Properties

Label 931.2.a.p.1.10
Level $931$
Weight $2$
Character 931.1
Self dual yes
Analytic conductor $7.434$
Analytic rank $1$
Dimension $10$
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] = [931,2,Mod(1,931)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(931, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("931.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 931 = 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 931.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: \(7.43407242818\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} - 13x^{8} + 24x^{7} + 57x^{6} - 98x^{5} - 93x^{4} + 152x^{3} + 39x^{2} - 58x - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(-2.27289\) of defining polynomial
Character \(\chi\) \(=\) 931.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.27289 q^{2} -1.88926 q^{3} +3.16604 q^{4} -1.72209 q^{5} -4.29408 q^{6} +2.65028 q^{8} +0.569302 q^{9} +O(q^{10})\) \(q+2.27289 q^{2} -1.88926 q^{3} +3.16604 q^{4} -1.72209 q^{5} -4.29408 q^{6} +2.65028 q^{8} +0.569302 q^{9} -3.91413 q^{10} -2.87184 q^{11} -5.98147 q^{12} +1.56496 q^{13} +3.25348 q^{15} -0.308281 q^{16} -5.43502 q^{17} +1.29396 q^{18} -1.00000 q^{19} -5.45221 q^{20} -6.52738 q^{22} -6.09715 q^{23} -5.00706 q^{24} -2.03440 q^{25} +3.55699 q^{26} +4.59222 q^{27} +3.17391 q^{29} +7.39480 q^{30} +1.03449 q^{31} -6.00124 q^{32} +5.42565 q^{33} -12.3532 q^{34} +1.80243 q^{36} +8.44727 q^{37} -2.27289 q^{38} -2.95662 q^{39} -4.56402 q^{40} -10.4075 q^{41} -3.21975 q^{43} -9.09235 q^{44} -0.980390 q^{45} -13.8582 q^{46} +5.50514 q^{47} +0.582422 q^{48} -4.62398 q^{50} +10.2682 q^{51} +4.95473 q^{52} -12.9804 q^{53} +10.4376 q^{54} +4.94557 q^{55} +1.88926 q^{57} +7.21396 q^{58} +6.21573 q^{59} +10.3006 q^{60} -9.76784 q^{61} +2.35128 q^{62} -13.0236 q^{64} -2.69501 q^{65} +12.3319 q^{66} -2.55851 q^{67} -17.2075 q^{68} +11.5191 q^{69} +12.3595 q^{71} +1.50881 q^{72} +10.5517 q^{73} +19.1997 q^{74} +3.84351 q^{75} -3.16604 q^{76} -6.72008 q^{78} +12.9952 q^{79} +0.530887 q^{80} -10.3838 q^{81} -23.6552 q^{82} -5.38474 q^{83} +9.35961 q^{85} -7.31815 q^{86} -5.99634 q^{87} -7.61117 q^{88} -7.45375 q^{89} -2.22832 q^{90} -19.3038 q^{92} -1.95442 q^{93} +12.5126 q^{94} +1.72209 q^{95} +11.3379 q^{96} +2.45162 q^{97} -1.63494 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{2} - 4 q^{3} + 10 q^{4} - 16 q^{5} - 8 q^{6} - 6 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 2 q^{2} - 4 q^{3} + 10 q^{4} - 16 q^{5} - 8 q^{6} - 6 q^{8} + 10 q^{9} + 12 q^{10} - 12 q^{12} - 12 q^{13} + 2 q^{16} - 16 q^{17} + 2 q^{18} - 10 q^{19} - 32 q^{20} + 4 q^{22} + 12 q^{23} + 8 q^{24} + 14 q^{25} - 24 q^{26} - 16 q^{27} - 12 q^{29} - 12 q^{30} - 8 q^{31} - 34 q^{32} + 4 q^{33} - 16 q^{34} - 6 q^{36} + 4 q^{37} + 2 q^{38} - 4 q^{39} + 20 q^{40} - 40 q^{41} + 4 q^{43} - 20 q^{44} - 24 q^{45} - 32 q^{46} - 16 q^{47} - 12 q^{48} - 34 q^{50} - 28 q^{51} + 40 q^{52} - 8 q^{54} - 16 q^{55} + 4 q^{57} - 8 q^{58} - 36 q^{59} + 32 q^{60} - 16 q^{61} + 16 q^{62} + 18 q^{64} + 8 q^{65} - 8 q^{66} - 28 q^{67} - 40 q^{68} - 48 q^{69} - 12 q^{71} + 34 q^{72} + 24 q^{74} + 32 q^{75} - 10 q^{76} + 28 q^{78} - 8 q^{79} - 8 q^{80} + 14 q^{81} + 8 q^{82} + 40 q^{85} - 52 q^{86} + 8 q^{87} - 4 q^{88} - 48 q^{89} + 64 q^{90} + 28 q^{92} + 40 q^{93} + 36 q^{94} + 16 q^{95} + 8 q^{96} + 16 q^{97} - 8 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.27289 1.60718 0.803589 0.595185i \(-0.202921\pi\)
0.803589 + 0.595185i \(0.202921\pi\)
\(3\) −1.88926 −1.09076 −0.545382 0.838187i \(-0.683616\pi\)
−0.545382 + 0.838187i \(0.683616\pi\)
\(4\) 3.16604 1.58302
\(5\) −1.72209 −0.770143 −0.385071 0.922887i \(-0.625823\pi\)
−0.385071 + 0.922887i \(0.625823\pi\)
\(6\) −4.29408 −1.75305
\(7\) 0 0
\(8\) 2.65028 0.937015
\(9\) 0.569302 0.189767
\(10\) −3.91413 −1.23776
\(11\) −2.87184 −0.865892 −0.432946 0.901420i \(-0.642526\pi\)
−0.432946 + 0.901420i \(0.642526\pi\)
\(12\) −5.98147 −1.72670
\(13\) 1.56496 0.434043 0.217021 0.976167i \(-0.430366\pi\)
0.217021 + 0.976167i \(0.430366\pi\)
\(14\) 0 0
\(15\) 3.25348 0.840044
\(16\) −0.308281 −0.0770702
\(17\) −5.43502 −1.31819 −0.659094 0.752061i \(-0.729060\pi\)
−0.659094 + 0.752061i \(0.729060\pi\)
\(18\) 1.29396 0.304990
\(19\) −1.00000 −0.229416
\(20\) −5.45221 −1.21915
\(21\) 0 0
\(22\) −6.52738 −1.39164
\(23\) −6.09715 −1.27134 −0.635672 0.771959i \(-0.719277\pi\)
−0.635672 + 0.771959i \(0.719277\pi\)
\(24\) −5.00706 −1.02206
\(25\) −2.03440 −0.406880
\(26\) 3.55699 0.697583
\(27\) 4.59222 0.883773
\(28\) 0 0
\(29\) 3.17391 0.589380 0.294690 0.955593i \(-0.404784\pi\)
0.294690 + 0.955593i \(0.404784\pi\)
\(30\) 7.39480 1.35010
\(31\) 1.03449 0.185800 0.0929000 0.995675i \(-0.470386\pi\)
0.0929000 + 0.995675i \(0.470386\pi\)
\(32\) −6.00124 −1.06088
\(33\) 5.42565 0.944484
\(34\) −12.3532 −2.11856
\(35\) 0 0
\(36\) 1.80243 0.300405
\(37\) 8.44727 1.38872 0.694361 0.719626i \(-0.255687\pi\)
0.694361 + 0.719626i \(0.255687\pi\)
\(38\) −2.27289 −0.368712
\(39\) −2.95662 −0.473438
\(40\) −4.56402 −0.721635
\(41\) −10.4075 −1.62538 −0.812691 0.582694i \(-0.801999\pi\)
−0.812691 + 0.582694i \(0.801999\pi\)
\(42\) 0 0
\(43\) −3.21975 −0.491008 −0.245504 0.969396i \(-0.578953\pi\)
−0.245504 + 0.969396i \(0.578953\pi\)
\(44\) −9.09235 −1.37072
\(45\) −0.980390 −0.146148
\(46\) −13.8582 −2.04327
\(47\) 5.50514 0.803008 0.401504 0.915857i \(-0.368488\pi\)
0.401504 + 0.915857i \(0.368488\pi\)
\(48\) 0.582422 0.0840654
\(49\) 0 0
\(50\) −4.62398 −0.653929
\(51\) 10.2682 1.43783
\(52\) 4.95473 0.687097
\(53\) −12.9804 −1.78299 −0.891495 0.453030i \(-0.850343\pi\)
−0.891495 + 0.453030i \(0.850343\pi\)
\(54\) 10.4376 1.42038
\(55\) 4.94557 0.666860
\(56\) 0 0
\(57\) 1.88926 0.250239
\(58\) 7.21396 0.947239
\(59\) 6.21573 0.809219 0.404609 0.914490i \(-0.367407\pi\)
0.404609 + 0.914490i \(0.367407\pi\)
\(60\) 10.3006 1.32981
\(61\) −9.76784 −1.25064 −0.625322 0.780367i \(-0.715032\pi\)
−0.625322 + 0.780367i \(0.715032\pi\)
\(62\) 2.35128 0.298613
\(63\) 0 0
\(64\) −13.0236 −1.62795
\(65\) −2.69501 −0.334275
\(66\) 12.3319 1.51795
\(67\) −2.55851 −0.312572 −0.156286 0.987712i \(-0.549952\pi\)
−0.156286 + 0.987712i \(0.549952\pi\)
\(68\) −17.2075 −2.08672
\(69\) 11.5191 1.38674
\(70\) 0 0
\(71\) 12.3595 1.46681 0.733404 0.679793i \(-0.237930\pi\)
0.733404 + 0.679793i \(0.237930\pi\)
\(72\) 1.50881 0.177815
\(73\) 10.5517 1.23498 0.617491 0.786578i \(-0.288149\pi\)
0.617491 + 0.786578i \(0.288149\pi\)
\(74\) 19.1997 2.23192
\(75\) 3.84351 0.443811
\(76\) −3.16604 −0.363169
\(77\) 0 0
\(78\) −6.72008 −0.760899
\(79\) 12.9952 1.46208 0.731040 0.682335i \(-0.239035\pi\)
0.731040 + 0.682335i \(0.239035\pi\)
\(80\) 0.530887 0.0593550
\(81\) −10.3838 −1.15376
\(82\) −23.6552 −2.61228
\(83\) −5.38474 −0.591052 −0.295526 0.955335i \(-0.595495\pi\)
−0.295526 + 0.955335i \(0.595495\pi\)
\(84\) 0 0
\(85\) 9.35961 1.01519
\(86\) −7.31815 −0.789136
\(87\) −5.99634 −0.642875
\(88\) −7.61117 −0.811354
\(89\) −7.45375 −0.790096 −0.395048 0.918661i \(-0.629272\pi\)
−0.395048 + 0.918661i \(0.629272\pi\)
\(90\) −2.22832 −0.234886
\(91\) 0 0
\(92\) −19.3038 −2.01256
\(93\) −1.95442 −0.202664
\(94\) 12.5126 1.29058
\(95\) 1.72209 0.176683
\(96\) 11.3379 1.15717
\(97\) 2.45162 0.248925 0.124462 0.992224i \(-0.460279\pi\)
0.124462 + 0.992224i \(0.460279\pi\)
\(98\) 0 0
\(99\) −1.63494 −0.164318
\(100\) −6.44099 −0.644099
\(101\) 1.46770 0.146041 0.0730207 0.997330i \(-0.476736\pi\)
0.0730207 + 0.997330i \(0.476736\pi\)
\(102\) 23.3384 2.31085
\(103\) 11.2451 1.10802 0.554008 0.832511i \(-0.313098\pi\)
0.554008 + 0.832511i \(0.313098\pi\)
\(104\) 4.14759 0.406704
\(105\) 0 0
\(106\) −29.5030 −2.86558
\(107\) −0.243353 −0.0235258 −0.0117629 0.999931i \(-0.503744\pi\)
−0.0117629 + 0.999931i \(0.503744\pi\)
\(108\) 14.5391 1.39903
\(109\) 16.1331 1.54527 0.772636 0.634849i \(-0.218938\pi\)
0.772636 + 0.634849i \(0.218938\pi\)
\(110\) 11.2407 1.07176
\(111\) −15.9591 −1.51477
\(112\) 0 0
\(113\) 17.0613 1.60499 0.802495 0.596659i \(-0.203505\pi\)
0.802495 + 0.596659i \(0.203505\pi\)
\(114\) 4.29408 0.402178
\(115\) 10.4998 0.979116
\(116\) 10.0487 0.933000
\(117\) 0.890937 0.0823671
\(118\) 14.1277 1.30056
\(119\) 0 0
\(120\) 8.62262 0.787134
\(121\) −2.75254 −0.250231
\(122\) −22.2013 −2.01001
\(123\) 19.6625 1.77291
\(124\) 3.27524 0.294125
\(125\) 12.1139 1.08350
\(126\) 0 0
\(127\) −3.51261 −0.311693 −0.155847 0.987781i \(-0.549811\pi\)
−0.155847 + 0.987781i \(0.549811\pi\)
\(128\) −17.5988 −1.55553
\(129\) 6.08295 0.535574
\(130\) −6.12546 −0.537239
\(131\) −9.87558 −0.862834 −0.431417 0.902153i \(-0.641986\pi\)
−0.431417 + 0.902153i \(0.641986\pi\)
\(132\) 17.1778 1.49514
\(133\) 0 0
\(134\) −5.81521 −0.502358
\(135\) −7.90822 −0.680631
\(136\) −14.4043 −1.23516
\(137\) 0.428635 0.0366207 0.0183104 0.999832i \(-0.494171\pi\)
0.0183104 + 0.999832i \(0.494171\pi\)
\(138\) 26.1817 2.22873
\(139\) −20.1401 −1.70826 −0.854130 0.520059i \(-0.825910\pi\)
−0.854130 + 0.520059i \(0.825910\pi\)
\(140\) 0 0
\(141\) −10.4006 −0.875892
\(142\) 28.0919 2.35742
\(143\) −4.49432 −0.375834
\(144\) −0.175505 −0.0146254
\(145\) −5.46576 −0.453907
\(146\) 23.9829 1.98484
\(147\) 0 0
\(148\) 26.7444 2.19837
\(149\) −9.30332 −0.762158 −0.381079 0.924543i \(-0.624447\pi\)
−0.381079 + 0.924543i \(0.624447\pi\)
\(150\) 8.73589 0.713282
\(151\) −13.1140 −1.06720 −0.533602 0.845735i \(-0.679162\pi\)
−0.533602 + 0.845735i \(0.679162\pi\)
\(152\) −2.65028 −0.214966
\(153\) −3.09417 −0.250149
\(154\) 0 0
\(155\) −1.78149 −0.143092
\(156\) −9.36077 −0.749462
\(157\) −16.6033 −1.32509 −0.662543 0.749024i \(-0.730523\pi\)
−0.662543 + 0.749024i \(0.730523\pi\)
\(158\) 29.5368 2.34982
\(159\) 24.5233 1.94482
\(160\) 10.3347 0.817029
\(161\) 0 0
\(162\) −23.6013 −1.85429
\(163\) −4.67202 −0.365941 −0.182970 0.983118i \(-0.558571\pi\)
−0.182970 + 0.983118i \(0.558571\pi\)
\(164\) −32.9506 −2.57301
\(165\) −9.34346 −0.727388
\(166\) −12.2389 −0.949926
\(167\) −3.21131 −0.248499 −0.124249 0.992251i \(-0.539652\pi\)
−0.124249 + 0.992251i \(0.539652\pi\)
\(168\) 0 0
\(169\) −10.5509 −0.811607
\(170\) 21.2734 1.63159
\(171\) −0.569302 −0.0435356
\(172\) −10.1939 −0.777274
\(173\) −7.38250 −0.561281 −0.280641 0.959813i \(-0.590547\pi\)
−0.280641 + 0.959813i \(0.590547\pi\)
\(174\) −13.6290 −1.03321
\(175\) 0 0
\(176\) 0.885332 0.0667344
\(177\) −11.7431 −0.882667
\(178\) −16.9416 −1.26982
\(179\) −26.1139 −1.95185 −0.975924 0.218111i \(-0.930010\pi\)
−0.975924 + 0.218111i \(0.930010\pi\)
\(180\) −3.10395 −0.231355
\(181\) 3.30215 0.245447 0.122723 0.992441i \(-0.460837\pi\)
0.122723 + 0.992441i \(0.460837\pi\)
\(182\) 0 0
\(183\) 18.4540 1.36416
\(184\) −16.1591 −1.19127
\(185\) −14.5470 −1.06951
\(186\) −4.44219 −0.325717
\(187\) 15.6085 1.14141
\(188\) 17.4295 1.27118
\(189\) 0 0
\(190\) 3.91413 0.283961
\(191\) 7.00807 0.507086 0.253543 0.967324i \(-0.418404\pi\)
0.253543 + 0.967324i \(0.418404\pi\)
\(192\) 24.6050 1.77571
\(193\) 6.61313 0.476024 0.238012 0.971262i \(-0.423504\pi\)
0.238012 + 0.971262i \(0.423504\pi\)
\(194\) 5.57227 0.400066
\(195\) 5.09157 0.364615
\(196\) 0 0
\(197\) −5.23496 −0.372976 −0.186488 0.982457i \(-0.559710\pi\)
−0.186488 + 0.982457i \(0.559710\pi\)
\(198\) −3.71605 −0.264088
\(199\) 13.3552 0.946724 0.473362 0.880868i \(-0.343040\pi\)
0.473362 + 0.880868i \(0.343040\pi\)
\(200\) −5.39173 −0.381253
\(201\) 4.83369 0.340942
\(202\) 3.33592 0.234714
\(203\) 0 0
\(204\) 32.5094 2.27611
\(205\) 17.9227 1.25178
\(206\) 25.5590 1.78078
\(207\) −3.47112 −0.241260
\(208\) −0.482448 −0.0334517
\(209\) 2.87184 0.198649
\(210\) 0 0
\(211\) 13.1468 0.905061 0.452531 0.891749i \(-0.350521\pi\)
0.452531 + 0.891749i \(0.350521\pi\)
\(212\) −41.0963 −2.82251
\(213\) −23.3504 −1.59994
\(214\) −0.553114 −0.0378101
\(215\) 5.54471 0.378146
\(216\) 12.1707 0.828108
\(217\) 0 0
\(218\) 36.6688 2.48353
\(219\) −19.9349 −1.34707
\(220\) 15.6579 1.05565
\(221\) −8.50561 −0.572149
\(222\) −36.2733 −2.43450
\(223\) −11.1267 −0.745099 −0.372549 0.928012i \(-0.621516\pi\)
−0.372549 + 0.928012i \(0.621516\pi\)
\(224\) 0 0
\(225\) −1.15819 −0.0772126
\(226\) 38.7784 2.57950
\(227\) 25.2738 1.67748 0.838740 0.544532i \(-0.183293\pi\)
0.838740 + 0.544532i \(0.183293\pi\)
\(228\) 5.98147 0.396132
\(229\) 4.09982 0.270923 0.135462 0.990783i \(-0.456748\pi\)
0.135462 + 0.990783i \(0.456748\pi\)
\(230\) 23.8650 1.57361
\(231\) 0 0
\(232\) 8.41175 0.552258
\(233\) −28.7409 −1.88288 −0.941439 0.337183i \(-0.890526\pi\)
−0.941439 + 0.337183i \(0.890526\pi\)
\(234\) 2.02500 0.132379
\(235\) −9.48036 −0.618430
\(236\) 19.6792 1.28101
\(237\) −24.5514 −1.59478
\(238\) 0 0
\(239\) 13.1082 0.847900 0.423950 0.905686i \(-0.360643\pi\)
0.423950 + 0.905686i \(0.360643\pi\)
\(240\) −1.00298 −0.0647423
\(241\) 0.00249380 0.000160640 0 8.03200e−5 1.00000i \(-0.499974\pi\)
8.03200e−5 1.00000i \(0.499974\pi\)
\(242\) −6.25623 −0.402165
\(243\) 5.84104 0.374703
\(244\) −30.9254 −1.97979
\(245\) 0 0
\(246\) 44.6908 2.84938
\(247\) −1.56496 −0.0995762
\(248\) 2.74169 0.174097
\(249\) 10.1732 0.644699
\(250\) 27.5335 1.74137
\(251\) −13.5644 −0.856175 −0.428087 0.903737i \(-0.640812\pi\)
−0.428087 + 0.903737i \(0.640812\pi\)
\(252\) 0 0
\(253\) 17.5100 1.10085
\(254\) −7.98377 −0.500947
\(255\) −17.6827 −1.10734
\(256\) −13.9529 −0.872057
\(257\) −25.9001 −1.61560 −0.807801 0.589456i \(-0.799342\pi\)
−0.807801 + 0.589456i \(0.799342\pi\)
\(258\) 13.8259 0.860762
\(259\) 0 0
\(260\) −8.53250 −0.529163
\(261\) 1.80691 0.111845
\(262\) −22.4461 −1.38673
\(263\) 19.2508 1.18705 0.593526 0.804814i \(-0.297735\pi\)
0.593526 + 0.804814i \(0.297735\pi\)
\(264\) 14.3795 0.884996
\(265\) 22.3534 1.37316
\(266\) 0 0
\(267\) 14.0821 0.861809
\(268\) −8.10033 −0.494807
\(269\) −19.2999 −1.17673 −0.588367 0.808594i \(-0.700229\pi\)
−0.588367 + 0.808594i \(0.700229\pi\)
\(270\) −17.9745 −1.09390
\(271\) 18.4313 1.11962 0.559812 0.828620i \(-0.310874\pi\)
0.559812 + 0.828620i \(0.310874\pi\)
\(272\) 1.67551 0.101593
\(273\) 0 0
\(274\) 0.974241 0.0588560
\(275\) 5.84247 0.352314
\(276\) 36.4699 2.19523
\(277\) −15.7856 −0.948463 −0.474231 0.880400i \(-0.657274\pi\)
−0.474231 + 0.880400i \(0.657274\pi\)
\(278\) −45.7763 −2.74548
\(279\) 0.588938 0.0352588
\(280\) 0 0
\(281\) −24.2255 −1.44517 −0.722587 0.691280i \(-0.757047\pi\)
−0.722587 + 0.691280i \(0.757047\pi\)
\(282\) −23.6395 −1.40771
\(283\) −7.99693 −0.475368 −0.237684 0.971343i \(-0.576388\pi\)
−0.237684 + 0.971343i \(0.576388\pi\)
\(284\) 39.1308 2.32198
\(285\) −3.25348 −0.192719
\(286\) −10.2151 −0.604032
\(287\) 0 0
\(288\) −3.41652 −0.201320
\(289\) 12.5395 0.737617
\(290\) −12.4231 −0.729509
\(291\) −4.63175 −0.271518
\(292\) 33.4071 1.95500
\(293\) 17.3236 1.01206 0.506028 0.862517i \(-0.331113\pi\)
0.506028 + 0.862517i \(0.331113\pi\)
\(294\) 0 0
\(295\) −10.7040 −0.623214
\(296\) 22.3876 1.30125
\(297\) −13.1881 −0.765252
\(298\) −21.1454 −1.22492
\(299\) −9.54181 −0.551817
\(300\) 12.1687 0.702561
\(301\) 0 0
\(302\) −29.8068 −1.71519
\(303\) −2.77286 −0.159297
\(304\) 0.308281 0.0176811
\(305\) 16.8211 0.963174
\(306\) −7.03272 −0.402034
\(307\) −5.30000 −0.302487 −0.151244 0.988497i \(-0.548328\pi\)
−0.151244 + 0.988497i \(0.548328\pi\)
\(308\) 0 0
\(309\) −21.2450 −1.20859
\(310\) −4.04913 −0.229975
\(311\) −6.02144 −0.341444 −0.170722 0.985319i \(-0.554610\pi\)
−0.170722 + 0.985319i \(0.554610\pi\)
\(312\) −7.83587 −0.443619
\(313\) −17.5310 −0.990908 −0.495454 0.868634i \(-0.664998\pi\)
−0.495454 + 0.868634i \(0.664998\pi\)
\(314\) −37.7374 −2.12965
\(315\) 0 0
\(316\) 41.1434 2.31450
\(317\) −20.3083 −1.14063 −0.570313 0.821427i \(-0.693178\pi\)
−0.570313 + 0.821427i \(0.693178\pi\)
\(318\) 55.7388 3.12568
\(319\) −9.11496 −0.510340
\(320\) 22.4279 1.25376
\(321\) 0.459756 0.0256611
\(322\) 0 0
\(323\) 5.43502 0.302413
\(324\) −32.8755 −1.82642
\(325\) −3.18376 −0.176603
\(326\) −10.6190 −0.588132
\(327\) −30.4796 −1.68553
\(328\) −27.5828 −1.52301
\(329\) 0 0
\(330\) −21.2367 −1.16904
\(331\) −13.9894 −0.768929 −0.384464 0.923140i \(-0.625614\pi\)
−0.384464 + 0.923140i \(0.625614\pi\)
\(332\) −17.0483 −0.935647
\(333\) 4.80905 0.263534
\(334\) −7.29897 −0.399382
\(335\) 4.40598 0.240725
\(336\) 0 0
\(337\) −25.6696 −1.39831 −0.699157 0.714969i \(-0.746441\pi\)
−0.699157 + 0.714969i \(0.746441\pi\)
\(338\) −23.9810 −1.30440
\(339\) −32.2332 −1.75067
\(340\) 29.6329 1.60707
\(341\) −2.97089 −0.160883
\(342\) −1.29396 −0.0699695
\(343\) 0 0
\(344\) −8.53324 −0.460081
\(345\) −19.8369 −1.06798
\(346\) −16.7796 −0.902078
\(347\) 26.5340 1.42442 0.712210 0.701966i \(-0.247694\pi\)
0.712210 + 0.701966i \(0.247694\pi\)
\(348\) −18.9846 −1.01768
\(349\) −32.7352 −1.75228 −0.876139 0.482059i \(-0.839889\pi\)
−0.876139 + 0.482059i \(0.839889\pi\)
\(350\) 0 0
\(351\) 7.18665 0.383595
\(352\) 17.2346 0.918608
\(353\) 6.36333 0.338686 0.169343 0.985557i \(-0.445835\pi\)
0.169343 + 0.985557i \(0.445835\pi\)
\(354\) −26.6909 −1.41860
\(355\) −21.2843 −1.12965
\(356\) −23.5989 −1.25074
\(357\) 0 0
\(358\) −59.3542 −3.13697
\(359\) 17.9081 0.945152 0.472576 0.881290i \(-0.343324\pi\)
0.472576 + 0.881290i \(0.343324\pi\)
\(360\) −2.59831 −0.136943
\(361\) 1.00000 0.0526316
\(362\) 7.50543 0.394477
\(363\) 5.20026 0.272943
\(364\) 0 0
\(365\) −18.1710 −0.951113
\(366\) 41.9439 2.19244
\(367\) 18.9798 0.990736 0.495368 0.868683i \(-0.335033\pi\)
0.495368 + 0.868683i \(0.335033\pi\)
\(368\) 1.87963 0.0979826
\(369\) −5.92503 −0.308445
\(370\) −33.0637 −1.71890
\(371\) 0 0
\(372\) −6.18777 −0.320821
\(373\) 1.89525 0.0981321 0.0490660 0.998796i \(-0.484376\pi\)
0.0490660 + 0.998796i \(0.484376\pi\)
\(374\) 35.4765 1.83444
\(375\) −22.8863 −1.18184
\(376\) 14.5902 0.752430
\(377\) 4.96705 0.255816
\(378\) 0 0
\(379\) −17.8480 −0.916790 −0.458395 0.888748i \(-0.651576\pi\)
−0.458395 + 0.888748i \(0.651576\pi\)
\(380\) 5.45221 0.279692
\(381\) 6.63623 0.339984
\(382\) 15.9286 0.814977
\(383\) 24.9132 1.27301 0.636503 0.771274i \(-0.280380\pi\)
0.636503 + 0.771274i \(0.280380\pi\)
\(384\) 33.2487 1.69671
\(385\) 0 0
\(386\) 15.0309 0.765054
\(387\) −1.83301 −0.0931773
\(388\) 7.76193 0.394052
\(389\) 20.4631 1.03752 0.518760 0.854920i \(-0.326394\pi\)
0.518760 + 0.854920i \(0.326394\pi\)
\(390\) 11.5726 0.586001
\(391\) 33.1382 1.67587
\(392\) 0 0
\(393\) 18.6575 0.941149
\(394\) −11.8985 −0.599438
\(395\) −22.3790 −1.12601
\(396\) −5.17630 −0.260119
\(397\) −34.7628 −1.74469 −0.872347 0.488888i \(-0.837403\pi\)
−0.872347 + 0.488888i \(0.837403\pi\)
\(398\) 30.3549 1.52155
\(399\) 0 0
\(400\) 0.627167 0.0313583
\(401\) 21.9956 1.09841 0.549205 0.835688i \(-0.314931\pi\)
0.549205 + 0.835688i \(0.314931\pi\)
\(402\) 10.9864 0.547954
\(403\) 1.61894 0.0806451
\(404\) 4.64679 0.231186
\(405\) 17.8819 0.888556
\(406\) 0 0
\(407\) −24.2592 −1.20248
\(408\) 27.2135 1.34727
\(409\) −3.51585 −0.173847 −0.0869237 0.996215i \(-0.527704\pi\)
−0.0869237 + 0.996215i \(0.527704\pi\)
\(410\) 40.7364 2.01183
\(411\) −0.809802 −0.0399446
\(412\) 35.6025 1.75401
\(413\) 0 0
\(414\) −7.88948 −0.387747
\(415\) 9.27302 0.455194
\(416\) −9.39172 −0.460467
\(417\) 38.0499 1.86331
\(418\) 6.52738 0.319265
\(419\) −38.5514 −1.88336 −0.941679 0.336514i \(-0.890752\pi\)
−0.941679 + 0.336514i \(0.890752\pi\)
\(420\) 0 0
\(421\) −8.44685 −0.411674 −0.205837 0.978586i \(-0.565992\pi\)
−0.205837 + 0.978586i \(0.565992\pi\)
\(422\) 29.8812 1.45459
\(423\) 3.13409 0.152385
\(424\) −34.4016 −1.67069
\(425\) 11.0570 0.536344
\(426\) −53.0729 −2.57139
\(427\) 0 0
\(428\) −0.770463 −0.0372418
\(429\) 8.49094 0.409946
\(430\) 12.6025 0.607748
\(431\) −3.55584 −0.171279 −0.0856394 0.996326i \(-0.527293\pi\)
−0.0856394 + 0.996326i \(0.527293\pi\)
\(432\) −1.41569 −0.0681125
\(433\) −1.09300 −0.0525264 −0.0262632 0.999655i \(-0.508361\pi\)
−0.0262632 + 0.999655i \(0.508361\pi\)
\(434\) 0 0
\(435\) 10.3262 0.495106
\(436\) 51.0781 2.44620
\(437\) 6.09715 0.291666
\(438\) −45.3098 −2.16499
\(439\) 8.19098 0.390934 0.195467 0.980710i \(-0.437378\pi\)
0.195467 + 0.980710i \(0.437378\pi\)
\(440\) 13.1071 0.624858
\(441\) 0 0
\(442\) −19.3323 −0.919545
\(443\) 10.2973 0.489239 0.244620 0.969619i \(-0.421337\pi\)
0.244620 + 0.969619i \(0.421337\pi\)
\(444\) −50.5271 −2.39791
\(445\) 12.8360 0.608486
\(446\) −25.2898 −1.19751
\(447\) 17.5764 0.831335
\(448\) 0 0
\(449\) 22.0244 1.03940 0.519699 0.854350i \(-0.326044\pi\)
0.519699 + 0.854350i \(0.326044\pi\)
\(450\) −2.63244 −0.124094
\(451\) 29.8887 1.40741
\(452\) 54.0166 2.54073
\(453\) 24.7758 1.16407
\(454\) 57.4446 2.69601
\(455\) 0 0
\(456\) 5.00706 0.234477
\(457\) −11.7275 −0.548591 −0.274296 0.961645i \(-0.588445\pi\)
−0.274296 + 0.961645i \(0.588445\pi\)
\(458\) 9.31844 0.435422
\(459\) −24.9588 −1.16498
\(460\) 33.2429 1.54996
\(461\) 19.1133 0.890194 0.445097 0.895482i \(-0.353169\pi\)
0.445097 + 0.895482i \(0.353169\pi\)
\(462\) 0 0
\(463\) 3.75469 0.174495 0.0872475 0.996187i \(-0.472193\pi\)
0.0872475 + 0.996187i \(0.472193\pi\)
\(464\) −0.978455 −0.0454236
\(465\) 3.36569 0.156080
\(466\) −65.3249 −3.02612
\(467\) −3.15207 −0.145860 −0.0729302 0.997337i \(-0.523235\pi\)
−0.0729302 + 0.997337i \(0.523235\pi\)
\(468\) 2.82074 0.130389
\(469\) 0 0
\(470\) −21.5478 −0.993927
\(471\) 31.3679 1.44536
\(472\) 16.4734 0.758250
\(473\) 9.24661 0.425160
\(474\) −55.8027 −2.56310
\(475\) 2.03440 0.0933448
\(476\) 0 0
\(477\) −7.38975 −0.338354
\(478\) 29.7936 1.36273
\(479\) −6.14080 −0.280580 −0.140290 0.990110i \(-0.544804\pi\)
−0.140290 + 0.990110i \(0.544804\pi\)
\(480\) −19.5249 −0.891186
\(481\) 13.2197 0.602765
\(482\) 0.00566815 0.000258177 0
\(483\) 0 0
\(484\) −8.71465 −0.396120
\(485\) −4.22192 −0.191707
\(486\) 13.2761 0.602214
\(487\) 40.4699 1.83387 0.916934 0.399038i \(-0.130656\pi\)
0.916934 + 0.399038i \(0.130656\pi\)
\(488\) −25.8875 −1.17187
\(489\) 8.82666 0.399155
\(490\) 0 0
\(491\) −18.0490 −0.814538 −0.407269 0.913308i \(-0.633519\pi\)
−0.407269 + 0.913308i \(0.633519\pi\)
\(492\) 62.2523 2.80655
\(493\) −17.2503 −0.776914
\(494\) −3.55699 −0.160037
\(495\) 2.81552 0.126548
\(496\) −0.318913 −0.0143196
\(497\) 0 0
\(498\) 23.1225 1.03615
\(499\) −35.5880 −1.59314 −0.796569 0.604548i \(-0.793354\pi\)
−0.796569 + 0.604548i \(0.793354\pi\)
\(500\) 38.3530 1.71520
\(501\) 6.06700 0.271054
\(502\) −30.8303 −1.37602
\(503\) 27.3502 1.21949 0.609744 0.792599i \(-0.291272\pi\)
0.609744 + 0.792599i \(0.291272\pi\)
\(504\) 0 0
\(505\) −2.52751 −0.112473
\(506\) 39.7984 1.76926
\(507\) 19.9334 0.885272
\(508\) −11.1210 −0.493417
\(509\) −41.1163 −1.82245 −0.911225 0.411909i \(-0.864862\pi\)
−0.911225 + 0.411909i \(0.864862\pi\)
\(510\) −40.1909 −1.77968
\(511\) 0 0
\(512\) 3.48413 0.153978
\(513\) −4.59222 −0.202751
\(514\) −58.8680 −2.59656
\(515\) −19.3652 −0.853331
\(516\) 19.2588 0.847823
\(517\) −15.8099 −0.695318
\(518\) 0 0
\(519\) 13.9475 0.612226
\(520\) −7.14252 −0.313220
\(521\) 12.8696 0.563828 0.281914 0.959440i \(-0.409031\pi\)
0.281914 + 0.959440i \(0.409031\pi\)
\(522\) 4.10692 0.179755
\(523\) 14.5744 0.637294 0.318647 0.947873i \(-0.396772\pi\)
0.318647 + 0.947873i \(0.396772\pi\)
\(524\) −31.2665 −1.36588
\(525\) 0 0
\(526\) 43.7549 1.90780
\(527\) −5.62248 −0.244919
\(528\) −1.67262 −0.0727916
\(529\) 14.1752 0.616315
\(530\) 50.8068 2.20691
\(531\) 3.53863 0.153563
\(532\) 0 0
\(533\) −16.2874 −0.705485
\(534\) 32.0070 1.38508
\(535\) 0.419075 0.0181182
\(536\) −6.78076 −0.292884
\(537\) 49.3360 2.12901
\(538\) −43.8665 −1.89122
\(539\) 0 0
\(540\) −25.0377 −1.07745
\(541\) −17.5194 −0.753219 −0.376609 0.926372i \(-0.622910\pi\)
−0.376609 + 0.926372i \(0.622910\pi\)
\(542\) 41.8924 1.79943
\(543\) −6.23862 −0.267725
\(544\) 32.6169 1.39844
\(545\) −27.7827 −1.19008
\(546\) 0 0
\(547\) −1.50159 −0.0642032 −0.0321016 0.999485i \(-0.510220\pi\)
−0.0321016 + 0.999485i \(0.510220\pi\)
\(548\) 1.35707 0.0579713
\(549\) −5.56086 −0.237332
\(550\) 13.2793 0.566232
\(551\) −3.17391 −0.135213
\(552\) 30.5288 1.29939
\(553\) 0 0
\(554\) −35.8789 −1.52435
\(555\) 27.4830 1.16659
\(556\) −63.7643 −2.70421
\(557\) −29.5856 −1.25358 −0.626790 0.779188i \(-0.715632\pi\)
−0.626790 + 0.779188i \(0.715632\pi\)
\(558\) 1.33859 0.0566671
\(559\) −5.03879 −0.213118
\(560\) 0 0
\(561\) −29.4885 −1.24501
\(562\) −55.0620 −2.32265
\(563\) 5.40462 0.227778 0.113889 0.993493i \(-0.463669\pi\)
0.113889 + 0.993493i \(0.463669\pi\)
\(564\) −32.9288 −1.38655
\(565\) −29.3811 −1.23607
\(566\) −18.1762 −0.764001
\(567\) 0 0
\(568\) 32.7562 1.37442
\(569\) 14.9850 0.628205 0.314102 0.949389i \(-0.398296\pi\)
0.314102 + 0.949389i \(0.398296\pi\)
\(570\) −7.39480 −0.309734
\(571\) 30.1295 1.26088 0.630441 0.776237i \(-0.282874\pi\)
0.630441 + 0.776237i \(0.282874\pi\)
\(572\) −14.2292 −0.594952
\(573\) −13.2401 −0.553111
\(574\) 0 0
\(575\) 12.4041 0.517285
\(576\) −7.41438 −0.308932
\(577\) 22.0535 0.918101 0.459050 0.888410i \(-0.348190\pi\)
0.459050 + 0.888410i \(0.348190\pi\)
\(578\) 28.5009 1.18548
\(579\) −12.4939 −0.519230
\(580\) −17.3048 −0.718543
\(581\) 0 0
\(582\) −10.5275 −0.436378
\(583\) 37.2775 1.54388
\(584\) 27.9649 1.15720
\(585\) −1.53427 −0.0634344
\(586\) 39.3747 1.62655
\(587\) −23.6848 −0.977575 −0.488788 0.872403i \(-0.662561\pi\)
−0.488788 + 0.872403i \(0.662561\pi\)
\(588\) 0 0
\(589\) −1.03449 −0.0426254
\(590\) −24.3291 −1.00162
\(591\) 9.89020 0.406829
\(592\) −2.60413 −0.107029
\(593\) 28.5028 1.17047 0.585235 0.810864i \(-0.301002\pi\)
0.585235 + 0.810864i \(0.301002\pi\)
\(594\) −29.9752 −1.22990
\(595\) 0 0
\(596\) −29.4547 −1.20651
\(597\) −25.2314 −1.03265
\(598\) −21.6875 −0.886868
\(599\) 4.10267 0.167630 0.0838152 0.996481i \(-0.473289\pi\)
0.0838152 + 0.996481i \(0.473289\pi\)
\(600\) 10.1864 0.415857
\(601\) −10.6311 −0.433651 −0.216825 0.976210i \(-0.569570\pi\)
−0.216825 + 0.976210i \(0.569570\pi\)
\(602\) 0 0
\(603\) −1.45656 −0.0593159
\(604\) −41.5195 −1.68941
\(605\) 4.74012 0.192713
\(606\) −6.30242 −0.256018
\(607\) −28.7604 −1.16735 −0.583674 0.811988i \(-0.698385\pi\)
−0.583674 + 0.811988i \(0.698385\pi\)
\(608\) 6.00124 0.243383
\(609\) 0 0
\(610\) 38.2326 1.54799
\(611\) 8.61534 0.348539
\(612\) −9.79627 −0.395991
\(613\) 19.8663 0.802390 0.401195 0.915993i \(-0.368595\pi\)
0.401195 + 0.915993i \(0.368595\pi\)
\(614\) −12.0463 −0.486150
\(615\) −33.8607 −1.36539
\(616\) 0 0
\(617\) 36.6057 1.47369 0.736846 0.676061i \(-0.236314\pi\)
0.736846 + 0.676061i \(0.236314\pi\)
\(618\) −48.2876 −1.94241
\(619\) 40.2839 1.61915 0.809574 0.587018i \(-0.199698\pi\)
0.809574 + 0.587018i \(0.199698\pi\)
\(620\) −5.64025 −0.226518
\(621\) −27.9994 −1.12358
\(622\) −13.6861 −0.548761
\(623\) 0 0
\(624\) 0.911469 0.0364880
\(625\) −10.6892 −0.427568
\(626\) −39.8460 −1.59257
\(627\) −5.42565 −0.216680
\(628\) −52.5666 −2.09763
\(629\) −45.9111 −1.83060
\(630\) 0 0
\(631\) 16.7789 0.667956 0.333978 0.942581i \(-0.391609\pi\)
0.333978 + 0.942581i \(0.391609\pi\)
\(632\) 34.4410 1.36999
\(633\) −24.8377 −0.987209
\(634\) −46.1585 −1.83319
\(635\) 6.04903 0.240048
\(636\) 77.6416 3.07869
\(637\) 0 0
\(638\) −20.7173 −0.820207
\(639\) 7.03631 0.278352
\(640\) 30.3067 1.19798
\(641\) −33.0517 −1.30546 −0.652732 0.757589i \(-0.726377\pi\)
−0.652732 + 0.757589i \(0.726377\pi\)
\(642\) 1.04498 0.0412419
\(643\) 45.2538 1.78463 0.892317 0.451409i \(-0.149078\pi\)
0.892317 + 0.451409i \(0.149078\pi\)
\(644\) 0 0
\(645\) −10.4754 −0.412468
\(646\) 12.3532 0.486031
\(647\) 16.0011 0.629068 0.314534 0.949246i \(-0.398152\pi\)
0.314534 + 0.949246i \(0.398152\pi\)
\(648\) −27.5200 −1.08109
\(649\) −17.8506 −0.700696
\(650\) −7.23635 −0.283833
\(651\) 0 0
\(652\) −14.7918 −0.579291
\(653\) 13.3269 0.521520 0.260760 0.965404i \(-0.416027\pi\)
0.260760 + 0.965404i \(0.416027\pi\)
\(654\) −69.2769 −2.70894
\(655\) 17.0067 0.664505
\(656\) 3.20844 0.125269
\(657\) 6.00710 0.234359
\(658\) 0 0
\(659\) −27.7239 −1.07997 −0.539985 0.841675i \(-0.681570\pi\)
−0.539985 + 0.841675i \(0.681570\pi\)
\(660\) −29.5818 −1.15147
\(661\) 11.1407 0.433325 0.216662 0.976247i \(-0.430483\pi\)
0.216662 + 0.976247i \(0.430483\pi\)
\(662\) −31.7965 −1.23580
\(663\) 16.0693 0.624080
\(664\) −14.2711 −0.553824
\(665\) 0 0
\(666\) 10.9305 0.423546
\(667\) −19.3518 −0.749305
\(668\) −10.1671 −0.393378
\(669\) 21.0212 0.812727
\(670\) 10.0143 0.386887
\(671\) 28.0517 1.08292
\(672\) 0 0
\(673\) 13.3885 0.516087 0.258043 0.966133i \(-0.416922\pi\)
0.258043 + 0.966133i \(0.416922\pi\)
\(674\) −58.3443 −2.24734
\(675\) −9.34242 −0.359590
\(676\) −33.4045 −1.28479
\(677\) −16.3781 −0.629462 −0.314731 0.949181i \(-0.601914\pi\)
−0.314731 + 0.949181i \(0.601914\pi\)
\(678\) −73.2625 −2.81363
\(679\) 0 0
\(680\) 24.8056 0.951250
\(681\) −47.7488 −1.82974
\(682\) −6.75251 −0.258567
\(683\) −3.38340 −0.129462 −0.0647311 0.997903i \(-0.520619\pi\)
−0.0647311 + 0.997903i \(0.520619\pi\)
\(684\) −1.80243 −0.0689177
\(685\) −0.738148 −0.0282032
\(686\) 0 0
\(687\) −7.74562 −0.295514
\(688\) 0.992587 0.0378420
\(689\) −20.3138 −0.773894
\(690\) −45.0872 −1.71644
\(691\) 34.1321 1.29845 0.649223 0.760598i \(-0.275094\pi\)
0.649223 + 0.760598i \(0.275094\pi\)
\(692\) −23.3733 −0.888519
\(693\) 0 0
\(694\) 60.3089 2.28930
\(695\) 34.6831 1.31560
\(696\) −15.8920 −0.602384
\(697\) 56.5652 2.14256
\(698\) −74.4037 −2.81622
\(699\) 54.2990 2.05378
\(700\) 0 0
\(701\) 12.5338 0.473395 0.236697 0.971583i \(-0.423935\pi\)
0.236697 + 0.971583i \(0.423935\pi\)
\(702\) 16.3345 0.616505
\(703\) −8.44727 −0.318595
\(704\) 37.4017 1.40963
\(705\) 17.9109 0.674562
\(706\) 14.4632 0.544328
\(707\) 0 0
\(708\) −37.1792 −1.39728
\(709\) 16.7799 0.630184 0.315092 0.949061i \(-0.397965\pi\)
0.315092 + 0.949061i \(0.397965\pi\)
\(710\) −48.3768 −1.81555
\(711\) 7.39822 0.277455
\(712\) −19.7545 −0.740331
\(713\) −6.30744 −0.236216
\(714\) 0 0
\(715\) 7.73963 0.289446
\(716\) −82.6777 −3.08981
\(717\) −24.7648 −0.924860
\(718\) 40.7031 1.51903
\(719\) −40.0031 −1.49186 −0.745932 0.666022i \(-0.767996\pi\)
−0.745932 + 0.666022i \(0.767996\pi\)
\(720\) 0.302235 0.0112636
\(721\) 0 0
\(722\) 2.27289 0.0845883
\(723\) −0.00471144 −0.000175221 0
\(724\) 10.4547 0.388547
\(725\) −6.45701 −0.239807
\(726\) 11.8196 0.438668
\(727\) −17.6291 −0.653828 −0.326914 0.945054i \(-0.606009\pi\)
−0.326914 + 0.945054i \(0.606009\pi\)
\(728\) 0 0
\(729\) 20.1162 0.745043
\(730\) −41.3007 −1.52861
\(731\) 17.4994 0.647240
\(732\) 58.4260 2.15949
\(733\) −2.33685 −0.0863136 −0.0431568 0.999068i \(-0.513742\pi\)
−0.0431568 + 0.999068i \(0.513742\pi\)
\(734\) 43.1390 1.59229
\(735\) 0 0
\(736\) 36.5905 1.34874
\(737\) 7.34762 0.270653
\(738\) −13.4670 −0.495725
\(739\) −31.7072 −1.16637 −0.583184 0.812340i \(-0.698193\pi\)
−0.583184 + 0.812340i \(0.698193\pi\)
\(740\) −46.0563 −1.69306
\(741\) 2.95662 0.108614
\(742\) 0 0
\(743\) 22.1449 0.812418 0.406209 0.913780i \(-0.366851\pi\)
0.406209 + 0.913780i \(0.366851\pi\)
\(744\) −5.17976 −0.189899
\(745\) 16.0212 0.586970
\(746\) 4.30769 0.157716
\(747\) −3.06555 −0.112162
\(748\) 49.4172 1.80687
\(749\) 0 0
\(750\) −52.0180 −1.89943
\(751\) −49.6623 −1.81220 −0.906102 0.423059i \(-0.860956\pi\)
−0.906102 + 0.423059i \(0.860956\pi\)
\(752\) −1.69713 −0.0618879
\(753\) 25.6266 0.933885
\(754\) 11.2896 0.411142
\(755\) 22.5836 0.821900
\(756\) 0 0
\(757\) −26.1148 −0.949158 −0.474579 0.880213i \(-0.657400\pi\)
−0.474579 + 0.880213i \(0.657400\pi\)
\(758\) −40.5666 −1.47344
\(759\) −33.0810 −1.20076
\(760\) 4.56402 0.165554
\(761\) −31.4608 −1.14045 −0.570226 0.821488i \(-0.693144\pi\)
−0.570226 + 0.821488i \(0.693144\pi\)
\(762\) 15.0834 0.546415
\(763\) 0 0
\(764\) 22.1878 0.802726
\(765\) 5.32845 0.192650
\(766\) 56.6250 2.04595
\(767\) 9.72738 0.351235
\(768\) 26.3607 0.951209
\(769\) 31.4360 1.13361 0.566806 0.823851i \(-0.308179\pi\)
0.566806 + 0.823851i \(0.308179\pi\)
\(770\) 0 0
\(771\) 48.9319 1.76224
\(772\) 20.9374 0.753554
\(773\) 10.7968 0.388334 0.194167 0.980969i \(-0.437800\pi\)
0.194167 + 0.980969i \(0.437800\pi\)
\(774\) −4.16624 −0.149752
\(775\) −2.10457 −0.0755983
\(776\) 6.49748 0.233246
\(777\) 0 0
\(778\) 46.5104 1.66748
\(779\) 10.4075 0.372888
\(780\) 16.1201 0.577192
\(781\) −35.4946 −1.27010
\(782\) 75.3195 2.69342
\(783\) 14.5753 0.520879
\(784\) 0 0
\(785\) 28.5923 1.02050
\(786\) 42.4066 1.51259
\(787\) 46.7996 1.66823 0.834113 0.551594i \(-0.185980\pi\)
0.834113 + 0.551594i \(0.185980\pi\)
\(788\) −16.5741 −0.590427
\(789\) −36.3697 −1.29480
\(790\) −50.8651 −1.80970
\(791\) 0 0
\(792\) −4.33306 −0.153968
\(793\) −15.2863 −0.542833
\(794\) −79.0120 −2.80403
\(795\) −42.2313 −1.49779
\(796\) 42.2830 1.49868
\(797\) −16.9067 −0.598864 −0.299432 0.954118i \(-0.596797\pi\)
−0.299432 + 0.954118i \(0.596797\pi\)
\(798\) 0 0
\(799\) −29.9206 −1.05851
\(800\) 12.2089 0.431651
\(801\) −4.24344 −0.149934
\(802\) 49.9937 1.76534
\(803\) −30.3028 −1.06936
\(804\) 15.3036 0.539718
\(805\) 0 0
\(806\) 3.67967 0.129611
\(807\) 36.4625 1.28354
\(808\) 3.88981 0.136843
\(809\) −13.3420 −0.469081 −0.234540 0.972106i \(-0.575358\pi\)
−0.234540 + 0.972106i \(0.575358\pi\)
\(810\) 40.6435 1.42807
\(811\) −20.7826 −0.729777 −0.364888 0.931051i \(-0.618893\pi\)
−0.364888 + 0.931051i \(0.618893\pi\)
\(812\) 0 0
\(813\) −34.8216 −1.22125
\(814\) −55.1386 −1.93261
\(815\) 8.04564 0.281827
\(816\) −3.16548 −0.110814
\(817\) 3.21975 0.112645
\(818\) −7.99114 −0.279404
\(819\) 0 0
\(820\) 56.7440 1.98159
\(821\) −10.2489 −0.357690 −0.178845 0.983877i \(-0.557236\pi\)
−0.178845 + 0.983877i \(0.557236\pi\)
\(822\) −1.84059 −0.0641981
\(823\) −42.0328 −1.46517 −0.732585 0.680675i \(-0.761687\pi\)
−0.732585 + 0.680675i \(0.761687\pi\)
\(824\) 29.8027 1.03823
\(825\) −11.0380 −0.384292
\(826\) 0 0
\(827\) 13.3468 0.464112 0.232056 0.972702i \(-0.425455\pi\)
0.232056 + 0.972702i \(0.425455\pi\)
\(828\) −10.9897 −0.381919
\(829\) 33.9734 1.17994 0.589972 0.807424i \(-0.299139\pi\)
0.589972 + 0.807424i \(0.299139\pi\)
\(830\) 21.0766 0.731578
\(831\) 29.8230 1.03455
\(832\) −20.3815 −0.706600
\(833\) 0 0
\(834\) 86.4832 2.99467
\(835\) 5.53017 0.191380
\(836\) 9.09235 0.314466
\(837\) 4.75061 0.164205
\(838\) −87.6231 −3.02689
\(839\) −38.4347 −1.32691 −0.663457 0.748215i \(-0.730911\pi\)
−0.663457 + 0.748215i \(0.730911\pi\)
\(840\) 0 0
\(841\) −18.9263 −0.652631
\(842\) −19.1988 −0.661633
\(843\) 45.7683 1.57634
\(844\) 41.6232 1.43273
\(845\) 18.1696 0.625053
\(846\) 7.12345 0.244909
\(847\) 0 0
\(848\) 4.00160 0.137415
\(849\) 15.1083 0.518515
\(850\) 25.1314 0.862001
\(851\) −51.5043 −1.76554
\(852\) −73.9282 −2.53274
\(853\) 55.0017 1.88322 0.941611 0.336701i \(-0.109311\pi\)
0.941611 + 0.336701i \(0.109311\pi\)
\(854\) 0 0
\(855\) 0.980390 0.0335286
\(856\) −0.644952 −0.0220440
\(857\) −15.7299 −0.537325 −0.268662 0.963234i \(-0.586582\pi\)
−0.268662 + 0.963234i \(0.586582\pi\)
\(858\) 19.2990 0.658857
\(859\) 38.6992 1.32040 0.660200 0.751090i \(-0.270471\pi\)
0.660200 + 0.751090i \(0.270471\pi\)
\(860\) 17.5548 0.598612
\(861\) 0 0
\(862\) −8.08204 −0.275275
\(863\) 49.0091 1.66829 0.834145 0.551545i \(-0.185961\pi\)
0.834145 + 0.551545i \(0.185961\pi\)
\(864\) −27.5590 −0.937577
\(865\) 12.7133 0.432267
\(866\) −2.48428 −0.0844192
\(867\) −23.6904 −0.804567
\(868\) 0 0
\(869\) −37.3203 −1.26600
\(870\) 23.4704 0.795723
\(871\) −4.00397 −0.135669
\(872\) 42.7572 1.44794
\(873\) 1.39571 0.0472378
\(874\) 13.8582 0.468759
\(875\) 0 0
\(876\) −63.1146 −2.13245
\(877\) 45.1810 1.52565 0.762827 0.646602i \(-0.223811\pi\)
0.762827 + 0.646602i \(0.223811\pi\)
\(878\) 18.6172 0.628300
\(879\) −32.7288 −1.10392
\(880\) −1.52462 −0.0513950
\(881\) −3.43242 −0.115641 −0.0578206 0.998327i \(-0.518415\pi\)
−0.0578206 + 0.998327i \(0.518415\pi\)
\(882\) 0 0
\(883\) 19.3441 0.650980 0.325490 0.945545i \(-0.394471\pi\)
0.325490 + 0.945545i \(0.394471\pi\)
\(884\) −26.9291 −0.905723
\(885\) 20.2227 0.679780
\(886\) 23.4046 0.786294
\(887\) 35.3049 1.18542 0.592712 0.805415i \(-0.298057\pi\)
0.592712 + 0.805415i \(0.298057\pi\)
\(888\) −42.2960 −1.41936
\(889\) 0 0
\(890\) 29.1749 0.977946
\(891\) 29.8206 0.999028
\(892\) −35.2275 −1.17951
\(893\) −5.50514 −0.184223
\(894\) 39.9492 1.33610
\(895\) 44.9706 1.50320
\(896\) 0 0
\(897\) 18.0270 0.601903
\(898\) 50.0591 1.67050
\(899\) 3.28338 0.109507
\(900\) −3.66687 −0.122229
\(901\) 70.5486 2.35032
\(902\) 67.9339 2.26195
\(903\) 0 0
\(904\) 45.2171 1.50390
\(905\) −5.68660 −0.189029
\(906\) 56.3127 1.87087
\(907\) −24.5319 −0.814569 −0.407284 0.913301i \(-0.633524\pi\)
−0.407284 + 0.913301i \(0.633524\pi\)
\(908\) 80.0178 2.65548
\(909\) 0.835564 0.0277139
\(910\) 0 0
\(911\) −52.4707 −1.73843 −0.869216 0.494433i \(-0.835376\pi\)
−0.869216 + 0.494433i \(0.835376\pi\)
\(912\) −0.582422 −0.0192859
\(913\) 15.4641 0.511787
\(914\) −26.6554 −0.881683
\(915\) −31.7795 −1.05060
\(916\) 12.9802 0.428877
\(917\) 0 0
\(918\) −56.7287 −1.87233
\(919\) −5.27938 −0.174151 −0.0870753 0.996202i \(-0.527752\pi\)
−0.0870753 + 0.996202i \(0.527752\pi\)
\(920\) 27.8275 0.917446
\(921\) 10.0131 0.329942
\(922\) 43.4424 1.43070
\(923\) 19.3422 0.636657
\(924\) 0 0
\(925\) −17.1851 −0.565044
\(926\) 8.53399 0.280444
\(927\) 6.40188 0.210265
\(928\) −19.0474 −0.625262
\(929\) 17.7720 0.583080 0.291540 0.956559i \(-0.405832\pi\)
0.291540 + 0.956559i \(0.405832\pi\)
\(930\) 7.64985 0.250849
\(931\) 0 0
\(932\) −90.9947 −2.98063
\(933\) 11.3761 0.372435
\(934\) −7.16431 −0.234423
\(935\) −26.8793 −0.879047
\(936\) 2.36123 0.0771792
\(937\) 37.1004 1.21202 0.606008 0.795458i \(-0.292770\pi\)
0.606008 + 0.795458i \(0.292770\pi\)
\(938\) 0 0
\(939\) 33.1205 1.08085
\(940\) −30.0152 −0.978987
\(941\) −8.01455 −0.261267 −0.130633 0.991431i \(-0.541701\pi\)
−0.130633 + 0.991431i \(0.541701\pi\)
\(942\) 71.2958 2.32294
\(943\) 63.4563 2.06642
\(944\) −1.91619 −0.0623666
\(945\) 0 0
\(946\) 21.0165 0.683307
\(947\) −36.1009 −1.17312 −0.586560 0.809905i \(-0.699518\pi\)
−0.586560 + 0.809905i \(0.699518\pi\)
\(948\) −77.7307 −2.52457
\(949\) 16.5130 0.536035
\(950\) 4.62398 0.150022
\(951\) 38.3676 1.24416
\(952\) 0 0
\(953\) −45.8308 −1.48461 −0.742303 0.670065i \(-0.766266\pi\)
−0.742303 + 0.670065i \(0.766266\pi\)
\(954\) −16.7961 −0.543794
\(955\) −12.0685 −0.390528
\(956\) 41.5011 1.34224
\(957\) 17.2205 0.556661
\(958\) −13.9574 −0.450942
\(959\) 0 0
\(960\) −42.3720 −1.36755
\(961\) −29.9298 −0.965478
\(962\) 30.0469 0.968750
\(963\) −0.138541 −0.00446443
\(964\) 0.00789548 0.000254296 0
\(965\) −11.3884 −0.366606
\(966\) 0 0
\(967\) −14.3040 −0.459985 −0.229993 0.973192i \(-0.573870\pi\)
−0.229993 + 0.973192i \(0.573870\pi\)
\(968\) −7.29500 −0.234470
\(969\) −10.2682 −0.329861
\(970\) −9.59596 −0.308108
\(971\) 27.0679 0.868651 0.434325 0.900756i \(-0.356987\pi\)
0.434325 + 0.900756i \(0.356987\pi\)
\(972\) 18.4930 0.593162
\(973\) 0 0
\(974\) 91.9838 2.94735
\(975\) 6.01495 0.192633
\(976\) 3.01124 0.0963873
\(977\) 28.6282 0.915898 0.457949 0.888978i \(-0.348584\pi\)
0.457949 + 0.888978i \(0.348584\pi\)
\(978\) 20.0620 0.641513
\(979\) 21.4060 0.684138
\(980\) 0 0
\(981\) 9.18462 0.293242
\(982\) −41.0233 −1.30911
\(983\) −39.6198 −1.26368 −0.631838 0.775100i \(-0.717699\pi\)
−0.631838 + 0.775100i \(0.717699\pi\)
\(984\) 52.1112 1.66124
\(985\) 9.01508 0.287244
\(986\) −39.2080 −1.24864
\(987\) 0 0
\(988\) −4.95473 −0.157631
\(989\) 19.6313 0.624239
\(990\) 6.39938 0.203386
\(991\) −4.20741 −0.133653 −0.0668264 0.997765i \(-0.521287\pi\)
−0.0668264 + 0.997765i \(0.521287\pi\)
\(992\) −6.20823 −0.197111
\(993\) 26.4297 0.838720
\(994\) 0 0
\(995\) −22.9989 −0.729113
\(996\) 32.2087 1.02057
\(997\) 21.5597 0.682803 0.341401 0.939918i \(-0.389098\pi\)
0.341401 + 0.939918i \(0.389098\pi\)
\(998\) −80.8877 −2.56045
\(999\) 38.7917 1.22732
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 931.2.a.p.1.10 10
3.2 odd 2 8379.2.a.ct.1.1 10
7.2 even 3 931.2.f.r.704.1 20
7.3 odd 6 931.2.f.q.324.1 20
7.4 even 3 931.2.f.r.324.1 20
7.5 odd 6 931.2.f.q.704.1 20
7.6 odd 2 931.2.a.q.1.10 yes 10
21.20 even 2 8379.2.a.cs.1.1 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
931.2.a.p.1.10 10 1.1 even 1 trivial
931.2.a.q.1.10 yes 10 7.6 odd 2
931.2.f.q.324.1 20 7.3 odd 6
931.2.f.q.704.1 20 7.5 odd 6
931.2.f.r.324.1 20 7.4 even 3
931.2.f.r.704.1 20 7.2 even 3
8379.2.a.cs.1.1 10 21.20 even 2
8379.2.a.ct.1.1 10 3.2 odd 2