 0
\(306\) 0 0
\(307\) 27.7710 1.58497 0.792486 0.609890i \(-0.208786\pi\)
0.792486 + 0.609890i \(0.208786\pi\)
\(308\) −8.13152 −0.463336
\(309\) 0 0
\(310\) 0 0
\(311\) −17.5213 −0.993545 −0.496772 0.867881i \(-0.665482\pi\)
−0.496772 + 0.867881i \(0.665482\pi\)
\(312\) 0 0
\(313\) −5.41530 −0.306091 −0.153045 0.988219i \(-0.548908\pi\)
−0.153045 + 0.988219i \(0.548908\pi\)
\(314\) 48.2269 2.72160
\(315\) 0 0
\(316\) 35.5878 2.00197
\(317\) −6.79679 −0.381746 −0.190873 0.981615i \(-0.561132\pi\)
−0.190873 + 0.981615i \(0.561132\pi\)
\(318\) 0 0
\(319\) 0.921150 0.0515745
\(320\) 0 0
\(321\) 0 0
\(322\) −25.6612 −1.43004
\(323\) −34.9545 −1.94492
\(324\) 0 0
\(325\) 0 0
\(326\) −17.4938 −0.968892
\(327\) 0 0
\(328\) 28.2459 1.55962
\(329\) −20.4242 −1.12602
\(330\) 0 0
\(331\) −7.11845 −0.391266 −0.195633 0.980677i \(-0.562676\pi\)
−0.195633 + 0.980677i \(0.562676\pi\)
\(332\) 27.4563 1.50686
\(333\) 0 0
\(334\) −7.98482 −0.436910
\(335\) 0 0
\(336\) 0 0
\(337\) −24.9105 −1.35696 −0.678480 0.734619i \(-0.737361\pi\)
−0.678480 + 0.734619i \(0.737361\pi\)
\(338\) −54.0466 −2.93975
\(339\) 0 0
\(340\) 0 0
\(341\) 3.03925 0.164584
\(342\) 0 0
\(343\) 19.2789 1.04096
\(344\) 4.15386 0.223961
\(345\) 0 0
\(346\) −38.8084 −2.08636
\(347\) 23.6253 1.26827 0.634137 0.773221i \(-0.281356\pi\)
0.634137 + 0.773221i \(0.281356\pi\)
\(348\) 0 0
\(349\) −31.1721 −1.66861 −0.834303 0.551306i \(-0.814130\pi\)
−0.834303 + 0.551306i \(0.814130\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 5.70163 0.303898
\(353\) 0.533044 0.0283711 0.0141855 0.999899i \(-0.495484\pi\)
0.0141855 + 0.999899i \(0.495484\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 48.8780 2.59053
\(357\) 0 0
\(358\) −20.9091 −1.10508
\(359\) −12.0268 −0.634752 −0.317376 0.948300i \(-0.602802\pi\)
−0.317376 + 0.948300i \(0.602802\pi\)
\(360\) 0 0
\(361\) 30.3815 1.59903
\(362\) −51.7603 −2.72046
\(363\) 0 0
\(364\) 49.2203 2.57984
\(365\) 0 0
\(366\) 0 0
\(367\) 14.9608 0.780945 0.390472 0.920615i \(-0.372312\pi\)
0.390472 + 0.920615i \(0.372312\pi\)
\(368\) 0.174828 0.00911357
\(369\) 0 0
\(370\) 0 0
\(371\) −3.89395 −0.202164
\(372\) 0 0
\(373\) −20.1315 −1.04237 −0.521185 0.853444i \(-0.674510\pi\)
−0.521185 + 0.853444i \(0.674510\pi\)
\(374\) −11.3726 −0.588062
\(375\) 0 0
\(376\) −22.7451 −1.17299
\(377\) −5.57574 −0.287165
\(378\) 0 0
\(379\) 15.8967 0.816558 0.408279 0.912857i \(-0.366129\pi\)
0.408279 + 0.912857i \(0.366129\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −27.9759 −1.43137
\(383\) 2.63909 0.134851 0.0674256 0.997724i \(-0.478521\pi\)
0.0674256 + 0.997724i \(0.478521\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 2.37292 0.120779
\(387\) 0 0
\(388\) 37.4359 1.90052
\(389\) 13.8423 0.701832 0.350916 0.936407i \(-0.385870\pi\)
0.350916 + 0.936407i \(0.385870\pi\)
\(390\) 0 0
\(391\) −22.1577 −1.12056
\(392\) 1.82807 0.0923313
\(393\) 0 0
\(394\) −18.4394 −0.928962
\(395\) 0 0
\(396\) 0 0
\(397\) 25.6783 1.28876 0.644379 0.764706i \(-0.277116\pi\)
0.644379 + 0.764706i \(0.277116\pi\)
\(398\) 30.0544 1.50649
\(399\) 0 0
\(400\) 0 0
\(401\) 31.1212 1.55412 0.777059 0.629428i \(-0.216711\pi\)
0.777059 + 0.629428i \(0.216711\pi\)
\(402\) 0 0
\(403\) −18.3966 −0.916402
\(404\) −0.381133 −0.0189621
\(405\) 0 0
\(406\) 5.30644 0.263354
\(407\) 3.49380 0.173181
\(408\) 0 0
\(409\) 12.7514 0.630516 0.315258 0.949006i \(-0.397909\pi\)
0.315258 + 0.949006i \(0.397909\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −34.3353 −1.69158
\(413\) 19.1487 0.942248
\(414\) 0 0
\(415\) 0 0
\(416\) −34.5121 −1.69209
\(417\) 0 0
\(418\) 16.0664 0.785835
\(419\) 0.236195 0.0115389 0.00576943 0.999983i \(-0.498164\pi\)
0.00576943 + 0.999983i \(0.498164\pi\)
\(420\) 0 0
\(421\) −21.7300 −1.05905 −0.529527 0.848293i \(-0.677631\pi\)
−0.529527 + 0.848293i \(0.677631\pi\)
\(422\) 6.97625 0.339599
\(423\) 0 0
\(424\) −4.33645 −0.210597
\(425\) 0 0
\(426\) 0 0
\(427\) −25.4634 −1.23226
\(428\) −60.3246 −2.91590
\(429\) 0 0
\(430\) 0 0
\(431\) −16.2121 −0.780909 −0.390455 0.920622i \(-0.627682\pi\)
−0.390455 + 0.920622i \(0.627682\pi\)
\(432\) 0 0
\(433\) −13.5213 −0.649795 −0.324897 0.945749i \(-0.605330\pi\)
−0.324897 + 0.945749i \(0.605330\pi\)
\(434\) 17.5081 0.840416
\(435\) 0 0
\(436\) −23.0598 −1.10437
\(437\) 31.3030 1.49743
\(438\) 0 0
\(439\) 0.0664435 0.00317118 0.00158559 0.999999i \(-0.499495\pi\)
0.00158559 + 0.999999i \(0.499495\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 68.8384 3.27431
\(443\) 0.560596 0.0266347 0.0133174 0.999911i \(-0.495761\pi\)
0.0133174 + 0.999911i \(0.495761\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −31.8904 −1.51006
\(447\) 0 0
\(448\) 32.6474 1.54245
\(449\) 9.92079 0.468191 0.234096 0.972214i \(-0.424787\pi\)
0.234096 + 0.972214i \(0.424787\pi\)
\(450\) 0 0
\(451\) −10.0664 −0.474010
\(452\) 8.34153 0.392353
\(453\) 0 0
\(454\) −67.7960 −3.18183
\(455\) 0 0
\(456\) 0 0
\(457\) −8.93457 −0.417942 −0.208971 0.977922i \(-0.567011\pi\)
−0.208971 + 0.977922i \(0.567011\pi\)
\(458\) −41.6387 −1.94565
\(459\) 0 0
\(460\) 0 0
\(461\) 34.1484 1.59045 0.795225 0.606315i \(-0.207353\pi\)
0.795225 + 0.606315i \(0.207353\pi\)
\(462\) 0 0
\(463\) 24.9359 1.15887 0.579436 0.815018i \(-0.303273\pi\)
0.579436 + 0.815018i \(0.303273\pi\)
\(464\) −0.0361525 −0.00167834
\(465\) 0 0
\(466\) 29.9056 1.38535
\(467\) 2.84844 0.131810 0.0659051 0.997826i \(-0.479007\pi\)
0.0659051 + 0.997826i \(0.479007\pi\)
\(468\) 0 0
\(469\) −21.8974 −1.01113
\(470\) 0 0
\(471\) 0 0
\(472\) 21.3248 0.981552
\(473\) −1.48038 −0.0680678
\(474\) 0 0
\(475\) 0 0
\(476\) −40.4476 −1.85391
\(477\) 0 0
\(478\) −1.83604 −0.0839784
\(479\) −15.1177 −0.690747 −0.345374 0.938465i \(-0.612248\pi\)
−0.345374 + 0.938465i \(0.612248\pi\)
\(480\) 0 0
\(481\) −21.1480 −0.964267
\(482\) −6.94870 −0.316505
\(483\) 0 0
\(484\) 3.22727 0.146694
\(485\) 0 0
\(486\) 0 0
\(487\) 4.53376 0.205444 0.102722 0.994710i \(-0.467245\pi\)
0.102722 + 0.994710i \(0.467245\pi\)
\(488\) −28.3570 −1.28366
\(489\) 0 0
\(490\) 0 0
\(491\) −15.0909 −0.681043 −0.340521 0.940237i \(-0.610604\pi\)
−0.340521 + 0.940237i \(0.610604\pi\)
\(492\) 0 0
\(493\) 4.58196 0.206361
\(494\) −97.2506 −4.37551
\(495\) 0 0
\(496\) −0.119282 −0.00535591
\(497\) −3.89395 −0.174668
\(498\) 0 0
\(499\) −8.93320 −0.399905 −0.199952 0.979806i \(-0.564079\pi\)
−0.199952 + 0.979806i \(0.564079\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −16.8357 −0.751413
\(503\) −26.7162 −1.19122 −0.595609 0.803275i \(-0.703089\pi\)
−0.595609 + 0.803275i \(0.703089\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 10.1845 0.452758
\(507\) 0 0
\(508\) −36.8836 −1.63645
\(509\) −10.9876 −0.487017 −0.243508 0.969899i \(-0.578298\pi\)
−0.243508 + 0.969899i \(0.578298\pi\)
\(510\) 0 0
\(511\) 15.2513 0.674680
\(512\) −0.444024 −0.0196233
\(513\) 0 0
\(514\) −9.02720 −0.398173
\(515\) 0 0
\(516\) 0 0
\(517\) 8.10605 0.356504
\(518\) 20.1266 0.884313
\(519\) 0 0
\(520\) 0 0
\(521\) −23.2782 −1.01984 −0.509918 0.860223i \(-0.670324\pi\)
−0.509918 + 0.860223i \(0.670324\pi\)
\(522\) 0 0
\(523\) 18.4163 0.805289 0.402645 0.915356i \(-0.368091\pi\)
0.402645 + 0.915356i \(0.368091\pi\)
\(524\) 26.1604 1.14282
\(525\) 0 0
\(526\) 65.4814 2.85512
\(527\) 15.1177 0.658539
\(528\) 0 0
\(529\) −3.15699 −0.137260
\(530\) 0 0
\(531\) 0 0
\(532\) 57.1418 2.47741
\(533\) 60.9324 2.63928
\(534\) 0 0
\(535\) 0 0
\(536\) −24.3858 −1.05330
\(537\) 0 0
\(538\) −64.5020 −2.78088
\(539\) −0.651497 −0.0280620
\(540\) 0 0
\(541\) −31.3815 −1.34920 −0.674598 0.738186i \(-0.735683\pi\)
−0.674598 + 0.738186i \(0.735683\pi\)
\(542\) −25.7518 −1.10613
\(543\) 0 0
\(544\) 28.3609 1.21596
\(545\) 0 0
\(546\) 0 0
\(547\) 3.74412 0.160087 0.0800436 0.996791i \(-0.474494\pi\)
0.0800436 + 0.996791i \(0.474494\pi\)
\(548\) −19.3636 −0.827174
\(549\) 0 0
\(550\) 0 0
\(551\) −6.47310 −0.275763
\(552\) 0 0
\(553\) −27.7844 −1.18151
\(554\) −66.3411 −2.81856
\(555\) 0 0
\(556\) −64.7599 −2.74643
\(557\) 25.3164 1.07269 0.536345 0.843999i \(-0.319804\pi\)
0.536345 + 0.843999i \(0.319804\pi\)
\(558\) 0 0
\(559\) 8.96075 0.378999
\(560\) 0 0
\(561\) 0 0
\(562\) −70.8201 −2.98737
\(563\) 24.7465 1.04294 0.521470 0.853269i \(-0.325384\pi\)
0.521470 + 0.853269i \(0.325384\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −54.4032 −2.28674
\(567\) 0 0
\(568\) −4.33645 −0.181953
\(569\) −44.0148 −1.84520 −0.922598 0.385763i \(-0.873938\pi\)
−0.922598 + 0.385763i \(0.873938\pi\)
\(570\) 0 0
\(571\) −2.01205 −0.0842017 −0.0421009 0.999113i \(-0.513405\pi\)
−0.0421009 + 0.999113i \(0.513405\pi\)
\(572\) −19.5348 −0.816790
\(573\) 0 0
\(574\) −57.9895 −2.42044
\(575\) 0 0
\(576\) 0 0
\(577\) −26.3181 −1.09564 −0.547819 0.836597i \(-0.684542\pi\)
−0.547819 + 0.836597i \(0.684542\pi\)
\(578\) −17.7016 −0.736291
\(579\) 0 0
\(580\) 0 0
\(581\) −21.4359 −0.889310
\(582\) 0 0
\(583\) 1.54545 0.0640060
\(584\) 16.9845 0.702822
\(585\) 0 0
\(586\) −2.28238 −0.0942842
\(587\) −35.8112 −1.47809 −0.739044 0.673657i \(-0.764722\pi\)
−0.739044 + 0.673657i \(0.764722\pi\)
\(588\) 0 0
\(589\) −21.3574 −0.880016
\(590\) 0 0
\(591\) 0 0
\(592\) −0.137122 −0.00563567
\(593\) 7.61953 0.312896 0.156448 0.987686i \(-0.449996\pi\)
0.156448 + 0.987686i \(0.449996\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 68.6228 2.81090
\(597\) 0 0
\(598\) −61.6473 −2.52094
\(599\) 40.2121 1.64302 0.821511 0.570193i \(-0.193132\pi\)
0.821511 + 0.570193i \(0.193132\pi\)
\(600\) 0 0
\(601\) 26.3181 1.07354 0.536770 0.843729i \(-0.319644\pi\)
0.536770 + 0.843729i \(0.319644\pi\)
\(602\) −8.52797 −0.347574
\(603\) 0 0
\(604\) −22.6787 −0.922783
\(605\) 0 0
\(606\) 0 0
\(607\) −11.4287 −0.463878 −0.231939 0.972730i \(-0.574507\pi\)
−0.231939 + 0.972730i \(0.574507\pi\)
\(608\) −40.0664 −1.62491
\(609\) 0 0
\(610\) 0 0
\(611\) −49.0661 −1.98500
\(612\) 0 0
\(613\) 21.1198 0.853022 0.426511 0.904482i \(-0.359743\pi\)
0.426511 + 0.904482i \(0.359743\pi\)
\(614\) −63.4934 −2.56239
\(615\) 0 0
\(616\) 7.06993 0.284856
\(617\) −31.3905 −1.26373 −0.631867 0.775077i \(-0.717711\pi\)
−0.631867 + 0.775077i \(0.717711\pi\)
\(618\) 0 0
\(619\) −7.11845 −0.286115 −0.143057 0.989714i \(-0.545693\pi\)
−0.143057 + 0.989714i \(0.545693\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 40.0595 1.60624
\(623\) −38.1604 −1.52887
\(624\) 0 0
\(625\) 0 0
\(626\) 12.3811 0.494850
\(627\) 0 0
\(628\) −68.0750 −2.71649
\(629\) 17.3787 0.692936
\(630\) 0 0
\(631\) 21.3574 0.850224 0.425112 0.905141i \(-0.360235\pi\)
0.425112 + 0.905141i \(0.360235\pi\)
\(632\) −30.9417 −1.23080
\(633\) 0 0
\(634\) 15.5397 0.617159
\(635\) 0 0
\(636\) 0 0
\(637\) 3.94353 0.156248
\(638\) −2.10605 −0.0833792
\(639\) 0 0
\(640\) 0 0
\(641\) −26.1329 −1.03219 −0.516093 0.856532i \(-0.672614\pi\)
−0.516093 + 0.856532i \(0.672614\pi\)
\(642\) 0 0
\(643\) −30.5358 −1.20421 −0.602107 0.798416i \(-0.705672\pi\)
−0.602107 + 0.798416i \(0.705672\pi\)
\(644\) 36.2222 1.42736
\(645\) 0 0
\(646\) 79.9173 3.14430
\(647\) −11.5482 −0.454006 −0.227003 0.973894i \(-0.572893\pi\)
−0.227003 + 0.973894i \(0.572893\pi\)
\(648\) 0 0
\(649\) −7.59984 −0.298320
\(650\) 0 0
\(651\) 0 0
\(652\) 24.6935 0.967071
\(653\) −22.2389 −0.870277 −0.435138 0.900364i \(-0.643301\pi\)
−0.435138 + 0.900364i \(0.643301\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0.395079 0.0154253
\(657\) 0 0
\(658\) 46.6963 1.82041
\(659\) −39.4359 −1.53620 −0.768102 0.640328i \(-0.778799\pi\)
−0.768102 + 0.640328i \(0.778799\pi\)
\(660\) 0 0
\(661\) 21.8967 0.851683 0.425841 0.904798i \(-0.359978\pi\)
0.425841 + 0.904798i \(0.359978\pi\)
\(662\) 16.2751 0.632549
\(663\) 0 0
\(664\) −23.8718 −0.926405
\(665\) 0 0
\(666\) 0 0
\(667\) −4.10331 −0.158881
\(668\) 11.2710 0.436089
\(669\) 0 0
\(670\) 0 0
\(671\) 10.1060 0.390140
\(672\) 0 0
\(673\) 18.2920 0.705103 0.352552 0.935792i \(-0.385314\pi\)
0.352552 + 0.935792i \(0.385314\pi\)
\(674\) 56.9534 2.19376
\(675\) 0 0
\(676\) 76.2898 2.93422
\(677\) 0.817184 0.0314069 0.0157035 0.999877i \(-0.495001\pi\)
0.0157035 + 0.999877i \(0.495001\pi\)
\(678\) 0 0
\(679\) −29.2272 −1.12164
\(680\) 0 0
\(681\) 0 0
\(682\) −6.94870 −0.266080
\(683\) −12.8031 −0.489895 −0.244948 0.969536i \(-0.578771\pi\)
−0.244948 + 0.969536i \(0.578771\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −44.0778 −1.68290
\(687\) 0 0
\(688\) 0.0581006 0.00221506
\(689\) −9.35465 −0.356384
\(690\) 0 0
\(691\) −22.9394 −0.872656 −0.436328 0.899788i \(-0.643721\pi\)
−0.436328 + 0.899788i \(0.643721\pi\)
\(692\) 54.7803 2.08244
\(693\) 0 0
\(694\) −54.0151 −2.05039
\(695\) 0 0
\(696\) 0 0
\(697\) −50.0722 −1.89662
\(698\) 71.2696 2.69759
\(699\) 0 0
\(700\) 0 0
\(701\) 32.3329 1.22120 0.610599 0.791940i \(-0.290929\pi\)
0.610599 + 0.791940i \(0.290929\pi\)
\(702\) 0 0
\(703\) −24.5516 −0.925981
\(704\) −12.9573 −0.488345
\(705\) 0 0
\(706\) −1.21871 −0.0458668
\(707\) 0.297561 0.0111909
\(708\) 0 0
\(709\) −9.25760 −0.347677 −0.173838 0.984774i \(-0.555617\pi\)
−0.173838 + 0.984774i \(0.555617\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −42.4969 −1.59264
\(713\) −13.5385 −0.507020
\(714\) 0 0
\(715\) 0 0
\(716\) 29.5144 1.10300
\(717\) 0 0
\(718\) 27.4972 1.02619
\(719\) 7.01170 0.261492 0.130746 0.991416i \(-0.458263\pi\)
0.130746 + 0.991416i \(0.458263\pi\)
\(720\) 0 0
\(721\) 26.8065 0.998326
\(722\) −69.4619 −2.58510
\(723\) 0 0
\(724\) 73.0626 2.71535
\(725\) 0 0
\(726\) 0 0
\(727\) −18.0303 −0.668706 −0.334353 0.942448i \(-0.608518\pi\)
−0.334353 + 0.942448i \(0.608518\pi\)
\(728\) −42.7945 −1.58607
\(729\) 0 0
\(730\) 0 0
\(731\) −7.36365 −0.272354
\(732\) 0 0
\(733\) 22.9890 0.849117 0.424558 0.905401i \(-0.360429\pi\)
0.424558 + 0.905401i \(0.360429\pi\)
\(734\) −34.2051 −1.26253
\(735\) 0 0
\(736\) −25.3982 −0.936189
\(737\) 8.69074 0.320128
\(738\) 0 0
\(739\) −30.0967 −1.10713 −0.553563 0.832807i \(-0.686732\pi\)
−0.553563 + 0.832807i \(0.686732\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 8.90284 0.326833
\(743\) 32.7706 1.20224 0.601119 0.799160i \(-0.294722\pi\)
0.601119 + 0.799160i \(0.294722\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 46.0272 1.68517
\(747\) 0 0
\(748\) 16.0530 0.586957
\(749\) 47.0971 1.72089
\(750\) 0 0
\(751\) −0.132887 −0.00484912 −0.00242456 0.999997i \(-0.500772\pi\)
−0.00242456 + 0.999997i \(0.500772\pi\)
\(752\) −0.318139 −0.0116014
\(753\) 0 0
\(754\) 12.7479 0.464253
\(755\) 0 0
\(756\) 0 0
\(757\) 40.1363 1.45878 0.729390 0.684098i \(-0.239804\pi\)
0.729390 + 0.684098i \(0.239804\pi\)
\(758\) −36.3450 −1.32011
\(759\) 0 0
\(760\) 0 0
\(761\) 30.9487 1.12189 0.560945 0.827853i \(-0.310438\pi\)
0.560945 + 0.827853i \(0.310438\pi\)
\(762\) 0 0
\(763\) 18.0035 0.651769
\(764\) 39.4896 1.42868
\(765\) 0 0
\(766\) −6.03381 −0.218011
\(767\) 46.0020 1.66104
\(768\) 0 0
\(769\) −7.17213 −0.258634 −0.129317 0.991603i \(-0.541278\pi\)
−0.129317 + 0.991603i \(0.541278\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −3.34952 −0.120552
\(773\) 15.3546 0.552268 0.276134 0.961119i \(-0.410947\pi\)
0.276134 + 0.961119i \(0.410947\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −32.5485 −1.16842
\(777\) 0 0
\(778\) −31.6480 −1.13463
\(779\) 70.7389 2.53448
\(780\) 0 0
\(781\) 1.54545 0.0553006
\(782\) 50.6597 1.81159
\(783\) 0 0
\(784\) 0.0255694 0.000913193 0
\(785\) 0 0
\(786\) 0 0
\(787\) −10.8619 −0.387184 −0.193592 0.981082i \(-0.562014\pi\)
−0.193592 + 0.981082i \(0.562014\pi\)
\(788\) 26.0282 0.927217
\(789\) 0 0
\(790\) 0 0
\(791\) −6.51247 −0.231557
\(792\) 0 0
\(793\) −61.1721 −2.17229
\(794\) −58.7090 −2.08350
\(795\) 0 0
\(796\) −42.4235 −1.50366
\(797\) 40.7720 1.44422 0.722109 0.691780i \(-0.243173\pi\)
0.722109 + 0.691780i \(0.243173\pi\)
\(798\) 0 0
\(799\) 40.3209 1.42645
\(800\) 0 0
\(801\) 0 0
\(802\) −71.1531 −2.51250
\(803\) −6.05302 −0.213607
\(804\) 0 0
\(805\) 0 0
\(806\) 42.0607 1.48152
\(807\) 0 0
\(808\) 0.331375 0.0116577
\(809\) −14.7634 −0.519055 −0.259528 0.965736i \(-0.583567\pi\)
−0.259528 + 0.965736i \(0.583567\pi\)
\(810\) 0 0
\(811\) 3.93356 0.138126 0.0690629 0.997612i \(-0.477999\pi\)
0.0690629 + 0.997612i \(0.477999\pi\)
\(812\) −7.49035 −0.262860
\(813\) 0 0
\(814\) −7.98795 −0.279977
\(815\) 0 0
\(816\) 0 0
\(817\) 10.4029 0.363951
\(818\) −29.1538 −1.01934
\(819\) 0 0
\(820\) 0 0
\(821\) 20.4603 0.714071 0.357035 0.934091i \(-0.383788\pi\)
0.357035 + 0.934091i \(0.383788\pi\)
\(822\) 0 0
\(823\) −23.7603 −0.828231 −0.414116 0.910224i \(-0.635909\pi\)
−0.414116 + 0.910224i \(0.635909\pi\)
\(824\) 29.8527 1.03997
\(825\) 0 0
\(826\) −43.7802 −1.52331
\(827\) −30.1253 −1.04756 −0.523779 0.851854i \(-0.675478\pi\)
−0.523779 + 0.851854i \(0.675478\pi\)
\(828\) 0 0
\(829\) 37.8388 1.31420 0.657098 0.753806i \(-0.271784\pi\)
0.657098 + 0.753806i \(0.271784\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 78.4306 2.71909
\(833\) −3.24066 −0.112282
\(834\) 0 0
\(835\) 0 0
\(836\) −22.6787 −0.784359
\(837\) 0 0
\(838\) −0.540017 −0.0186546
\(839\) 45.7871 1.58075 0.790374 0.612625i \(-0.209886\pi\)
0.790374 + 0.612625i \(0.209886\pi\)
\(840\) 0 0
\(841\) −28.1515 −0.970741
\(842\) 49.6818 1.71215
\(843\) 0 0
\(844\) −9.84738 −0.338961
\(845\) 0 0
\(846\) 0 0
\(847\) −2.51962 −0.0865753
\(848\) −0.0606546 −0.00208289
\(849\) 0 0
\(850\) 0 0
\(851\) −15.5633 −0.533503
\(852\) 0 0
\(853\) −23.3312 −0.798845 −0.399423 0.916767i \(-0.630789\pi\)
−0.399423 + 0.916767i \(0.630789\pi\)
\(854\) 58.2176 1.99217
\(855\) 0 0
\(856\) 52.4491 1.79267
\(857\) −16.0892 −0.549596 −0.274798 0.961502i \(-0.588611\pi\)
−0.274798 + 0.961502i \(0.588611\pi\)
\(858\) 0 0
\(859\) 13.1969 0.450274 0.225137 0.974327i \(-0.427717\pi\)
0.225137 + 0.974327i \(0.427717\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 37.0661 1.26248
\(863\) −48.5606 −1.65302 −0.826511 0.562921i \(-0.809678\pi\)
−0.826511 + 0.562921i \(0.809678\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 30.9142 1.05051
\(867\) 0 0
\(868\) −24.7137 −0.838837
\(869\) 11.0272 0.374072
\(870\) 0 0
\(871\) −52.6053 −1.78246
\(872\) 20.0493 0.678955
\(873\) 0 0
\(874\) −71.5688 −2.42085
\(875\) 0 0
\(876\) 0 0
\(877\) −17.4862 −0.590466 −0.295233 0.955425i \(-0.595397\pi\)
−0.295233 + 0.955425i \(0.595397\pi\)
\(878\) −0.151911 −0.00512676
\(879\) 0 0
\(880\) 0 0
\(881\) 23.2238 0.782429 0.391215 0.920299i \(-0.372055\pi\)
0.391215 + 0.920299i \(0.372055\pi\)
\(882\) 0 0
\(883\) −39.8967 −1.34263 −0.671315 0.741172i \(-0.734270\pi\)
−0.671315 + 0.741172i \(0.734270\pi\)
\(884\) −97.1693 −3.26816
\(885\) 0 0
\(886\) −1.28170 −0.0430597
\(887\) 13.6770 0.459228 0.229614 0.973282i \(-0.426254\pi\)
0.229614 + 0.973282i \(0.426254\pi\)
\(888\) 0 0
\(889\) 28.7961 0.965789
\(890\) 0 0
\(891\) 0 0
\(892\) 45.0151 1.50722
\(893\) −56.9628 −1.90619
\(894\) 0 0
\(895\) 0 0
\(896\) −45.9107 −1.53377
\(897\) 0 0
\(898\) −22.6821 −0.756913
\(899\) 2.79960 0.0933720
\(900\) 0 0
\(901\) 7.68734 0.256102
\(902\) 23.0151 0.766321
\(903\) 0 0
\(904\) −7.25252 −0.241215
\(905\) 0 0
\(906\) 0 0
\(907\) −1.73625 −0.0576513 −0.0288257 0.999584i \(-0.509177\pi\)
−0.0288257 + 0.999584i \(0.509177\pi\)
\(908\) 95.6979 3.17585
\(909\) 0 0
\(910\) 0 0
\(911\) −49.4662 −1.63889 −0.819444 0.573160i \(-0.805717\pi\)
−0.819444 + 0.573160i \(0.805717\pi\)
\(912\) 0 0
\(913\) 8.50757 0.281560
\(914\) 20.4273 0.675676
\(915\) 0 0
\(916\) 58.7754 1.94199
\(917\) −20.4242 −0.674466
\(918\) 0 0
\(919\) 40.0423 1.32087 0.660437 0.750881i \(-0.270371\pi\)
0.660437 + 0.750881i \(0.270371\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −78.0743 −2.57124
\(923\) −9.35465 −0.307912
\(924\) 0 0
\(925\) 0 0
\(926\) −57.0116 −1.87352
\(927\) 0 0
\(928\) 5.25205 0.172407
\(929\) 29.0420 0.952837 0.476418 0.879219i \(-0.341935\pi\)
0.476418 + 0.879219i \(0.341935\pi\)
\(930\) 0 0
\(931\) 4.57820 0.150044
\(932\) −42.2135 −1.38275
\(933\) 0 0
\(934\) −6.51247 −0.213094
\(935\) 0 0
\(936\) 0 0
\(937\) −40.0282 −1.30766 −0.653832 0.756639i \(-0.726840\pi\)
−0.653832 + 0.756639i \(0.726840\pi\)
\(938\) 50.0645 1.63467
\(939\) 0 0
\(940\) 0 0
\(941\) 13.9604 0.455096 0.227548 0.973767i \(-0.426929\pi\)
0.227548 + 0.973767i \(0.426929\pi\)
\(942\) 0 0
\(943\) 44.8415 1.46024
\(944\) 0.298272 0.00970794
\(945\) 0 0
\(946\) 3.38462 0.110043
\(947\) −17.4875 −0.568269 −0.284134 0.958785i \(-0.591706\pi\)
−0.284134 + 0.958785i \(0.591706\pi\)
\(948\) 0 0
\(949\) 36.6391 1.18936
\(950\) 0 0
\(951\) 0 0
\(952\) 35.1671 1.13977
\(953\) 49.1890 1.59339 0.796694 0.604383i \(-0.206580\pi\)
0.796694 + 0.604383i \(0.206580\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 2.59167 0.0838206
\(957\) 0 0
\(958\) 34.5640 1.11671
\(959\) 15.1177 0.488177
\(960\) 0 0
\(961\) −21.7630 −0.702032
\(962\) 48.3512 1.55891
\(963\) 0 0
\(964\) 9.80849 0.315910
\(965\) 0 0
\(966\) 0 0
\(967\) 23.5589 0.757602 0.378801 0.925478i \(-0.376336\pi\)
0.378801 + 0.925478i \(0.376336\pi\)
\(968\) −2.80595 −0.0901866
\(969\) 0 0
\(970\) 0 0
\(971\) −5.16940 −0.165894 −0.0829469 0.996554i \(-0.526433\pi\)
−0.0829469 + 0.996554i \(0.526433\pi\)
\(972\) 0 0
\(973\) 50.5599 1.62088
\(974\) −10.3656 −0.332136
\(975\) 0 0
\(976\) −0.396634 −0.0126959
\(977\) 9.73970 0.311601 0.155800 0.987789i \(-0.450204\pi\)
0.155800 + 0.987789i \(0.450204\pi\)
\(978\) 0 0
\(979\) 15.1453 0.484046
\(980\) 0 0
\(981\) 0 0
\(982\) 34.5027 1.10102
\(983\) −2.16670 −0.0691070 −0.0345535 0.999403i \(-0.511001\pi\)
−0.0345535 + 0.999403i \(0.511001\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −10.4758 −0.333619
\(987\) 0 0
\(988\) 137.275 4.36729
\(989\) 6.59441 0.209690
\(990\) 0 0
\(991\) −25.2265 −0.801347 −0.400674 0.916221i \(-0.631224\pi\)
−0.400674 + 0.916221i \(0.631224\pi\)
\(992\) 17.3286 0.550185
\(993\) 0 0
\(994\) 8.90284 0.282381
\(995\) 0 0
\(996\) 0 0
\(997\) −29.5950 −0.937281 −0.468641 0.883389i \(-0.655256\pi\)
−0.468641 + 0.883389i \(0.655256\pi\)
\(998\) 20.4242 0.646516
\(999\) 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 2475.2.a.bj.1.1 4
3.2 odd 2 2475.2.a.bf.1.4 4
5.2 odd 4 2475.2.c.t.199.2 8
5.3 odd 4 2475.2.c.t.199.7 8
5.4 even 2 495.2.a.f.1.4 4
15.2 even 4 2475.2.c.s.199.7 8
15.8 even 4 2475.2.c.s.199.2 8
15.14 odd 2 495.2.a.g.1.1 yes 4
20.19 odd 2 7920.2.a.cm.1.2 4
55.54 odd 2 5445.2.a.bs.1.1 4
60.59 even 2 7920.2.a.cn.1.2 4
165.164 even 2 5445.2.a.bh.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
495.2.a.f.1.4 4 5.4 even 2
495.2.a.g.1.1 yes 4 15.14 odd 2
2475.2.a.bf.1.4 4 3.2 odd 2
2475.2.a.bj.1.1 4 1.1 even 1 trivial
2475.2.c.s.199.2 8 15.8 even 4
2475.2.c.s.199.7 8 15.2 even 4
2475.2.c.t.199.2 8 5.2 odd 4
2475.2.c.t.199.7 8 5.3 odd 4
5445.2.a.bh.1.4 4 165.164 even 2
5445.2.a.bs.1.1 4 55.54 odd 2
7920.2.a.cm.1.2 4 20.19 odd 2
7920.2.a.cn.1.2 4 60.59 even 2