Properties

Label 1449.2.a.r.1.3
Level $1449$
Weight $2$
Character 1449.1
Self dual yes
Analytic conductor $11.570$
Analytic rank $0$
Dimension $5$
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] = [1449,2,Mod(1,1449)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1449, 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("1449.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1449 = 3^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1449.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: \(11.5703232529\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.2147108.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 9x^{3} + 17x^{2} + 16x - 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 161)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.23828\) of defining polynomial
Character \(\chi\) \(=\) 1449.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.23828 q^{2} -0.466664 q^{4} +1.86253 q^{5} +1.00000 q^{7} +3.05442 q^{8} +O(q^{10})\) \(q-1.23828 q^{2} -0.466664 q^{4} +1.86253 q^{5} +1.00000 q^{7} +3.05442 q^{8} -2.30633 q^{10} +0.846153 q^{11} +2.55110 q^{13} -1.23828 q^{14} -2.84890 q^{16} +7.07080 q^{17} -0.476559 q^{19} -0.869177 q^{20} -1.04777 q^{22} +1.00000 q^{23} -1.53098 q^{25} -3.15897 q^{26} -0.466664 q^{28} -8.63827 q^{29} +3.31143 q^{31} -2.58111 q^{32} -8.75562 q^{34} +1.86253 q^{35} +7.85369 q^{37} +0.590113 q^{38} +5.68895 q^{40} -2.82603 q^{41} -0.274938 q^{43} -0.394869 q^{44} -1.23828 q^{46} +13.4756 q^{47} +1.00000 q^{49} +1.89578 q^{50} -1.19051 q^{52} -8.93333 q^{53} +1.57599 q^{55} +3.05442 q^{56} +10.6966 q^{58} -1.66091 q^{59} -11.7162 q^{61} -4.10048 q^{62} +8.89393 q^{64} +4.75150 q^{65} -2.82636 q^{67} -3.29969 q^{68} -2.30633 q^{70} -9.92823 q^{71} +7.31556 q^{73} -9.72506 q^{74} +0.222393 q^{76} +0.846153 q^{77} +11.7795 q^{79} -5.30616 q^{80} +3.49942 q^{82} -3.72506 q^{83} +13.1696 q^{85} +0.340450 q^{86} +2.58451 q^{88} +8.76310 q^{89} +2.55110 q^{91} -0.466664 q^{92} -16.6865 q^{94} -0.887605 q^{95} +1.82229 q^{97} -1.23828 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{2} + 12 q^{4} + 4 q^{5} + 5 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 2 q^{2} + 12 q^{4} + 4 q^{5} + 5 q^{7} - 3 q^{8} - 8 q^{10} + 4 q^{11} - 6 q^{13} - 2 q^{14} + 10 q^{16} + 12 q^{17} + 6 q^{19} + 14 q^{22} + 5 q^{23} + 19 q^{25} - q^{26} + 12 q^{28} + 4 q^{29} + 30 q^{31} - 8 q^{32} + 6 q^{34} + 4 q^{35} + 4 q^{37} + 40 q^{38} - 50 q^{40} - 6 q^{41} - 12 q^{43} + 26 q^{44} - 2 q^{46} - 10 q^{47} + 5 q^{49} + 2 q^{50} - 21 q^{52} - 16 q^{53} + 18 q^{55} - 3 q^{56} + 13 q^{58} - 22 q^{59} - 18 q^{61} - 15 q^{62} + 25 q^{64} + 26 q^{65} - 2 q^{67} - 12 q^{68} - 8 q^{70} - 4 q^{71} - 2 q^{73} - 38 q^{74} + 10 q^{76} + 4 q^{77} + 30 q^{79} + 10 q^{80} - 7 q^{82} - 8 q^{83} - 12 q^{85} - 8 q^{86} + 4 q^{88} + 20 q^{89} - 6 q^{91} + 12 q^{92} - 25 q^{94} - 8 q^{95} - 12 q^{97} - 2 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.23828 −0.875596 −0.437798 0.899073i \(-0.644241\pi\)
−0.437798 + 0.899073i \(0.644241\pi\)
\(3\) 0 0
\(4\) −0.466664 −0.233332
\(5\) 1.86253 0.832949 0.416475 0.909147i \(-0.363265\pi\)
0.416475 + 0.909147i \(0.363265\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 3.05442 1.07990
\(9\) 0 0
\(10\) −2.30633 −0.729327
\(11\) 0.846153 0.255125 0.127562 0.991831i \(-0.459285\pi\)
0.127562 + 0.991831i \(0.459285\pi\)
\(12\) 0 0
\(13\) 2.55110 0.707547 0.353773 0.935331i \(-0.384898\pi\)
0.353773 + 0.935331i \(0.384898\pi\)
\(14\) −1.23828 −0.330944
\(15\) 0 0
\(16\) −2.84890 −0.712224
\(17\) 7.07080 1.71492 0.857460 0.514550i \(-0.172041\pi\)
0.857460 + 0.514550i \(0.172041\pi\)
\(18\) 0 0
\(19\) −0.476559 −0.109330 −0.0546650 0.998505i \(-0.517409\pi\)
−0.0546650 + 0.998505i \(0.517409\pi\)
\(20\) −0.869177 −0.194354
\(21\) 0 0
\(22\) −1.04777 −0.223386
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −1.53098 −0.306196
\(26\) −3.15897 −0.619525
\(27\) 0 0
\(28\) −0.466664 −0.0881912
\(29\) −8.63827 −1.60409 −0.802043 0.597266i \(-0.796254\pi\)
−0.802043 + 0.597266i \(0.796254\pi\)
\(30\) 0 0
\(31\) 3.31143 0.594751 0.297376 0.954761i \(-0.403889\pi\)
0.297376 + 0.954761i \(0.403889\pi\)
\(32\) −2.58111 −0.456280
\(33\) 0 0
\(34\) −8.75562 −1.50158
\(35\) 1.86253 0.314825
\(36\) 0 0
\(37\) 7.85369 1.29114 0.645569 0.763702i \(-0.276620\pi\)
0.645569 + 0.763702i \(0.276620\pi\)
\(38\) 0.590113 0.0957289
\(39\) 0 0
\(40\) 5.68895 0.899502
\(41\) −2.82603 −0.441352 −0.220676 0.975347i \(-0.570826\pi\)
−0.220676 + 0.975347i \(0.570826\pi\)
\(42\) 0 0
\(43\) −0.274938 −0.0419276 −0.0209638 0.999780i \(-0.506673\pi\)
−0.0209638 + 0.999780i \(0.506673\pi\)
\(44\) −0.394869 −0.0595288
\(45\) 0 0
\(46\) −1.23828 −0.182574
\(47\) 13.4756 1.96562 0.982808 0.184631i \(-0.0591089\pi\)
0.982808 + 0.184631i \(0.0591089\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 1.89578 0.268104
\(51\) 0 0
\(52\) −1.19051 −0.165093
\(53\) −8.93333 −1.22709 −0.613544 0.789661i \(-0.710256\pi\)
−0.613544 + 0.789661i \(0.710256\pi\)
\(54\) 0 0
\(55\) 1.57599 0.212506
\(56\) 3.05442 0.408164
\(57\) 0 0
\(58\) 10.6966 1.40453
\(59\) −1.66091 −0.216232 −0.108116 0.994138i \(-0.534482\pi\)
−0.108116 + 0.994138i \(0.534482\pi\)
\(60\) 0 0
\(61\) −11.7162 −1.50011 −0.750054 0.661376i \(-0.769973\pi\)
−0.750054 + 0.661376i \(0.769973\pi\)
\(62\) −4.10048 −0.520762
\(63\) 0 0
\(64\) 8.89393 1.11174
\(65\) 4.75150 0.589351
\(66\) 0 0
\(67\) −2.82636 −0.345295 −0.172648 0.984984i \(-0.555232\pi\)
−0.172648 + 0.984984i \(0.555232\pi\)
\(68\) −3.29969 −0.400146
\(69\) 0 0
\(70\) −2.30633 −0.275660
\(71\) −9.92823 −1.17826 −0.589132 0.808037i \(-0.700530\pi\)
−0.589132 + 0.808037i \(0.700530\pi\)
\(72\) 0 0
\(73\) 7.31556 0.856222 0.428111 0.903726i \(-0.359179\pi\)
0.428111 + 0.903726i \(0.359179\pi\)
\(74\) −9.72506 −1.13052
\(75\) 0 0
\(76\) 0.222393 0.0255102
\(77\) 0.846153 0.0964281
\(78\) 0 0
\(79\) 11.7795 1.32530 0.662648 0.748931i \(-0.269433\pi\)
0.662648 + 0.748931i \(0.269433\pi\)
\(80\) −5.30616 −0.593246
\(81\) 0 0
\(82\) 3.49942 0.386446
\(83\) −3.72506 −0.408879 −0.204439 0.978879i \(-0.565537\pi\)
−0.204439 + 0.978879i \(0.565537\pi\)
\(84\) 0 0
\(85\) 13.1696 1.42844
\(86\) 0.340450 0.0367117
\(87\) 0 0
\(88\) 2.58451 0.275509
\(89\) 8.76310 0.928887 0.464444 0.885603i \(-0.346254\pi\)
0.464444 + 0.885603i \(0.346254\pi\)
\(90\) 0 0
\(91\) 2.55110 0.267428
\(92\) −0.466664 −0.0486531
\(93\) 0 0
\(94\) −16.6865 −1.72108
\(95\) −0.887605 −0.0910664
\(96\) 0 0
\(97\) 1.82229 0.185026 0.0925130 0.995711i \(-0.470510\pi\)
0.0925130 + 0.995711i \(0.470510\pi\)
\(98\) −1.23828 −0.125085
\(99\) 0 0
\(100\) 0.714453 0.0714453
\(101\) 14.7870 1.47136 0.735682 0.677327i \(-0.236862\pi\)
0.735682 + 0.677327i \(0.236862\pi\)
\(102\) 0 0
\(103\) 10.9333 1.07729 0.538646 0.842532i \(-0.318936\pi\)
0.538646 + 0.842532i \(0.318936\pi\)
\(104\) 7.79212 0.764080
\(105\) 0 0
\(106\) 11.0620 1.07443
\(107\) 17.2636 1.66893 0.834466 0.551059i \(-0.185776\pi\)
0.834466 + 0.551059i \(0.185776\pi\)
\(108\) 0 0
\(109\) −13.2438 −1.26852 −0.634262 0.773118i \(-0.718696\pi\)
−0.634262 + 0.773118i \(0.718696\pi\)
\(110\) −1.95151 −0.186069
\(111\) 0 0
\(112\) −2.84890 −0.269195
\(113\) 7.10219 0.668118 0.334059 0.942552i \(-0.391581\pi\)
0.334059 + 0.942552i \(0.391581\pi\)
\(114\) 0 0
\(115\) 1.86253 0.173682
\(116\) 4.03117 0.374285
\(117\) 0 0
\(118\) 2.05667 0.189332
\(119\) 7.07080 0.648179
\(120\) 0 0
\(121\) −10.2840 −0.934911
\(122\) 14.5080 1.31349
\(123\) 0 0
\(124\) −1.54533 −0.138775
\(125\) −12.1641 −1.08799
\(126\) 0 0
\(127\) 16.1101 1.42954 0.714768 0.699361i \(-0.246532\pi\)
0.714768 + 0.699361i \(0.246532\pi\)
\(128\) −5.85095 −0.517156
\(129\) 0 0
\(130\) −5.88368 −0.516033
\(131\) 0.854665 0.0746724 0.0373362 0.999303i \(-0.488113\pi\)
0.0373362 + 0.999303i \(0.488113\pi\)
\(132\) 0 0
\(133\) −0.476559 −0.0413229
\(134\) 3.49983 0.302339
\(135\) 0 0
\(136\) 21.5972 1.85194
\(137\) −7.11516 −0.607889 −0.303945 0.952690i \(-0.598304\pi\)
−0.303945 + 0.952690i \(0.598304\pi\)
\(138\) 0 0
\(139\) 10.7111 0.908505 0.454253 0.890873i \(-0.349906\pi\)
0.454253 + 0.890873i \(0.349906\pi\)
\(140\) −0.869177 −0.0734588
\(141\) 0 0
\(142\) 12.2939 1.03168
\(143\) 2.15862 0.180513
\(144\) 0 0
\(145\) −16.0890 −1.33612
\(146\) −9.05871 −0.749704
\(147\) 0 0
\(148\) −3.66504 −0.301264
\(149\) 6.43632 0.527284 0.263642 0.964621i \(-0.415076\pi\)
0.263642 + 0.964621i \(0.415076\pi\)
\(150\) 0 0
\(151\) 0.803480 0.0653863 0.0326931 0.999465i \(-0.489592\pi\)
0.0326931 + 0.999465i \(0.489592\pi\)
\(152\) −1.45561 −0.118066
\(153\) 0 0
\(154\) −1.04777 −0.0844320
\(155\) 6.16765 0.495398
\(156\) 0 0
\(157\) −14.7959 −1.18084 −0.590419 0.807097i \(-0.701038\pi\)
−0.590419 + 0.807097i \(0.701038\pi\)
\(158\) −14.5863 −1.16042
\(159\) 0 0
\(160\) −4.80740 −0.380058
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) 3.13373 0.245453 0.122726 0.992441i \(-0.460836\pi\)
0.122726 + 0.992441i \(0.460836\pi\)
\(164\) 1.31881 0.102982
\(165\) 0 0
\(166\) 4.61267 0.358012
\(167\) 23.4168 1.81204 0.906022 0.423230i \(-0.139104\pi\)
0.906022 + 0.423230i \(0.139104\pi\)
\(168\) 0 0
\(169\) −6.49191 −0.499377
\(170\) −16.3076 −1.25074
\(171\) 0 0
\(172\) 0.128304 0.00978306
\(173\) 0.645424 0.0490706 0.0245353 0.999699i \(-0.492189\pi\)
0.0245353 + 0.999699i \(0.492189\pi\)
\(174\) 0 0
\(175\) −1.53098 −0.115731
\(176\) −2.41060 −0.181706
\(177\) 0 0
\(178\) −10.8512 −0.813330
\(179\) 16.9141 1.26422 0.632110 0.774879i \(-0.282189\pi\)
0.632110 + 0.774879i \(0.282189\pi\)
\(180\) 0 0
\(181\) −4.61403 −0.342958 −0.171479 0.985188i \(-0.554855\pi\)
−0.171479 + 0.985188i \(0.554855\pi\)
\(182\) −3.15897 −0.234158
\(183\) 0 0
\(184\) 3.05442 0.225175
\(185\) 14.6277 1.07545
\(186\) 0 0
\(187\) 5.98298 0.437519
\(188\) −6.28857 −0.458641
\(189\) 0 0
\(190\) 1.09910 0.0797373
\(191\) −6.44380 −0.466257 −0.233129 0.972446i \(-0.574896\pi\)
−0.233129 + 0.972446i \(0.574896\pi\)
\(192\) 0 0
\(193\) 10.1400 0.729897 0.364948 0.931028i \(-0.381087\pi\)
0.364948 + 0.931028i \(0.381087\pi\)
\(194\) −2.25651 −0.162008
\(195\) 0 0
\(196\) −0.466664 −0.0333332
\(197\) 2.37475 0.169194 0.0845970 0.996415i \(-0.473040\pi\)
0.0845970 + 0.996415i \(0.473040\pi\)
\(198\) 0 0
\(199\) 18.6052 1.31889 0.659443 0.751754i \(-0.270792\pi\)
0.659443 + 0.751754i \(0.270792\pi\)
\(200\) −4.67625 −0.330661
\(201\) 0 0
\(202\) −18.3105 −1.28832
\(203\) −8.63827 −0.606288
\(204\) 0 0
\(205\) −5.26358 −0.367624
\(206\) −13.5385 −0.943273
\(207\) 0 0
\(208\) −7.26781 −0.503932
\(209\) −0.403242 −0.0278928
\(210\) 0 0
\(211\) −21.1247 −1.45429 −0.727144 0.686485i \(-0.759153\pi\)
−0.727144 + 0.686485i \(0.759153\pi\)
\(212\) 4.16886 0.286319
\(213\) 0 0
\(214\) −21.3771 −1.46131
\(215\) −0.512080 −0.0349236
\(216\) 0 0
\(217\) 3.31143 0.224795
\(218\) 16.3995 1.11071
\(219\) 0 0
\(220\) −0.735456 −0.0495845
\(221\) 18.0383 1.21339
\(222\) 0 0
\(223\) −21.5426 −1.44260 −0.721301 0.692622i \(-0.756456\pi\)
−0.721301 + 0.692622i \(0.756456\pi\)
\(224\) −2.58111 −0.172458
\(225\) 0 0
\(226\) −8.79450 −0.585001
\(227\) −12.9135 −0.857102 −0.428551 0.903518i \(-0.640976\pi\)
−0.428551 + 0.903518i \(0.640976\pi\)
\(228\) 0 0
\(229\) 28.4835 1.88224 0.941120 0.338074i \(-0.109775\pi\)
0.941120 + 0.338074i \(0.109775\pi\)
\(230\) −2.30633 −0.152075
\(231\) 0 0
\(232\) −26.3849 −1.73225
\(233\) −2.58296 −0.169215 −0.0846077 0.996414i \(-0.526964\pi\)
−0.0846077 + 0.996414i \(0.526964\pi\)
\(234\) 0 0
\(235\) 25.0987 1.63726
\(236\) 0.775087 0.0504539
\(237\) 0 0
\(238\) −8.75562 −0.567543
\(239\) −11.9610 −0.773692 −0.386846 0.922144i \(-0.626435\pi\)
−0.386846 + 0.922144i \(0.626435\pi\)
\(240\) 0 0
\(241\) 2.03140 0.130854 0.0654269 0.997857i \(-0.479159\pi\)
0.0654269 + 0.997857i \(0.479159\pi\)
\(242\) 12.7345 0.818604
\(243\) 0 0
\(244\) 5.46754 0.350023
\(245\) 1.86253 0.118993
\(246\) 0 0
\(247\) −1.21575 −0.0773562
\(248\) 10.1145 0.642272
\(249\) 0 0
\(250\) 15.0626 0.952643
\(251\) −27.4454 −1.73234 −0.866169 0.499750i \(-0.833425\pi\)
−0.866169 + 0.499750i \(0.833425\pi\)
\(252\) 0 0
\(253\) 0.846153 0.0531972
\(254\) −19.9488 −1.25170
\(255\) 0 0
\(256\) −10.5427 −0.658922
\(257\) 25.1645 1.56972 0.784860 0.619673i \(-0.212735\pi\)
0.784860 + 0.619673i \(0.212735\pi\)
\(258\) 0 0
\(259\) 7.85369 0.488005
\(260\) −2.21735 −0.137514
\(261\) 0 0
\(262\) −1.05831 −0.0653828
\(263\) −10.8612 −0.669732 −0.334866 0.942266i \(-0.608691\pi\)
−0.334866 + 0.942266i \(0.608691\pi\)
\(264\) 0 0
\(265\) −16.6386 −1.02210
\(266\) 0.590113 0.0361821
\(267\) 0 0
\(268\) 1.31896 0.0805685
\(269\) 2.02133 0.123243 0.0616215 0.998100i \(-0.480373\pi\)
0.0616215 + 0.998100i \(0.480373\pi\)
\(270\) 0 0
\(271\) −4.05733 −0.246465 −0.123233 0.992378i \(-0.539326\pi\)
−0.123233 + 0.992378i \(0.539326\pi\)
\(272\) −20.1440 −1.22141
\(273\) 0 0
\(274\) 8.81056 0.532265
\(275\) −1.29544 −0.0781181
\(276\) 0 0
\(277\) −31.6071 −1.89909 −0.949544 0.313634i \(-0.898454\pi\)
−0.949544 + 0.313634i \(0.898454\pi\)
\(278\) −13.2634 −0.795483
\(279\) 0 0
\(280\) 5.68895 0.339980
\(281\) −25.4680 −1.51929 −0.759646 0.650337i \(-0.774628\pi\)
−0.759646 + 0.650337i \(0.774628\pi\)
\(282\) 0 0
\(283\) −8.06860 −0.479629 −0.239814 0.970819i \(-0.577087\pi\)
−0.239814 + 0.970819i \(0.577087\pi\)
\(284\) 4.63315 0.274927
\(285\) 0 0
\(286\) −2.67297 −0.158056
\(287\) −2.82603 −0.166816
\(288\) 0 0
\(289\) 32.9962 1.94095
\(290\) 19.9227 1.16990
\(291\) 0 0
\(292\) −3.41391 −0.199784
\(293\) 12.5766 0.734730 0.367365 0.930077i \(-0.380260\pi\)
0.367365 + 0.930077i \(0.380260\pi\)
\(294\) 0 0
\(295\) −3.09350 −0.180110
\(296\) 23.9885 1.39430
\(297\) 0 0
\(298\) −7.96996 −0.461688
\(299\) 2.55110 0.147534
\(300\) 0 0
\(301\) −0.274938 −0.0158472
\(302\) −0.994933 −0.0572519
\(303\) 0 0
\(304\) 1.35767 0.0778675
\(305\) −21.8218 −1.24951
\(306\) 0 0
\(307\) 0.588753 0.0336019 0.0168009 0.999859i \(-0.494652\pi\)
0.0168009 + 0.999859i \(0.494652\pi\)
\(308\) −0.394869 −0.0224998
\(309\) 0 0
\(310\) −7.63727 −0.433768
\(311\) −2.32490 −0.131833 −0.0659165 0.997825i \(-0.520997\pi\)
−0.0659165 + 0.997825i \(0.520997\pi\)
\(312\) 0 0
\(313\) −8.02863 −0.453805 −0.226903 0.973917i \(-0.572860\pi\)
−0.226903 + 0.973917i \(0.572860\pi\)
\(314\) 18.3214 1.03394
\(315\) 0 0
\(316\) −5.49706 −0.309234
\(317\) −27.0431 −1.51889 −0.759445 0.650572i \(-0.774529\pi\)
−0.759445 + 0.650572i \(0.774529\pi\)
\(318\) 0 0
\(319\) −7.30930 −0.409242
\(320\) 16.5652 0.926024
\(321\) 0 0
\(322\) −1.23828 −0.0690066
\(323\) −3.36965 −0.187492
\(324\) 0 0
\(325\) −3.90567 −0.216648
\(326\) −3.88043 −0.214917
\(327\) 0 0
\(328\) −8.63189 −0.476617
\(329\) 13.4756 0.742933
\(330\) 0 0
\(331\) 22.7555 1.25075 0.625377 0.780323i \(-0.284945\pi\)
0.625377 + 0.780323i \(0.284945\pi\)
\(332\) 1.73835 0.0954045
\(333\) 0 0
\(334\) −28.9965 −1.58662
\(335\) −5.26419 −0.287613
\(336\) 0 0
\(337\) −1.05919 −0.0576978 −0.0288489 0.999584i \(-0.509184\pi\)
−0.0288489 + 0.999584i \(0.509184\pi\)
\(338\) 8.03879 0.437253
\(339\) 0 0
\(340\) −6.14577 −0.333301
\(341\) 2.80198 0.151736
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −0.839776 −0.0452777
\(345\) 0 0
\(346\) −0.799215 −0.0429660
\(347\) −14.2245 −0.763611 −0.381806 0.924243i \(-0.624698\pi\)
−0.381806 + 0.924243i \(0.624698\pi\)
\(348\) 0 0
\(349\) −23.4865 −1.25721 −0.628603 0.777727i \(-0.716373\pi\)
−0.628603 + 0.777727i \(0.716373\pi\)
\(350\) 1.89578 0.101334
\(351\) 0 0
\(352\) −2.18401 −0.116408
\(353\) 1.19652 0.0636843 0.0318422 0.999493i \(-0.489863\pi\)
0.0318422 + 0.999493i \(0.489863\pi\)
\(354\) 0 0
\(355\) −18.4916 −0.981434
\(356\) −4.08943 −0.216739
\(357\) 0 0
\(358\) −20.9444 −1.10695
\(359\) −8.62376 −0.455145 −0.227572 0.973761i \(-0.573079\pi\)
−0.227572 + 0.973761i \(0.573079\pi\)
\(360\) 0 0
\(361\) −18.7729 −0.988047
\(362\) 5.71346 0.300293
\(363\) 0 0
\(364\) −1.19051 −0.0623994
\(365\) 13.6255 0.713189
\(366\) 0 0
\(367\) 1.40241 0.0732050 0.0366025 0.999330i \(-0.488346\pi\)
0.0366025 + 0.999330i \(0.488346\pi\)
\(368\) −2.84890 −0.148509
\(369\) 0 0
\(370\) −18.1132 −0.941662
\(371\) −8.93333 −0.463795
\(372\) 0 0
\(373\) 5.38461 0.278805 0.139402 0.990236i \(-0.455482\pi\)
0.139402 + 0.990236i \(0.455482\pi\)
\(374\) −7.40860 −0.383089
\(375\) 0 0
\(376\) 41.1601 2.12267
\(377\) −22.0371 −1.13497
\(378\) 0 0
\(379\) 26.3413 1.35306 0.676532 0.736413i \(-0.263482\pi\)
0.676532 + 0.736413i \(0.263482\pi\)
\(380\) 0.414214 0.0212487
\(381\) 0 0
\(382\) 7.97923 0.408253
\(383\) −23.4277 −1.19710 −0.598550 0.801085i \(-0.704256\pi\)
−0.598550 + 0.801085i \(0.704256\pi\)
\(384\) 0 0
\(385\) 1.57599 0.0803197
\(386\) −12.5562 −0.639094
\(387\) 0 0
\(388\) −0.850399 −0.0431725
\(389\) 13.5760 0.688330 0.344165 0.938909i \(-0.388162\pi\)
0.344165 + 0.938909i \(0.388162\pi\)
\(390\) 0 0
\(391\) 7.07080 0.357586
\(392\) 3.05442 0.154271
\(393\) 0 0
\(394\) −2.94060 −0.148146
\(395\) 21.9396 1.10390
\(396\) 0 0
\(397\) 3.24696 0.162960 0.0814801 0.996675i \(-0.474035\pi\)
0.0814801 + 0.996675i \(0.474035\pi\)
\(398\) −23.0384 −1.15481
\(399\) 0 0
\(400\) 4.36160 0.218080
\(401\) −30.2805 −1.51213 −0.756067 0.654494i \(-0.772882\pi\)
−0.756067 + 0.654494i \(0.772882\pi\)
\(402\) 0 0
\(403\) 8.44779 0.420814
\(404\) −6.90057 −0.343316
\(405\) 0 0
\(406\) 10.6966 0.530863
\(407\) 6.64542 0.329401
\(408\) 0 0
\(409\) 12.3775 0.612029 0.306014 0.952027i \(-0.401004\pi\)
0.306014 + 0.952027i \(0.401004\pi\)
\(410\) 6.51778 0.321890
\(411\) 0 0
\(412\) −5.10219 −0.251367
\(413\) −1.66091 −0.0817280
\(414\) 0 0
\(415\) −6.93804 −0.340575
\(416\) −6.58466 −0.322840
\(417\) 0 0
\(418\) 0.499326 0.0244228
\(419\) −5.46520 −0.266992 −0.133496 0.991049i \(-0.542620\pi\)
−0.133496 + 0.991049i \(0.542620\pi\)
\(420\) 0 0
\(421\) −6.55232 −0.319340 −0.159670 0.987170i \(-0.551043\pi\)
−0.159670 + 0.987170i \(0.551043\pi\)
\(422\) 26.1583 1.27337
\(423\) 0 0
\(424\) −27.2861 −1.32513
\(425\) −10.8252 −0.525101
\(426\) 0 0
\(427\) −11.7162 −0.566988
\(428\) −8.05629 −0.389416
\(429\) 0 0
\(430\) 0.634099 0.0305789
\(431\) 7.88645 0.379877 0.189938 0.981796i \(-0.439171\pi\)
0.189938 + 0.981796i \(0.439171\pi\)
\(432\) 0 0
\(433\) −33.7313 −1.62102 −0.810511 0.585723i \(-0.800811\pi\)
−0.810511 + 0.585723i \(0.800811\pi\)
\(434\) −4.10048 −0.196829
\(435\) 0 0
\(436\) 6.18040 0.295988
\(437\) −0.476559 −0.0227969
\(438\) 0 0
\(439\) −13.6800 −0.652909 −0.326455 0.945213i \(-0.605854\pi\)
−0.326455 + 0.945213i \(0.605854\pi\)
\(440\) 4.81372 0.229485
\(441\) 0 0
\(442\) −22.3364 −1.06244
\(443\) 17.2726 0.820646 0.410323 0.911940i \(-0.365416\pi\)
0.410323 + 0.911940i \(0.365416\pi\)
\(444\) 0 0
\(445\) 16.3216 0.773716
\(446\) 26.6758 1.26314
\(447\) 0 0
\(448\) 8.89393 0.420199
\(449\) −8.26286 −0.389949 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(450\) 0 0
\(451\) −2.39126 −0.112600
\(452\) −3.31434 −0.155893
\(453\) 0 0
\(454\) 15.9906 0.750475
\(455\) 4.75150 0.222754
\(456\) 0 0
\(457\) 2.80779 0.131343 0.0656714 0.997841i \(-0.479081\pi\)
0.0656714 + 0.997841i \(0.479081\pi\)
\(458\) −35.2705 −1.64808
\(459\) 0 0
\(460\) −0.869177 −0.0405256
\(461\) 0.220189 0.0102552 0.00512761 0.999987i \(-0.498368\pi\)
0.00512761 + 0.999987i \(0.498368\pi\)
\(462\) 0 0
\(463\) 25.5262 1.18630 0.593152 0.805091i \(-0.297883\pi\)
0.593152 + 0.805091i \(0.297883\pi\)
\(464\) 24.6095 1.14247
\(465\) 0 0
\(466\) 3.19843 0.148164
\(467\) −6.43084 −0.297584 −0.148792 0.988869i \(-0.547538\pi\)
−0.148792 + 0.988869i \(0.547538\pi\)
\(468\) 0 0
\(469\) −2.82636 −0.130509
\(470\) −31.0792 −1.43358
\(471\) 0 0
\(472\) −5.07312 −0.233509
\(473\) −0.232640 −0.0106968
\(474\) 0 0
\(475\) 0.729601 0.0334764
\(476\) −3.29969 −0.151241
\(477\) 0 0
\(478\) 14.8110 0.677441
\(479\) −25.1810 −1.15055 −0.575276 0.817960i \(-0.695105\pi\)
−0.575276 + 0.817960i \(0.695105\pi\)
\(480\) 0 0
\(481\) 20.0355 0.913541
\(482\) −2.51544 −0.114575
\(483\) 0 0
\(484\) 4.79919 0.218145
\(485\) 3.39408 0.154117
\(486\) 0 0
\(487\) 37.4074 1.69509 0.847545 0.530724i \(-0.178080\pi\)
0.847545 + 0.530724i \(0.178080\pi\)
\(488\) −35.7863 −1.61997
\(489\) 0 0
\(490\) −2.30633 −0.104190
\(491\) 8.61145 0.388629 0.194315 0.980939i \(-0.437752\pi\)
0.194315 + 0.980939i \(0.437752\pi\)
\(492\) 0 0
\(493\) −61.0795 −2.75088
\(494\) 1.50543 0.0677327
\(495\) 0 0
\(496\) −9.43393 −0.423596
\(497\) −9.92823 −0.445342
\(498\) 0 0
\(499\) −31.2249 −1.39782 −0.698909 0.715211i \(-0.746331\pi\)
−0.698909 + 0.715211i \(0.746331\pi\)
\(500\) 5.67657 0.253864
\(501\) 0 0
\(502\) 33.9851 1.51683
\(503\) −12.5421 −0.559223 −0.279612 0.960113i \(-0.590206\pi\)
−0.279612 + 0.960113i \(0.590206\pi\)
\(504\) 0 0
\(505\) 27.5413 1.22557
\(506\) −1.04777 −0.0465792
\(507\) 0 0
\(508\) −7.51799 −0.333557
\(509\) −12.5436 −0.555986 −0.277993 0.960583i \(-0.589669\pi\)
−0.277993 + 0.960583i \(0.589669\pi\)
\(510\) 0 0
\(511\) 7.31556 0.323621
\(512\) 24.7568 1.09410
\(513\) 0 0
\(514\) −31.1607 −1.37444
\(515\) 20.3637 0.897330
\(516\) 0 0
\(517\) 11.4024 0.501477
\(518\) −9.72506 −0.427295
\(519\) 0 0
\(520\) 14.5131 0.636440
\(521\) −18.3842 −0.805426 −0.402713 0.915326i \(-0.631933\pi\)
−0.402713 + 0.915326i \(0.631933\pi\)
\(522\) 0 0
\(523\) 21.0572 0.920769 0.460384 0.887720i \(-0.347712\pi\)
0.460384 + 0.887720i \(0.347712\pi\)
\(524\) −0.398841 −0.0174235
\(525\) 0 0
\(526\) 13.4492 0.586414
\(527\) 23.4145 1.01995
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 20.6032 0.894947
\(531\) 0 0
\(532\) 0.222393 0.00964196
\(533\) −7.20949 −0.312278
\(534\) 0 0
\(535\) 32.1539 1.39014
\(536\) −8.63290 −0.372885
\(537\) 0 0
\(538\) −2.50298 −0.107911
\(539\) 0.846153 0.0364464
\(540\) 0 0
\(541\) 30.0230 1.29079 0.645395 0.763849i \(-0.276693\pi\)
0.645395 + 0.763849i \(0.276693\pi\)
\(542\) 5.02411 0.215804
\(543\) 0 0
\(544\) −18.2505 −0.782484
\(545\) −24.6670 −1.05662
\(546\) 0 0
\(547\) −9.39847 −0.401849 −0.200925 0.979607i \(-0.564395\pi\)
−0.200925 + 0.979607i \(0.564395\pi\)
\(548\) 3.32039 0.141840
\(549\) 0 0
\(550\) 1.60412 0.0683999
\(551\) 4.11664 0.175375
\(552\) 0 0
\(553\) 11.7795 0.500915
\(554\) 39.1384 1.66283
\(555\) 0 0
\(556\) −4.99850 −0.211983
\(557\) 22.6305 0.958885 0.479443 0.877573i \(-0.340839\pi\)
0.479443 + 0.877573i \(0.340839\pi\)
\(558\) 0 0
\(559\) −0.701393 −0.0296658
\(560\) −5.30616 −0.224226
\(561\) 0 0
\(562\) 31.5365 1.33029
\(563\) 21.0667 0.887854 0.443927 0.896063i \(-0.353585\pi\)
0.443927 + 0.896063i \(0.353585\pi\)
\(564\) 0 0
\(565\) 13.2281 0.556508
\(566\) 9.99118 0.419961
\(567\) 0 0
\(568\) −30.3250 −1.27241
\(569\) 40.2625 1.68789 0.843945 0.536430i \(-0.180227\pi\)
0.843945 + 0.536430i \(0.180227\pi\)
\(570\) 0 0
\(571\) 7.61151 0.318532 0.159266 0.987236i \(-0.449087\pi\)
0.159266 + 0.987236i \(0.449087\pi\)
\(572\) −1.00735 −0.0421194
\(573\) 0 0
\(574\) 3.49942 0.146063
\(575\) −1.53098 −0.0638462
\(576\) 0 0
\(577\) 18.6364 0.775845 0.387923 0.921692i \(-0.373193\pi\)
0.387923 + 0.921692i \(0.373193\pi\)
\(578\) −40.8585 −1.69949
\(579\) 0 0
\(580\) 7.50818 0.311760
\(581\) −3.72506 −0.154542
\(582\) 0 0
\(583\) −7.55896 −0.313060
\(584\) 22.3448 0.924634
\(585\) 0 0
\(586\) −15.5733 −0.643327
\(587\) −40.9445 −1.68996 −0.844981 0.534797i \(-0.820388\pi\)
−0.844981 + 0.534797i \(0.820388\pi\)
\(588\) 0 0
\(589\) −1.57809 −0.0650242
\(590\) 3.83061 0.157704
\(591\) 0 0
\(592\) −22.3743 −0.919580
\(593\) −20.5649 −0.844501 −0.422251 0.906479i \(-0.638760\pi\)
−0.422251 + 0.906479i \(0.638760\pi\)
\(594\) 0 0
\(595\) 13.1696 0.539900
\(596\) −3.00360 −0.123032
\(597\) 0 0
\(598\) −3.15897 −0.129180
\(599\) −2.94630 −0.120382 −0.0601912 0.998187i \(-0.519171\pi\)
−0.0601912 + 0.998187i \(0.519171\pi\)
\(600\) 0 0
\(601\) 37.8860 1.54540 0.772702 0.634769i \(-0.218905\pi\)
0.772702 + 0.634769i \(0.218905\pi\)
\(602\) 0.340450 0.0138757
\(603\) 0 0
\(604\) −0.374955 −0.0152567
\(605\) −19.1543 −0.778734
\(606\) 0 0
\(607\) −7.00707 −0.284408 −0.142204 0.989837i \(-0.545419\pi\)
−0.142204 + 0.989837i \(0.545419\pi\)
\(608\) 1.23005 0.0498851
\(609\) 0 0
\(610\) 27.0215 1.09407
\(611\) 34.3775 1.39077
\(612\) 0 0
\(613\) −18.9408 −0.765012 −0.382506 0.923953i \(-0.624939\pi\)
−0.382506 + 0.923953i \(0.624939\pi\)
\(614\) −0.729040 −0.0294217
\(615\) 0 0
\(616\) 2.58451 0.104133
\(617\) 2.22511 0.0895797 0.0447898 0.998996i \(-0.485738\pi\)
0.0447898 + 0.998996i \(0.485738\pi\)
\(618\) 0 0
\(619\) −15.2083 −0.611272 −0.305636 0.952148i \(-0.598869\pi\)
−0.305636 + 0.952148i \(0.598869\pi\)
\(620\) −2.87822 −0.115592
\(621\) 0 0
\(622\) 2.87888 0.115432
\(623\) 8.76310 0.351086
\(624\) 0 0
\(625\) −15.0012 −0.600049
\(626\) 9.94169 0.397350
\(627\) 0 0
\(628\) 6.90470 0.275527
\(629\) 55.5319 2.21420
\(630\) 0 0
\(631\) 6.16049 0.245245 0.122623 0.992453i \(-0.460870\pi\)
0.122623 + 0.992453i \(0.460870\pi\)
\(632\) 35.9795 1.43119
\(633\) 0 0
\(634\) 33.4869 1.32993
\(635\) 30.0055 1.19073
\(636\) 0 0
\(637\) 2.55110 0.101078
\(638\) 9.05096 0.358331
\(639\) 0 0
\(640\) −10.8976 −0.430764
\(641\) −16.3586 −0.646126 −0.323063 0.946377i \(-0.604713\pi\)
−0.323063 + 0.946377i \(0.604713\pi\)
\(642\) 0 0
\(643\) −30.8553 −1.21681 −0.608407 0.793625i \(-0.708191\pi\)
−0.608407 + 0.793625i \(0.708191\pi\)
\(644\) −0.466664 −0.0183891
\(645\) 0 0
\(646\) 4.17257 0.164168
\(647\) −10.3205 −0.405742 −0.202871 0.979206i \(-0.565027\pi\)
−0.202871 + 0.979206i \(0.565027\pi\)
\(648\) 0 0
\(649\) −1.40538 −0.0551662
\(650\) 4.83631 0.189696
\(651\) 0 0
\(652\) −1.46240 −0.0572720
\(653\) −1.58679 −0.0620961 −0.0310480 0.999518i \(-0.509884\pi\)
−0.0310480 + 0.999518i \(0.509884\pi\)
\(654\) 0 0
\(655\) 1.59184 0.0621983
\(656\) 8.05108 0.314342
\(657\) 0 0
\(658\) −16.6865 −0.650509
\(659\) 43.3271 1.68779 0.843893 0.536512i \(-0.180258\pi\)
0.843893 + 0.536512i \(0.180258\pi\)
\(660\) 0 0
\(661\) 18.2262 0.708917 0.354458 0.935072i \(-0.384665\pi\)
0.354458 + 0.935072i \(0.384665\pi\)
\(662\) −28.1776 −1.09516
\(663\) 0 0
\(664\) −11.3779 −0.441548
\(665\) −0.887605 −0.0344199
\(666\) 0 0
\(667\) −8.63827 −0.334475
\(668\) −10.9278 −0.422808
\(669\) 0 0
\(670\) 6.51854 0.251833
\(671\) −9.91372 −0.382715
\(672\) 0 0
\(673\) 6.47655 0.249653 0.124826 0.992179i \(-0.460163\pi\)
0.124826 + 0.992179i \(0.460163\pi\)
\(674\) 1.31157 0.0505199
\(675\) 0 0
\(676\) 3.02954 0.116521
\(677\) 8.19344 0.314899 0.157450 0.987527i \(-0.449673\pi\)
0.157450 + 0.987527i \(0.449673\pi\)
\(678\) 0 0
\(679\) 1.82229 0.0699332
\(680\) 40.2254 1.54257
\(681\) 0 0
\(682\) −3.46963 −0.132859
\(683\) 7.95826 0.304514 0.152257 0.988341i \(-0.451346\pi\)
0.152257 + 0.988341i \(0.451346\pi\)
\(684\) 0 0
\(685\) −13.2522 −0.506341
\(686\) −1.23828 −0.0472777
\(687\) 0 0
\(688\) 0.783270 0.0298619
\(689\) −22.7898 −0.868222
\(690\) 0 0
\(691\) −3.66850 −0.139556 −0.0697782 0.997563i \(-0.522229\pi\)
−0.0697782 + 0.997563i \(0.522229\pi\)
\(692\) −0.301196 −0.0114498
\(693\) 0 0
\(694\) 17.6139 0.668615
\(695\) 19.9498 0.756739
\(696\) 0 0
\(697\) −19.9823 −0.756884
\(698\) 29.0829 1.10080
\(699\) 0 0
\(700\) 0.714453 0.0270038
\(701\) −40.6182 −1.53413 −0.767063 0.641571i \(-0.778283\pi\)
−0.767063 + 0.641571i \(0.778283\pi\)
\(702\) 0 0
\(703\) −3.74274 −0.141160
\(704\) 7.52562 0.283633
\(705\) 0 0
\(706\) −1.48163 −0.0557617
\(707\) 14.7870 0.556123
\(708\) 0 0
\(709\) −38.7523 −1.45537 −0.727687 0.685910i \(-0.759405\pi\)
−0.727687 + 0.685910i \(0.759405\pi\)
\(710\) 22.8978 0.859339
\(711\) 0 0
\(712\) 26.7662 1.00311
\(713\) 3.31143 0.124014
\(714\) 0 0
\(715\) 4.02049 0.150358
\(716\) −7.89321 −0.294983
\(717\) 0 0
\(718\) 10.6786 0.398523
\(719\) −34.5902 −1.29000 −0.644998 0.764184i \(-0.723142\pi\)
−0.644998 + 0.764184i \(0.723142\pi\)
\(720\) 0 0
\(721\) 10.9333 0.407178
\(722\) 23.2461 0.865130
\(723\) 0 0
\(724\) 2.15320 0.0800231
\(725\) 13.2250 0.491164
\(726\) 0 0
\(727\) −8.51989 −0.315985 −0.157993 0.987440i \(-0.550502\pi\)
−0.157993 + 0.987440i \(0.550502\pi\)
\(728\) 7.79212 0.288795
\(729\) 0 0
\(730\) −16.8721 −0.624465
\(731\) −1.94403 −0.0719026
\(732\) 0 0
\(733\) −35.1739 −1.29918 −0.649590 0.760285i \(-0.725059\pi\)
−0.649590 + 0.760285i \(0.725059\pi\)
\(734\) −1.73657 −0.0640980
\(735\) 0 0
\(736\) −2.58111 −0.0951410
\(737\) −2.39154 −0.0880934
\(738\) 0 0
\(739\) 15.1355 0.556769 0.278384 0.960470i \(-0.410201\pi\)
0.278384 + 0.960470i \(0.410201\pi\)
\(740\) −6.82624 −0.250938
\(741\) 0 0
\(742\) 11.0620 0.406097
\(743\) 20.2165 0.741670 0.370835 0.928699i \(-0.379072\pi\)
0.370835 + 0.928699i \(0.379072\pi\)
\(744\) 0 0
\(745\) 11.9878 0.439201
\(746\) −6.66765 −0.244120
\(747\) 0 0
\(748\) −2.79204 −0.102087
\(749\) 17.2636 0.630797
\(750\) 0 0
\(751\) 31.1017 1.13492 0.567459 0.823401i \(-0.307926\pi\)
0.567459 + 0.823401i \(0.307926\pi\)
\(752\) −38.3905 −1.39996
\(753\) 0 0
\(754\) 27.2880 0.993772
\(755\) 1.49651 0.0544634
\(756\) 0 0
\(757\) −11.9728 −0.435158 −0.217579 0.976043i \(-0.569816\pi\)
−0.217579 + 0.976043i \(0.569816\pi\)
\(758\) −32.6179 −1.18474
\(759\) 0 0
\(760\) −2.71112 −0.0983426
\(761\) −17.8372 −0.646597 −0.323299 0.946297i \(-0.604792\pi\)
−0.323299 + 0.946297i \(0.604792\pi\)
\(762\) 0 0
\(763\) −13.2438 −0.479457
\(764\) 3.00709 0.108793
\(765\) 0 0
\(766\) 29.0101 1.04818
\(767\) −4.23714 −0.152994
\(768\) 0 0
\(769\) −47.9412 −1.72880 −0.864401 0.502802i \(-0.832302\pi\)
−0.864401 + 0.502802i \(0.832302\pi\)
\(770\) −1.95151 −0.0703276
\(771\) 0 0
\(772\) −4.73200 −0.170308
\(773\) −22.7079 −0.816748 −0.408374 0.912815i \(-0.633904\pi\)
−0.408374 + 0.912815i \(0.633904\pi\)
\(774\) 0 0
\(775\) −5.06973 −0.182110
\(776\) 5.56605 0.199810
\(777\) 0 0
\(778\) −16.8109 −0.602699
\(779\) 1.34677 0.0482531
\(780\) 0 0
\(781\) −8.40080 −0.300604
\(782\) −8.75562 −0.313100
\(783\) 0 0
\(784\) −2.84890 −0.101746
\(785\) −27.5577 −0.983578
\(786\) 0 0
\(787\) −20.2521 −0.721910 −0.360955 0.932583i \(-0.617549\pi\)
−0.360955 + 0.932583i \(0.617549\pi\)
\(788\) −1.10821 −0.0394784
\(789\) 0 0
\(790\) −27.1674 −0.966573
\(791\) 7.10219 0.252525
\(792\) 0 0
\(793\) −29.8892 −1.06140
\(794\) −4.02064 −0.142687
\(795\) 0 0
\(796\) −8.68237 −0.307739
\(797\) 22.1099 0.783172 0.391586 0.920142i \(-0.371927\pi\)
0.391586 + 0.920142i \(0.371927\pi\)
\(798\) 0 0
\(799\) 95.2831 3.37087
\(800\) 3.95162 0.139711
\(801\) 0 0
\(802\) 37.4957 1.32402
\(803\) 6.19008 0.218443
\(804\) 0 0
\(805\) 1.86253 0.0656456
\(806\) −10.4607 −0.368463
\(807\) 0 0
\(808\) 45.1658 1.58893
\(809\) 35.3424 1.24257 0.621287 0.783583i \(-0.286610\pi\)
0.621287 + 0.783583i \(0.286610\pi\)
\(810\) 0 0
\(811\) −15.3968 −0.540654 −0.270327 0.962769i \(-0.587132\pi\)
−0.270327 + 0.962769i \(0.587132\pi\)
\(812\) 4.03117 0.141466
\(813\) 0 0
\(814\) −8.22889 −0.288422
\(815\) 5.83667 0.204450
\(816\) 0 0
\(817\) 0.131024 0.00458395
\(818\) −15.3268 −0.535890
\(819\) 0 0
\(820\) 2.45632 0.0857785
\(821\) 47.6244 1.66210 0.831052 0.556195i \(-0.187739\pi\)
0.831052 + 0.556195i \(0.187739\pi\)
\(822\) 0 0
\(823\) 24.1554 0.842006 0.421003 0.907059i \(-0.361678\pi\)
0.421003 + 0.907059i \(0.361678\pi\)
\(824\) 33.3950 1.16337
\(825\) 0 0
\(826\) 2.05667 0.0715607
\(827\) 14.8594 0.516710 0.258355 0.966050i \(-0.416820\pi\)
0.258355 + 0.966050i \(0.416820\pi\)
\(828\) 0 0
\(829\) 20.2699 0.704003 0.352001 0.935999i \(-0.385501\pi\)
0.352001 + 0.935999i \(0.385501\pi\)
\(830\) 8.59124 0.298206
\(831\) 0 0
\(832\) 22.6893 0.786609
\(833\) 7.07080 0.244989
\(834\) 0 0
\(835\) 43.6145 1.50934
\(836\) 0.188178 0.00650829
\(837\) 0 0
\(838\) 6.76744 0.233777
\(839\) 20.9307 0.722609 0.361304 0.932448i \(-0.382332\pi\)
0.361304 + 0.932448i \(0.382332\pi\)
\(840\) 0 0
\(841\) 45.6197 1.57309
\(842\) 8.11360 0.279613
\(843\) 0 0
\(844\) 9.85816 0.339332
\(845\) −12.0914 −0.415956
\(846\) 0 0
\(847\) −10.2840 −0.353363
\(848\) 25.4501 0.873961
\(849\) 0 0
\(850\) 13.4047 0.459776
\(851\) 7.85369 0.269221
\(852\) 0 0
\(853\) −28.3729 −0.971470 −0.485735 0.874106i \(-0.661448\pi\)
−0.485735 + 0.874106i \(0.661448\pi\)
\(854\) 14.5080 0.496452
\(855\) 0 0
\(856\) 52.7302 1.80228
\(857\) 32.4441 1.10827 0.554134 0.832428i \(-0.313050\pi\)
0.554134 + 0.832428i \(0.313050\pi\)
\(858\) 0 0
\(859\) 32.7552 1.11759 0.558797 0.829305i \(-0.311263\pi\)
0.558797 + 0.829305i \(0.311263\pi\)
\(860\) 0.238970 0.00814880
\(861\) 0 0
\(862\) −9.76562 −0.332618
\(863\) −5.88556 −0.200347 −0.100173 0.994970i \(-0.531940\pi\)
−0.100173 + 0.994970i \(0.531940\pi\)
\(864\) 0 0
\(865\) 1.20212 0.0408734
\(866\) 41.7688 1.41936
\(867\) 0 0
\(868\) −1.54533 −0.0524518
\(869\) 9.96724 0.338116
\(870\) 0 0
\(871\) −7.21033 −0.244313
\(872\) −40.4521 −1.36988
\(873\) 0 0
\(874\) 0.590113 0.0199609
\(875\) −12.1641 −0.411223
\(876\) 0 0
\(877\) −53.7879 −1.81629 −0.908144 0.418658i \(-0.862501\pi\)
−0.908144 + 0.418658i \(0.862501\pi\)
\(878\) 16.9396 0.571685
\(879\) 0 0
\(880\) −4.48982 −0.151352
\(881\) −6.04127 −0.203536 −0.101768 0.994808i \(-0.532450\pi\)
−0.101768 + 0.994808i \(0.532450\pi\)
\(882\) 0 0
\(883\) 44.1976 1.48737 0.743683 0.668532i \(-0.233077\pi\)
0.743683 + 0.668532i \(0.233077\pi\)
\(884\) −8.41782 −0.283122
\(885\) 0 0
\(886\) −21.3883 −0.718554
\(887\) 36.4986 1.22550 0.612752 0.790275i \(-0.290063\pi\)
0.612752 + 0.790275i \(0.290063\pi\)
\(888\) 0 0
\(889\) 16.1101 0.540314
\(890\) −20.2106 −0.677462
\(891\) 0 0
\(892\) 10.0532 0.336605
\(893\) −6.42191 −0.214901
\(894\) 0 0
\(895\) 31.5030 1.05303
\(896\) −5.85095 −0.195466
\(897\) 0 0
\(898\) 10.2317 0.341437
\(899\) −28.6051 −0.954033
\(900\) 0 0
\(901\) −63.1658 −2.10436
\(902\) 2.96105 0.0985920
\(903\) 0 0
\(904\) 21.6931 0.721501
\(905\) −8.59377 −0.285667
\(906\) 0 0
\(907\) 8.22173 0.272998 0.136499 0.990640i \(-0.456415\pi\)
0.136499 + 0.990640i \(0.456415\pi\)
\(908\) 6.02629 0.199989
\(909\) 0 0
\(910\) −5.88368 −0.195042
\(911\) 21.5853 0.715154 0.357577 0.933884i \(-0.383603\pi\)
0.357577 + 0.933884i \(0.383603\pi\)
\(912\) 0 0
\(913\) −3.15197 −0.104315
\(914\) −3.47683 −0.115003
\(915\) 0 0
\(916\) −13.2922 −0.439187
\(917\) 0.854665 0.0282235
\(918\) 0 0
\(919\) −38.0934 −1.25658 −0.628292 0.777978i \(-0.716246\pi\)
−0.628292 + 0.777978i \(0.716246\pi\)
\(920\) 5.68895 0.187559
\(921\) 0 0
\(922\) −0.272656 −0.00897943
\(923\) −25.3279 −0.833677
\(924\) 0 0
\(925\) −12.0238 −0.395341
\(926\) −31.6086 −1.03872
\(927\) 0 0
\(928\) 22.2963 0.731913
\(929\) 9.15900 0.300497 0.150249 0.988648i \(-0.451993\pi\)
0.150249 + 0.988648i \(0.451993\pi\)
\(930\) 0 0
\(931\) −0.476559 −0.0156186
\(932\) 1.20538 0.0394834
\(933\) 0 0
\(934\) 7.96317 0.260563
\(935\) 11.1435 0.364431
\(936\) 0 0
\(937\) −18.8314 −0.615194 −0.307597 0.951517i \(-0.599525\pi\)
−0.307597 + 0.951517i \(0.599525\pi\)
\(938\) 3.49983 0.114273
\(939\) 0 0
\(940\) −11.7127 −0.382025
\(941\) −25.4216 −0.828721 −0.414360 0.910113i \(-0.635995\pi\)
−0.414360 + 0.910113i \(0.635995\pi\)
\(942\) 0 0
\(943\) −2.82603 −0.0920283
\(944\) 4.73176 0.154006
\(945\) 0 0
\(946\) 0.288073 0.00936605
\(947\) −33.3211 −1.08279 −0.541395 0.840768i \(-0.682104\pi\)
−0.541395 + 0.840768i \(0.682104\pi\)
\(948\) 0 0
\(949\) 18.6627 0.605817
\(950\) −0.903450 −0.0293118
\(951\) 0 0
\(952\) 21.5972 0.699969
\(953\) −30.5121 −0.988383 −0.494192 0.869353i \(-0.664536\pi\)
−0.494192 + 0.869353i \(0.664536\pi\)
\(954\) 0 0
\(955\) −12.0018 −0.388369
\(956\) 5.58176 0.180527
\(957\) 0 0
\(958\) 31.1812 1.00742
\(959\) −7.11516 −0.229761
\(960\) 0 0
\(961\) −20.0344 −0.646271
\(962\) −24.8096 −0.799893
\(963\) 0 0
\(964\) −0.947980 −0.0305324
\(965\) 18.8862 0.607967
\(966\) 0 0
\(967\) −23.0298 −0.740587 −0.370294 0.928915i \(-0.620743\pi\)
−0.370294 + 0.928915i \(0.620743\pi\)
\(968\) −31.4117 −1.00961
\(969\) 0 0
\(970\) −4.20282 −0.134944
\(971\) −6.95588 −0.223225 −0.111612 0.993752i \(-0.535602\pi\)
−0.111612 + 0.993752i \(0.535602\pi\)
\(972\) 0 0
\(973\) 10.7111 0.343383
\(974\) −46.3208 −1.48421
\(975\) 0 0
\(976\) 33.3783 1.06841
\(977\) 23.1596 0.740942 0.370471 0.928844i \(-0.379196\pi\)
0.370471 + 0.928844i \(0.379196\pi\)
\(978\) 0 0
\(979\) 7.41493 0.236982
\(980\) −0.869177 −0.0277648
\(981\) 0 0
\(982\) −10.6634 −0.340282
\(983\) −11.7384 −0.374397 −0.187198 0.982322i \(-0.559941\pi\)
−0.187198 + 0.982322i \(0.559941\pi\)
\(984\) 0 0
\(985\) 4.42305 0.140930
\(986\) 75.6334 2.40866
\(987\) 0 0
\(988\) 0.567346 0.0180497
\(989\) −0.274938 −0.00874252
\(990\) 0 0
\(991\) −14.8198 −0.470766 −0.235383 0.971903i \(-0.575634\pi\)
−0.235383 + 0.971903i \(0.575634\pi\)
\(992\) −8.54718 −0.271373
\(993\) 0 0
\(994\) 12.2939 0.389939
\(995\) 34.6527 1.09857
\(996\) 0 0
\(997\) −8.94358 −0.283246 −0.141623 0.989921i \(-0.545232\pi\)
−0.141623 + 0.989921i \(0.545232\pi\)
\(998\) 38.6651 1.22392
\(999\) 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 1449.2.a.r.1.3 5
3.2 odd 2 161.2.a.d.1.3 5
12.11 even 2 2576.2.a.bd.1.1 5
15.14 odd 2 4025.2.a.p.1.3 5
21.20 even 2 1127.2.a.h.1.3 5
69.68 even 2 3703.2.a.j.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
161.2.a.d.1.3 5 3.2 odd 2
1127.2.a.h.1.3 5 21.20 even 2
1449.2.a.r.1.3 5 1.1 even 1 trivial
2576.2.a.bd.1.1 5 12.11 even 2
3703.2.a.j.1.3 5 69.68 even 2
4025.2.a.p.1.3 5 15.14 odd 2