Properties

Label 1805.2.a.o.1.4
Level $1805$
Weight $2$
Character 1805.1
Self dual yes
Analytic conductor $14.413$
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] = [1805,2,Mod(1,1805)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1805, 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("1805.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1805 = 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1805.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: \(14.4129975648\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.7537.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 4x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.491918\) of defining polynomial
Character \(\chi\) \(=\) 1805.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.75802 q^{2} +1.49192 q^{3} +5.60665 q^{4} +1.00000 q^{5} +4.11474 q^{6} -2.84864 q^{7} +9.94721 q^{8} -0.774179 q^{9} +O(q^{10})\) \(q+2.75802 q^{2} +1.49192 q^{3} +5.60665 q^{4} +1.00000 q^{5} +4.11474 q^{6} -2.84864 q^{7} +9.94721 q^{8} -0.774179 q^{9} +2.75802 q^{10} -0.864801 q^{11} +8.36467 q^{12} +0.643281 q^{13} -7.85659 q^{14} +1.49192 q^{15} +16.2213 q^{16} +3.74185 q^{17} -2.13520 q^{18} +5.60665 q^{20} -4.24993 q^{21} -2.38513 q^{22} +0.417460 q^{23} +14.8404 q^{24} +1.00000 q^{25} +1.77418 q^{26} -5.63077 q^{27} -15.9713 q^{28} -9.70523 q^{29} +4.11474 q^{30} +4.93349 q^{31} +24.8441 q^{32} -1.29021 q^{33} +10.3201 q^{34} -2.84864 q^{35} -4.34056 q^{36} +6.36467 q^{37} +0.959723 q^{39} +9.94721 q^{40} -4.01372 q^{41} -11.7214 q^{42} -2.05829 q^{43} -4.84864 q^{44} -0.774179 q^{45} +1.15136 q^{46} -3.95396 q^{47} +24.2008 q^{48} +1.11474 q^{49} +2.75802 q^{50} +5.58254 q^{51} +3.60665 q^{52} -10.9875 q^{53} -15.5297 q^{54} -0.864801 q^{55} -28.3360 q^{56} -26.7672 q^{58} +2.45959 q^{59} +8.36467 q^{60} +6.33479 q^{61} +13.6067 q^{62} +2.20536 q^{63} +36.0778 q^{64} +0.643281 q^{65} -3.55843 q^{66} -2.53220 q^{67} +20.9793 q^{68} +0.622817 q^{69} -7.85659 q^{70} -1.78213 q^{71} -7.70092 q^{72} -7.13090 q^{73} +17.5539 q^{74} +1.49192 q^{75} +2.46350 q^{77} +2.64693 q^{78} +1.82452 q^{79} +16.2213 q^{80} -6.07811 q^{81} -11.0699 q^{82} -7.43913 q^{83} -23.8279 q^{84} +3.74185 q^{85} -5.67681 q^{86} -14.4794 q^{87} -8.60235 q^{88} +4.44588 q^{89} -2.13520 q^{90} -1.83247 q^{91} +2.34056 q^{92} +7.36037 q^{93} -10.9051 q^{94} +37.0653 q^{96} -10.8541 q^{97} +3.07446 q^{98} +0.669511 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} + 3 q^{3} + 5 q^{4} + 4 q^{5} + 2 q^{6} - 4 q^{7} + 12 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} + 3 q^{3} + 5 q^{4} + 4 q^{5} + 2 q^{6} - 4 q^{7} + 12 q^{8} + q^{9} + q^{10} - 2 q^{11} + 6 q^{12} + 7 q^{13} - q^{14} + 3 q^{15} + 7 q^{16} - q^{17} - 10 q^{18} + 5 q^{20} - 4 q^{21} + 2 q^{22} + 2 q^{23} + 23 q^{24} + 4 q^{25} + 3 q^{26} + 12 q^{27} - 19 q^{28} - q^{29} + 2 q^{30} + 30 q^{32} + 19 q^{33} + 15 q^{34} - 4 q^{35} - 7 q^{36} - 2 q^{37} + 15 q^{39} + 12 q^{40} - 8 q^{41} - 15 q^{42} + q^{43} - 12 q^{44} + q^{45} + 12 q^{46} - 12 q^{47} + 23 q^{48} - 10 q^{49} + q^{50} + 22 q^{51} - 3 q^{52} - 5 q^{53} - 34 q^{54} - 2 q^{55} - 41 q^{56} - 27 q^{58} - 5 q^{59} + 6 q^{60} + 37 q^{62} - 3 q^{63} + 56 q^{64} + 7 q^{65} - 31 q^{66} + 4 q^{67} + 16 q^{68} - 9 q^{69} - q^{70} + 20 q^{71} + 17 q^{72} - 20 q^{73} + 25 q^{74} + 3 q^{75} + 14 q^{77} - 18 q^{78} + 17 q^{79} + 7 q^{80} + 12 q^{81} + 21 q^{82} + q^{83} - 20 q^{84} - q^{85} + 8 q^{86} - 16 q^{87} - 7 q^{88} + 11 q^{89} - 10 q^{90} + 6 q^{91} - q^{92} - 8 q^{93} - 31 q^{94} + 21 q^{96} + q^{97} + 9 q^{98} + 38 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.75802 1.95021 0.975106 0.221739i \(-0.0711734\pi\)
0.975106 + 0.221739i \(0.0711734\pi\)
\(3\) 1.49192 0.861360 0.430680 0.902505i \(-0.358274\pi\)
0.430680 + 0.902505i \(0.358274\pi\)
\(4\) 5.60665 2.80333
\(5\) 1.00000 0.447214
\(6\) 4.11474 1.67983
\(7\) −2.84864 −1.07668 −0.538342 0.842727i \(-0.680949\pi\)
−0.538342 + 0.842727i \(0.680949\pi\)
\(8\) 9.94721 3.51687
\(9\) −0.774179 −0.258060
\(10\) 2.75802 0.872161
\(11\) −0.864801 −0.260747 −0.130374 0.991465i \(-0.541618\pi\)
−0.130374 + 0.991465i \(0.541618\pi\)
\(12\) 8.36467 2.41467
\(13\) 0.643281 0.178414 0.0892070 0.996013i \(-0.471567\pi\)
0.0892070 + 0.996013i \(0.471567\pi\)
\(14\) −7.85659 −2.09976
\(15\) 1.49192 0.385212
\(16\) 16.2213 4.05531
\(17\) 3.74185 0.907533 0.453766 0.891121i \(-0.350080\pi\)
0.453766 + 0.891121i \(0.350080\pi\)
\(18\) −2.13520 −0.503271
\(19\) 0 0
\(20\) 5.60665 1.25369
\(21\) −4.24993 −0.927412
\(22\) −2.38513 −0.508512
\(23\) 0.417460 0.0870465 0.0435233 0.999052i \(-0.486142\pi\)
0.0435233 + 0.999052i \(0.486142\pi\)
\(24\) 14.8404 3.02929
\(25\) 1.00000 0.200000
\(26\) 1.77418 0.347945
\(27\) −5.63077 −1.08364
\(28\) −15.9713 −3.01830
\(29\) −9.70523 −1.80222 −0.901108 0.433596i \(-0.857245\pi\)
−0.901108 + 0.433596i \(0.857245\pi\)
\(30\) 4.11474 0.751244
\(31\) 4.93349 0.886081 0.443041 0.896501i \(-0.353900\pi\)
0.443041 + 0.896501i \(0.353900\pi\)
\(32\) 24.8441 4.39185
\(33\) −1.29021 −0.224597
\(34\) 10.3201 1.76988
\(35\) −2.84864 −0.481508
\(36\) −4.34056 −0.723426
\(37\) 6.36467 1.04635 0.523173 0.852227i \(-0.324748\pi\)
0.523173 + 0.852227i \(0.324748\pi\)
\(38\) 0 0
\(39\) 0.959723 0.153679
\(40\) 9.94721 1.57279
\(41\) −4.01372 −0.626837 −0.313419 0.949615i \(-0.601474\pi\)
−0.313419 + 0.949615i \(0.601474\pi\)
\(42\) −11.7214 −1.80865
\(43\) −2.05829 −0.313887 −0.156944 0.987608i \(-0.550164\pi\)
−0.156944 + 0.987608i \(0.550164\pi\)
\(44\) −4.84864 −0.730960
\(45\) −0.774179 −0.115408
\(46\) 1.15136 0.169759
\(47\) −3.95396 −0.576744 −0.288372 0.957518i \(-0.593114\pi\)
−0.288372 + 0.957518i \(0.593114\pi\)
\(48\) 24.2008 3.49308
\(49\) 1.11474 0.159248
\(50\) 2.75802 0.390042
\(51\) 5.58254 0.781712
\(52\) 3.60665 0.500153
\(53\) −10.9875 −1.50925 −0.754624 0.656158i \(-0.772181\pi\)
−0.754624 + 0.656158i \(0.772181\pi\)
\(54\) −15.5297 −2.11333
\(55\) −0.864801 −0.116610
\(56\) −28.3360 −3.78656
\(57\) 0 0
\(58\) −26.7672 −3.51470
\(59\) 2.45959 0.320212 0.160106 0.987100i \(-0.448816\pi\)
0.160106 + 0.987100i \(0.448816\pi\)
\(60\) 8.36467 1.07987
\(61\) 6.33479 0.811087 0.405543 0.914076i \(-0.367082\pi\)
0.405543 + 0.914076i \(0.367082\pi\)
\(62\) 13.6067 1.72805
\(63\) 2.20536 0.277849
\(64\) 36.0778 4.50973
\(65\) 0.643281 0.0797892
\(66\) −3.55843 −0.438012
\(67\) −2.53220 −0.309357 −0.154678 0.987965i \(-0.549434\pi\)
−0.154678 + 0.987965i \(0.549434\pi\)
\(68\) 20.9793 2.54411
\(69\) 0.622817 0.0749783
\(70\) −7.85659 −0.939042
\(71\) −1.78213 −0.211500 −0.105750 0.994393i \(-0.533724\pi\)
−0.105750 + 0.994393i \(0.533724\pi\)
\(72\) −7.70092 −0.907563
\(73\) −7.13090 −0.834609 −0.417304 0.908767i \(-0.637025\pi\)
−0.417304 + 0.908767i \(0.637025\pi\)
\(74\) 17.5539 2.04060
\(75\) 1.49192 0.172272
\(76\) 0 0
\(77\) 2.46350 0.280742
\(78\) 2.64693 0.299706
\(79\) 1.82452 0.205275 0.102637 0.994719i \(-0.467272\pi\)
0.102637 + 0.994719i \(0.467272\pi\)
\(80\) 16.2213 1.81359
\(81\) −6.07811 −0.675345
\(82\) −11.0699 −1.22247
\(83\) −7.43913 −0.816550 −0.408275 0.912859i \(-0.633870\pi\)
−0.408275 + 0.912859i \(0.633870\pi\)
\(84\) −23.8279 −2.59984
\(85\) 3.74185 0.405861
\(86\) −5.67681 −0.612146
\(87\) −14.4794 −1.55236
\(88\) −8.60235 −0.917014
\(89\) 4.44588 0.471262 0.235631 0.971843i \(-0.424284\pi\)
0.235631 + 0.971843i \(0.424284\pi\)
\(90\) −2.13520 −0.225070
\(91\) −1.83247 −0.192096
\(92\) 2.34056 0.244020
\(93\) 7.36037 0.763235
\(94\) −10.9051 −1.12477
\(95\) 0 0
\(96\) 37.0653 3.78296
\(97\) −10.8541 −1.10207 −0.551036 0.834482i \(-0.685767\pi\)
−0.551036 + 0.834482i \(0.685767\pi\)
\(98\) 3.07446 0.310567
\(99\) 0.669511 0.0672884
\(100\) 5.60665 0.560665
\(101\) −5.29598 −0.526969 −0.263485 0.964664i \(-0.584872\pi\)
−0.263485 + 0.964664i \(0.584872\pi\)
\(102\) 15.3967 1.52450
\(103\) −0.385134 −0.0379484 −0.0189742 0.999820i \(-0.506040\pi\)
−0.0189742 + 0.999820i \(0.506040\pi\)
\(104\) 6.39885 0.627459
\(105\) −4.24993 −0.414751
\(106\) −30.3037 −2.94335
\(107\) −6.43336 −0.621937 −0.310968 0.950420i \(-0.600653\pi\)
−0.310968 + 0.950420i \(0.600653\pi\)
\(108\) −31.5698 −3.03780
\(109\) 6.56882 0.629179 0.314590 0.949228i \(-0.398133\pi\)
0.314590 + 0.949228i \(0.398133\pi\)
\(110\) −2.38513 −0.227414
\(111\) 9.49557 0.901279
\(112\) −46.2085 −4.36629
\(113\) 0.294513 0.0277054 0.0138527 0.999904i \(-0.495590\pi\)
0.0138527 + 0.999904i \(0.495590\pi\)
\(114\) 0 0
\(115\) 0.417460 0.0389284
\(116\) −54.4138 −5.05220
\(117\) −0.498015 −0.0460415
\(118\) 6.78360 0.624481
\(119\) −10.6592 −0.977126
\(120\) 14.8404 1.35474
\(121\) −10.2521 −0.932011
\(122\) 17.4715 1.58179
\(123\) −5.98814 −0.539932
\(124\) 27.6604 2.48398
\(125\) 1.00000 0.0894427
\(126\) 6.08241 0.541864
\(127\) 8.83492 0.783972 0.391986 0.919971i \(-0.371788\pi\)
0.391986 + 0.919971i \(0.371788\pi\)
\(128\) 49.8151 4.40308
\(129\) −3.07081 −0.270370
\(130\) 1.77418 0.155606
\(131\) 20.9128 1.82716 0.913578 0.406662i \(-0.133307\pi\)
0.913578 + 0.406662i \(0.133307\pi\)
\(132\) −7.23377 −0.629619
\(133\) 0 0
\(134\) −6.98384 −0.603312
\(135\) −5.63077 −0.484619
\(136\) 37.2210 3.19167
\(137\) 5.21477 0.445528 0.222764 0.974872i \(-0.428492\pi\)
0.222764 + 0.974872i \(0.428492\pi\)
\(138\) 1.71774 0.146224
\(139\) −10.7238 −0.909584 −0.454792 0.890598i \(-0.650286\pi\)
−0.454792 + 0.890598i \(0.650286\pi\)
\(140\) −15.9713 −1.34982
\(141\) −5.89898 −0.496784
\(142\) −4.91514 −0.412470
\(143\) −0.556310 −0.0465210
\(144\) −12.5582 −1.04651
\(145\) −9.70523 −0.805975
\(146\) −19.6671 −1.62766
\(147\) 1.66309 0.137170
\(148\) 35.6845 2.93325
\(149\) −14.9116 −1.22160 −0.610801 0.791784i \(-0.709153\pi\)
−0.610801 + 0.791784i \(0.709153\pi\)
\(150\) 4.11474 0.335967
\(151\) 21.4589 1.74630 0.873152 0.487448i \(-0.162072\pi\)
0.873152 + 0.487448i \(0.162072\pi\)
\(152\) 0 0
\(153\) −2.89687 −0.234198
\(154\) 6.79438 0.547507
\(155\) 4.93349 0.396268
\(156\) 5.38083 0.430811
\(157\) −2.43118 −0.194029 −0.0970145 0.995283i \(-0.530929\pi\)
−0.0970145 + 0.995283i \(0.530929\pi\)
\(158\) 5.03207 0.400330
\(159\) −16.3924 −1.30000
\(160\) 24.8441 1.96410
\(161\) −1.18919 −0.0937216
\(162\) −16.7635 −1.31707
\(163\) 17.8175 1.39558 0.697788 0.716305i \(-0.254168\pi\)
0.697788 + 0.716305i \(0.254168\pi\)
\(164\) −22.5035 −1.75723
\(165\) −1.29021 −0.100443
\(166\) −20.5172 −1.59245
\(167\) −0.405598 −0.0313861 −0.0156931 0.999877i \(-0.504995\pi\)
−0.0156931 + 0.999877i \(0.504995\pi\)
\(168\) −42.2750 −3.26159
\(169\) −12.5862 −0.968168
\(170\) 10.3201 0.791515
\(171\) 0 0
\(172\) −11.5401 −0.879928
\(173\) 18.0210 1.37011 0.685056 0.728490i \(-0.259778\pi\)
0.685056 + 0.728490i \(0.259778\pi\)
\(174\) −39.9344 −3.02742
\(175\) −2.84864 −0.215337
\(176\) −14.0282 −1.05741
\(177\) 3.66951 0.275817
\(178\) 12.2618 0.919061
\(179\) 20.1523 1.50625 0.753127 0.657875i \(-0.228545\pi\)
0.753127 + 0.657875i \(0.228545\pi\)
\(180\) −4.34056 −0.323526
\(181\) −17.1108 −1.27184 −0.635919 0.771756i \(-0.719379\pi\)
−0.635919 + 0.771756i \(0.719379\pi\)
\(182\) −5.05399 −0.374627
\(183\) 9.45099 0.698637
\(184\) 4.15257 0.306131
\(185\) 6.36467 0.467940
\(186\) 20.3000 1.48847
\(187\) −3.23596 −0.236637
\(188\) −22.1685 −1.61680
\(189\) 16.0400 1.16674
\(190\) 0 0
\(191\) 5.28080 0.382105 0.191053 0.981580i \(-0.438810\pi\)
0.191053 + 0.981580i \(0.438810\pi\)
\(192\) 53.8252 3.88450
\(193\) 18.0036 1.29593 0.647966 0.761670i \(-0.275620\pi\)
0.647966 + 0.761670i \(0.275620\pi\)
\(194\) −29.9359 −2.14927
\(195\) 0.959723 0.0687272
\(196\) 6.24993 0.446424
\(197\) 8.07785 0.575523 0.287761 0.957702i \(-0.407089\pi\)
0.287761 + 0.957702i \(0.407089\pi\)
\(198\) 1.84652 0.131227
\(199\) 1.40374 0.0995088 0.0497544 0.998761i \(-0.484156\pi\)
0.0497544 + 0.998761i \(0.484156\pi\)
\(200\) 9.94721 0.703374
\(201\) −3.77783 −0.266468
\(202\) −14.6064 −1.02770
\(203\) 27.6467 1.94042
\(204\) 31.2994 2.19139
\(205\) −4.01372 −0.280330
\(206\) −1.06221 −0.0740074
\(207\) −0.323189 −0.0224632
\(208\) 10.4348 0.723525
\(209\) 0 0
\(210\) −11.7214 −0.808853
\(211\) 18.9163 1.30226 0.651128 0.758968i \(-0.274296\pi\)
0.651128 + 0.758968i \(0.274296\pi\)
\(212\) −61.6030 −4.23091
\(213\) −2.65879 −0.182178
\(214\) −17.7433 −1.21291
\(215\) −2.05829 −0.140375
\(216\) −56.0104 −3.81103
\(217\) −14.0537 −0.954030
\(218\) 18.1169 1.22703
\(219\) −10.6387 −0.718898
\(220\) −4.84864 −0.326895
\(221\) 2.40706 0.161917
\(222\) 26.1889 1.75769
\(223\) −16.1480 −1.08135 −0.540675 0.841231i \(-0.681831\pi\)
−0.540675 + 0.841231i \(0.681831\pi\)
\(224\) −70.7718 −4.72864
\(225\) −0.774179 −0.0516120
\(226\) 0.812271 0.0540315
\(227\) 26.3186 1.74683 0.873414 0.486978i \(-0.161901\pi\)
0.873414 + 0.486978i \(0.161901\pi\)
\(228\) 0 0
\(229\) −13.3323 −0.881026 −0.440513 0.897746i \(-0.645203\pi\)
−0.440513 + 0.897746i \(0.645203\pi\)
\(230\) 1.15136 0.0759186
\(231\) 3.67535 0.241820
\(232\) −96.5399 −6.33816
\(233\) 25.3094 1.65808 0.829038 0.559192i \(-0.188889\pi\)
0.829038 + 0.559192i \(0.188889\pi\)
\(234\) −1.37353 −0.0897907
\(235\) −3.95396 −0.257928
\(236\) 13.7901 0.897658
\(237\) 2.72204 0.176815
\(238\) −29.3982 −1.90560
\(239\) −23.5500 −1.52332 −0.761660 0.647977i \(-0.775615\pi\)
−0.761660 + 0.647977i \(0.775615\pi\)
\(240\) 24.2008 1.56215
\(241\) 8.38415 0.540071 0.270035 0.962850i \(-0.412965\pi\)
0.270035 + 0.962850i \(0.412965\pi\)
\(242\) −28.2755 −1.81762
\(243\) 7.82426 0.501927
\(244\) 35.5170 2.27374
\(245\) 1.11474 0.0712178
\(246\) −16.5154 −1.05298
\(247\) 0 0
\(248\) 49.0745 3.11623
\(249\) −11.0986 −0.703343
\(250\) 2.75802 0.174432
\(251\) 18.2478 1.15179 0.575896 0.817523i \(-0.304653\pi\)
0.575896 + 0.817523i \(0.304653\pi\)
\(252\) 12.3647 0.778901
\(253\) −0.361020 −0.0226971
\(254\) 24.3669 1.52891
\(255\) 5.58254 0.349592
\(256\) 65.2353 4.07720
\(257\) 14.0998 0.879520 0.439760 0.898115i \(-0.355064\pi\)
0.439760 + 0.898115i \(0.355064\pi\)
\(258\) −8.46934 −0.527278
\(259\) −18.1306 −1.12658
\(260\) 3.60665 0.223675
\(261\) 7.51359 0.465079
\(262\) 57.6778 3.56334
\(263\) −6.41071 −0.395301 −0.197651 0.980273i \(-0.563331\pi\)
−0.197651 + 0.980273i \(0.563331\pi\)
\(264\) −12.8340 −0.789879
\(265\) −10.9875 −0.674956
\(266\) 0 0
\(267\) 6.63288 0.405926
\(268\) −14.1971 −0.867229
\(269\) 17.9911 1.09694 0.548469 0.836171i \(-0.315211\pi\)
0.548469 + 0.836171i \(0.315211\pi\)
\(270\) −15.5297 −0.945110
\(271\) 11.8819 0.721774 0.360887 0.932609i \(-0.382474\pi\)
0.360887 + 0.932609i \(0.382474\pi\)
\(272\) 60.6976 3.68033
\(273\) −2.73390 −0.165463
\(274\) 14.3824 0.868874
\(275\) −0.864801 −0.0521494
\(276\) 3.49192 0.210189
\(277\) 23.6240 1.41943 0.709715 0.704489i \(-0.248824\pi\)
0.709715 + 0.704489i \(0.248824\pi\)
\(278\) −29.5765 −1.77388
\(279\) −3.81941 −0.228662
\(280\) −28.3360 −1.69340
\(281\) 13.8093 0.823794 0.411897 0.911230i \(-0.364866\pi\)
0.411897 + 0.911230i \(0.364866\pi\)
\(282\) −16.2695 −0.968834
\(283\) −11.7574 −0.698903 −0.349451 0.936954i \(-0.613632\pi\)
−0.349451 + 0.936954i \(0.613632\pi\)
\(284\) −9.99179 −0.592904
\(285\) 0 0
\(286\) −1.53431 −0.0907257
\(287\) 11.4336 0.674905
\(288\) −19.2338 −1.13336
\(289\) −2.99854 −0.176384
\(290\) −26.7672 −1.57182
\(291\) −16.1935 −0.949279
\(292\) −39.9805 −2.33968
\(293\) −27.0576 −1.58072 −0.790362 0.612640i \(-0.790108\pi\)
−0.790362 + 0.612640i \(0.790108\pi\)
\(294\) 4.58684 0.267510
\(295\) 2.45959 0.143203
\(296\) 63.3107 3.67986
\(297\) 4.86949 0.282557
\(298\) −41.1263 −2.38238
\(299\) 0.268544 0.0155303
\(300\) 8.36467 0.482934
\(301\) 5.86334 0.337957
\(302\) 59.1841 3.40566
\(303\) −7.90117 −0.453910
\(304\) 0 0
\(305\) 6.33479 0.362729
\(306\) −7.98960 −0.456735
\(307\) 8.83824 0.504425 0.252212 0.967672i \(-0.418842\pi\)
0.252212 + 0.967672i \(0.418842\pi\)
\(308\) 13.8120 0.787012
\(309\) −0.574589 −0.0326872
\(310\) 13.6067 0.772806
\(311\) −0.651493 −0.0369428 −0.0184714 0.999829i \(-0.505880\pi\)
−0.0184714 + 0.999829i \(0.505880\pi\)
\(312\) 9.54656 0.540468
\(313\) −2.96556 −0.167623 −0.0838116 0.996482i \(-0.526709\pi\)
−0.0838116 + 0.996482i \(0.526709\pi\)
\(314\) −6.70523 −0.378398
\(315\) 2.20536 0.124258
\(316\) 10.2295 0.575453
\(317\) −10.3799 −0.582991 −0.291495 0.956572i \(-0.594153\pi\)
−0.291495 + 0.956572i \(0.594153\pi\)
\(318\) −45.2106 −2.53528
\(319\) 8.39309 0.469923
\(320\) 36.0778 2.01681
\(321\) −9.59805 −0.535711
\(322\) −3.27981 −0.182777
\(323\) 0 0
\(324\) −34.0778 −1.89321
\(325\) 0.643281 0.0356828
\(326\) 49.1410 2.72167
\(327\) 9.80015 0.541949
\(328\) −39.9253 −2.20450
\(329\) 11.2634 0.620971
\(330\) −3.55843 −0.195885
\(331\) 15.0922 0.829543 0.414772 0.909926i \(-0.363861\pi\)
0.414772 + 0.909926i \(0.363861\pi\)
\(332\) −41.7086 −2.28906
\(333\) −4.92740 −0.270020
\(334\) −1.11865 −0.0612096
\(335\) −2.53220 −0.138349
\(336\) −68.9393 −3.76095
\(337\) 15.7974 0.860541 0.430271 0.902700i \(-0.358418\pi\)
0.430271 + 0.902700i \(0.358418\pi\)
\(338\) −34.7129 −1.88813
\(339\) 0.439389 0.0238643
\(340\) 20.9793 1.13776
\(341\) −4.26649 −0.231043
\(342\) 0 0
\(343\) 16.7650 0.905224
\(344\) −20.4743 −1.10390
\(345\) 0.622817 0.0335313
\(346\) 49.7023 2.67201
\(347\) −21.3522 −1.14624 −0.573122 0.819470i \(-0.694268\pi\)
−0.573122 + 0.819470i \(0.694268\pi\)
\(348\) −81.1810 −4.35176
\(349\) −32.3897 −1.73378 −0.866891 0.498497i \(-0.833885\pi\)
−0.866891 + 0.498497i \(0.833885\pi\)
\(350\) −7.85659 −0.419952
\(351\) −3.62217 −0.193337
\(352\) −21.4852 −1.14516
\(353\) 0.730583 0.0388850 0.0194425 0.999811i \(-0.493811\pi\)
0.0194425 + 0.999811i \(0.493811\pi\)
\(354\) 10.1206 0.537902
\(355\) −1.78213 −0.0945857
\(356\) 24.9265 1.32110
\(357\) −15.9026 −0.841657
\(358\) 55.5804 2.93751
\(359\) −26.8496 −1.41707 −0.708533 0.705677i \(-0.750643\pi\)
−0.708533 + 0.705677i \(0.750643\pi\)
\(360\) −7.70092 −0.405874
\(361\) 0 0
\(362\) −47.1919 −2.48035
\(363\) −15.2953 −0.802796
\(364\) −10.2740 −0.538506
\(365\) −7.13090 −0.373248
\(366\) 26.0660 1.36249
\(367\) −22.9643 −1.19873 −0.599364 0.800477i \(-0.704580\pi\)
−0.599364 + 0.800477i \(0.704580\pi\)
\(368\) 6.77173 0.353001
\(369\) 3.10734 0.161761
\(370\) 17.5539 0.912582
\(371\) 31.2994 1.62498
\(372\) 41.2670 2.13960
\(373\) 29.5305 1.52903 0.764515 0.644606i \(-0.222979\pi\)
0.764515 + 0.644606i \(0.222979\pi\)
\(374\) −8.92482 −0.461492
\(375\) 1.49192 0.0770423
\(376\) −39.3308 −2.02833
\(377\) −6.24319 −0.321541
\(378\) 44.2386 2.27539
\(379\) 17.5117 0.899517 0.449759 0.893150i \(-0.351510\pi\)
0.449759 + 0.893150i \(0.351510\pi\)
\(380\) 0 0
\(381\) 13.1810 0.675282
\(382\) 14.5645 0.745186
\(383\) 8.10652 0.414224 0.207112 0.978317i \(-0.433594\pi\)
0.207112 + 0.978317i \(0.433594\pi\)
\(384\) 74.3201 3.79263
\(385\) 2.46350 0.125552
\(386\) 49.6544 2.52734
\(387\) 1.59349 0.0810016
\(388\) −60.8554 −3.08947
\(389\) 17.3078 0.877542 0.438771 0.898599i \(-0.355414\pi\)
0.438771 + 0.898599i \(0.355414\pi\)
\(390\) 2.64693 0.134033
\(391\) 1.56208 0.0789976
\(392\) 11.0885 0.560054
\(393\) 31.2001 1.57384
\(394\) 22.2788 1.12239
\(395\) 1.82452 0.0918017
\(396\) 3.75372 0.188631
\(397\) −11.3894 −0.571619 −0.285810 0.958286i \(-0.592263\pi\)
−0.285810 + 0.958286i \(0.592263\pi\)
\(398\) 3.87155 0.194063
\(399\) 0 0
\(400\) 16.2213 0.811063
\(401\) −8.93861 −0.446373 −0.223186 0.974776i \(-0.571646\pi\)
−0.223186 + 0.974776i \(0.571646\pi\)
\(402\) −10.4193 −0.519668
\(403\) 3.17362 0.158089
\(404\) −29.6927 −1.47727
\(405\) −6.07811 −0.302024
\(406\) 76.2500 3.78422
\(407\) −5.50417 −0.272832
\(408\) 55.5307 2.74918
\(409\) −6.54471 −0.323615 −0.161808 0.986822i \(-0.551732\pi\)
−0.161808 + 0.986822i \(0.551732\pi\)
\(410\) −11.0699 −0.546703
\(411\) 7.78001 0.383760
\(412\) −2.15931 −0.106382
\(413\) −7.00649 −0.344767
\(414\) −0.891361 −0.0438080
\(415\) −7.43913 −0.365172
\(416\) 15.9817 0.783568
\(417\) −15.9991 −0.783479
\(418\) 0 0
\(419\) 21.8441 1.06715 0.533576 0.845752i \(-0.320848\pi\)
0.533576 + 0.845752i \(0.320848\pi\)
\(420\) −23.8279 −1.16268
\(421\) −29.3434 −1.43011 −0.715054 0.699069i \(-0.753598\pi\)
−0.715054 + 0.699069i \(0.753598\pi\)
\(422\) 52.1716 2.53967
\(423\) 3.06107 0.148834
\(424\) −109.295 −5.30783
\(425\) 3.74185 0.181507
\(426\) −7.33299 −0.355285
\(427\) −18.0455 −0.873284
\(428\) −36.0696 −1.74349
\(429\) −0.829969 −0.0400713
\(430\) −5.67681 −0.273760
\(431\) −12.8867 −0.620732 −0.310366 0.950617i \(-0.600452\pi\)
−0.310366 + 0.950617i \(0.600452\pi\)
\(432\) −91.3381 −4.39451
\(433\) −13.8429 −0.665246 −0.332623 0.943060i \(-0.607934\pi\)
−0.332623 + 0.943060i \(0.607934\pi\)
\(434\) −38.7604 −1.86056
\(435\) −14.4794 −0.694234
\(436\) 36.8291 1.76379
\(437\) 0 0
\(438\) −29.3418 −1.40200
\(439\) −0.0708081 −0.00337948 −0.00168974 0.999999i \(-0.500538\pi\)
−0.00168974 + 0.999999i \(0.500538\pi\)
\(440\) −8.60235 −0.410101
\(441\) −0.863005 −0.0410955
\(442\) 6.63872 0.315772
\(443\) −3.78914 −0.180027 −0.0900137 0.995941i \(-0.528691\pi\)
−0.0900137 + 0.995941i \(0.528691\pi\)
\(444\) 53.2384 2.52658
\(445\) 4.44588 0.210755
\(446\) −44.5365 −2.10886
\(447\) −22.2468 −1.05224
\(448\) −102.773 −4.85555
\(449\) −26.5765 −1.25422 −0.627112 0.778929i \(-0.715763\pi\)
−0.627112 + 0.778929i \(0.715763\pi\)
\(450\) −2.13520 −0.100654
\(451\) 3.47106 0.163446
\(452\) 1.65123 0.0776674
\(453\) 32.0150 1.50420
\(454\) 72.5872 3.40669
\(455\) −1.83247 −0.0859077
\(456\) 0 0
\(457\) −33.1523 −1.55080 −0.775400 0.631471i \(-0.782452\pi\)
−0.775400 + 0.631471i \(0.782452\pi\)
\(458\) −36.7708 −1.71819
\(459\) −21.0695 −0.983440
\(460\) 2.34056 0.109129
\(461\) −19.2536 −0.896729 −0.448364 0.893851i \(-0.647993\pi\)
−0.448364 + 0.893851i \(0.647993\pi\)
\(462\) 10.1367 0.471600
\(463\) 39.1713 1.82044 0.910222 0.414120i \(-0.135911\pi\)
0.910222 + 0.414120i \(0.135911\pi\)
\(464\) −157.431 −7.30855
\(465\) 7.36037 0.341329
\(466\) 69.8038 3.23360
\(467\) −39.0650 −1.80771 −0.903856 0.427836i \(-0.859276\pi\)
−0.903856 + 0.427836i \(0.859276\pi\)
\(468\) −2.79220 −0.129069
\(469\) 7.21331 0.333080
\(470\) −10.9051 −0.503014
\(471\) −3.62712 −0.167129
\(472\) 24.4661 1.12614
\(473\) 1.78001 0.0818452
\(474\) 7.50743 0.344828
\(475\) 0 0
\(476\) −59.7623 −2.73920
\(477\) 8.50629 0.389476
\(478\) −64.9512 −2.97080
\(479\) −24.7550 −1.13109 −0.565543 0.824718i \(-0.691334\pi\)
−0.565543 + 0.824718i \(0.691334\pi\)
\(480\) 37.0653 1.69179
\(481\) 4.09427 0.186683
\(482\) 23.1236 1.05325
\(483\) −1.77418 −0.0807280
\(484\) −57.4801 −2.61273
\(485\) −10.8541 −0.492861
\(486\) 21.5794 0.978863
\(487\) 21.8871 0.991797 0.495899 0.868380i \(-0.334839\pi\)
0.495899 + 0.868380i \(0.334839\pi\)
\(488\) 63.0135 2.85249
\(489\) 26.5823 1.20209
\(490\) 3.07446 0.138890
\(491\) 9.39553 0.424014 0.212007 0.977268i \(-0.432000\pi\)
0.212007 + 0.977268i \(0.432000\pi\)
\(492\) −33.5734 −1.51361
\(493\) −36.3155 −1.63557
\(494\) 0 0
\(495\) 0.669511 0.0300923
\(496\) 80.0275 3.59334
\(497\) 5.07664 0.227719
\(498\) −30.6100 −1.37167
\(499\) 24.9115 1.11519 0.557596 0.830112i \(-0.311724\pi\)
0.557596 + 0.830112i \(0.311724\pi\)
\(500\) 5.60665 0.250737
\(501\) −0.605119 −0.0270347
\(502\) 50.3278 2.24624
\(503\) −31.3180 −1.39640 −0.698200 0.715903i \(-0.746015\pi\)
−0.698200 + 0.715903i \(0.746015\pi\)
\(504\) 21.9371 0.977158
\(505\) −5.29598 −0.235668
\(506\) −0.995699 −0.0442642
\(507\) −18.7776 −0.833941
\(508\) 49.5343 2.19773
\(509\) −9.66619 −0.428446 −0.214223 0.976785i \(-0.568722\pi\)
−0.214223 + 0.976785i \(0.568722\pi\)
\(510\) 15.3967 0.681779
\(511\) 20.3133 0.898609
\(512\) 80.2896 3.54833
\(513\) 0 0
\(514\) 38.8874 1.71525
\(515\) −0.385134 −0.0169710
\(516\) −17.2170 −0.757934
\(517\) 3.41938 0.150384
\(518\) −50.0046 −2.19708
\(519\) 26.8859 1.18016
\(520\) 6.39885 0.280608
\(521\) −0.982633 −0.0430499 −0.0215250 0.999768i \(-0.506852\pi\)
−0.0215250 + 0.999768i \(0.506852\pi\)
\(522\) 20.7226 0.907003
\(523\) 39.7209 1.73687 0.868436 0.495801i \(-0.165125\pi\)
0.868436 + 0.495801i \(0.165125\pi\)
\(524\) 117.251 5.12212
\(525\) −4.24993 −0.185482
\(526\) −17.6809 −0.770922
\(527\) 18.4604 0.804148
\(528\) −20.9289 −0.910812
\(529\) −22.8257 −0.992423
\(530\) −30.3037 −1.31631
\(531\) −1.90417 −0.0826337
\(532\) 0 0
\(533\) −2.58195 −0.111837
\(534\) 18.2936 0.791642
\(535\) −6.43336 −0.278139
\(536\) −25.1883 −1.08797
\(537\) 30.0656 1.29743
\(538\) 49.6198 2.13926
\(539\) −0.964024 −0.0415234
\(540\) −31.5698 −1.35855
\(541\) 30.7775 1.32323 0.661614 0.749845i \(-0.269872\pi\)
0.661614 + 0.749845i \(0.269872\pi\)
\(542\) 32.7705 1.40761
\(543\) −25.5280 −1.09551
\(544\) 92.9629 3.98575
\(545\) 6.56882 0.281377
\(546\) −7.54015 −0.322688
\(547\) −17.8657 −0.763884 −0.381942 0.924186i \(-0.624745\pi\)
−0.381942 + 0.924186i \(0.624745\pi\)
\(548\) 29.2374 1.24896
\(549\) −4.90426 −0.209309
\(550\) −2.38513 −0.101702
\(551\) 0 0
\(552\) 6.19529 0.263689
\(553\) −5.19741 −0.221016
\(554\) 65.1554 2.76819
\(555\) 9.49557 0.403064
\(556\) −60.1248 −2.54986
\(557\) −10.6576 −0.451575 −0.225787 0.974177i \(-0.572495\pi\)
−0.225787 + 0.974177i \(0.572495\pi\)
\(558\) −10.5340 −0.445939
\(559\) −1.32406 −0.0560019
\(560\) −46.2085 −1.95266
\(561\) −4.82778 −0.203829
\(562\) 38.0863 1.60657
\(563\) 7.75961 0.327029 0.163514 0.986541i \(-0.447717\pi\)
0.163514 + 0.986541i \(0.447717\pi\)
\(564\) −33.0735 −1.39265
\(565\) 0.294513 0.0123903
\(566\) −32.4270 −1.36301
\(567\) 17.3143 0.727133
\(568\) −17.7272 −0.743818
\(569\) 5.72754 0.240111 0.120056 0.992767i \(-0.461693\pi\)
0.120056 + 0.992767i \(0.461693\pi\)
\(570\) 0 0
\(571\) −20.8347 −0.871903 −0.435952 0.899970i \(-0.643588\pi\)
−0.435952 + 0.899970i \(0.643588\pi\)
\(572\) −3.11904 −0.130413
\(573\) 7.87852 0.329130
\(574\) 31.5341 1.31621
\(575\) 0.417460 0.0174093
\(576\) −27.9307 −1.16378
\(577\) −5.11190 −0.212811 −0.106406 0.994323i \(-0.533934\pi\)
−0.106406 + 0.994323i \(0.533934\pi\)
\(578\) −8.27001 −0.343987
\(579\) 26.8600 1.11626
\(580\) −54.4138 −2.25941
\(581\) 21.1914 0.879167
\(582\) −44.6619 −1.85130
\(583\) 9.50199 0.393532
\(584\) −70.9325 −2.93521
\(585\) −0.498015 −0.0205904
\(586\) −74.6254 −3.08275
\(587\) −10.6692 −0.440367 −0.220184 0.975458i \(-0.570666\pi\)
−0.220184 + 0.975458i \(0.570666\pi\)
\(588\) 9.32439 0.384531
\(589\) 0 0
\(590\) 6.78360 0.279276
\(591\) 12.0515 0.495732
\(592\) 103.243 4.24326
\(593\) −17.0027 −0.698216 −0.349108 0.937083i \(-0.613515\pi\)
−0.349108 + 0.937083i \(0.613515\pi\)
\(594\) 13.4301 0.551045
\(595\) −10.6592 −0.436984
\(596\) −83.6040 −3.42455
\(597\) 2.09427 0.0857128
\(598\) 0.740650 0.0302874
\(599\) 28.6751 1.17163 0.585816 0.810444i \(-0.300774\pi\)
0.585816 + 0.810444i \(0.300774\pi\)
\(600\) 14.8404 0.605858
\(601\) 27.4370 1.11918 0.559590 0.828770i \(-0.310959\pi\)
0.559590 + 0.828770i \(0.310959\pi\)
\(602\) 16.1712 0.659088
\(603\) 1.96037 0.0798326
\(604\) 120.313 4.89546
\(605\) −10.2521 −0.416808
\(606\) −21.7915 −0.885221
\(607\) −17.7547 −0.720639 −0.360320 0.932829i \(-0.617332\pi\)
−0.360320 + 0.932829i \(0.617332\pi\)
\(608\) 0 0
\(609\) 41.2466 1.67140
\(610\) 17.4715 0.707399
\(611\) −2.54351 −0.102899
\(612\) −16.2417 −0.656533
\(613\) 34.6391 1.39906 0.699530 0.714603i \(-0.253393\pi\)
0.699530 + 0.714603i \(0.253393\pi\)
\(614\) 24.3760 0.983736
\(615\) −5.98814 −0.241465
\(616\) 24.5050 0.987334
\(617\) 4.46569 0.179782 0.0898909 0.995952i \(-0.471348\pi\)
0.0898909 + 0.995952i \(0.471348\pi\)
\(618\) −1.58472 −0.0637470
\(619\) −17.9112 −0.719913 −0.359957 0.932969i \(-0.617209\pi\)
−0.359957 + 0.932969i \(0.617209\pi\)
\(620\) 27.6604 1.11087
\(621\) −2.35062 −0.0943272
\(622\) −1.79683 −0.0720463
\(623\) −12.6647 −0.507400
\(624\) 15.5679 0.623215
\(625\) 1.00000 0.0400000
\(626\) −8.17906 −0.326901
\(627\) 0 0
\(628\) −13.6308 −0.543927
\(629\) 23.8157 0.949593
\(630\) 6.08241 0.242329
\(631\) 4.96881 0.197805 0.0989026 0.995097i \(-0.468467\pi\)
0.0989026 + 0.995097i \(0.468467\pi\)
\(632\) 18.1489 0.721925
\(633\) 28.2216 1.12171
\(634\) −28.6278 −1.13696
\(635\) 8.83492 0.350603
\(636\) −91.9067 −3.64434
\(637\) 0.717088 0.0284121
\(638\) 23.1483 0.916449
\(639\) 1.37969 0.0545796
\(640\) 49.8151 1.96912
\(641\) −37.9521 −1.49902 −0.749508 0.661995i \(-0.769710\pi\)
−0.749508 + 0.661995i \(0.769710\pi\)
\(642\) −26.4716 −1.04475
\(643\) 35.2502 1.39013 0.695067 0.718945i \(-0.255375\pi\)
0.695067 + 0.718945i \(0.255375\pi\)
\(644\) −6.66740 −0.262732
\(645\) −3.07081 −0.120913
\(646\) 0 0
\(647\) 35.5219 1.39651 0.698254 0.715850i \(-0.253960\pi\)
0.698254 + 0.715850i \(0.253960\pi\)
\(648\) −60.4602 −2.37510
\(649\) −2.12706 −0.0834943
\(650\) 1.77418 0.0695890
\(651\) −20.9670 −0.821762
\(652\) 99.8966 3.91225
\(653\) 8.02411 0.314008 0.157004 0.987598i \(-0.449816\pi\)
0.157004 + 0.987598i \(0.449816\pi\)
\(654\) 27.0290 1.05692
\(655\) 20.9128 0.817129
\(656\) −65.1075 −2.54202
\(657\) 5.52059 0.215379
\(658\) 31.0646 1.21102
\(659\) −47.2195 −1.83941 −0.919706 0.392608i \(-0.871573\pi\)
−0.919706 + 0.392608i \(0.871573\pi\)
\(660\) −7.23377 −0.281574
\(661\) −26.1159 −1.01579 −0.507896 0.861418i \(-0.669577\pi\)
−0.507896 + 0.861418i \(0.669577\pi\)
\(662\) 41.6246 1.61778
\(663\) 3.59114 0.139468
\(664\) −73.9986 −2.87170
\(665\) 0 0
\(666\) −13.5898 −0.526596
\(667\) −4.05155 −0.156877
\(668\) −2.27405 −0.0879856
\(669\) −24.0915 −0.931431
\(670\) −6.98384 −0.269809
\(671\) −5.47833 −0.211489
\(672\) −105.586 −4.07306
\(673\) 15.3820 0.592931 0.296466 0.955044i \(-0.404192\pi\)
0.296466 + 0.955044i \(0.404192\pi\)
\(674\) 43.5696 1.67824
\(675\) −5.63077 −0.216728
\(676\) −70.5664 −2.71409
\(677\) 24.4763 0.940701 0.470350 0.882480i \(-0.344128\pi\)
0.470350 + 0.882480i \(0.344128\pi\)
\(678\) 1.21184 0.0465405
\(679\) 30.9195 1.18658
\(680\) 37.2210 1.42736
\(681\) 39.2652 1.50465
\(682\) −11.7670 −0.450583
\(683\) −17.8502 −0.683018 −0.341509 0.939879i \(-0.610938\pi\)
−0.341509 + 0.939879i \(0.610938\pi\)
\(684\) 0 0
\(685\) 5.21477 0.199246
\(686\) 46.2381 1.76538
\(687\) −19.8908 −0.758880
\(688\) −33.3881 −1.27291
\(689\) −7.06804 −0.269271
\(690\) 1.71774 0.0653932
\(691\) −9.27242 −0.352739 −0.176370 0.984324i \(-0.556435\pi\)
−0.176370 + 0.984324i \(0.556435\pi\)
\(692\) 101.038 3.84087
\(693\) −1.90719 −0.0724483
\(694\) −58.8896 −2.23542
\(695\) −10.7238 −0.406778
\(696\) −144.030 −5.45943
\(697\) −15.0187 −0.568875
\(698\) −89.3314 −3.38124
\(699\) 37.7596 1.42820
\(700\) −15.9713 −0.603659
\(701\) −7.68906 −0.290412 −0.145206 0.989401i \(-0.546384\pi\)
−0.145206 + 0.989401i \(0.546384\pi\)
\(702\) −9.98999 −0.377048
\(703\) 0 0
\(704\) −31.2001 −1.17590
\(705\) −5.89898 −0.222168
\(706\) 2.01496 0.0758340
\(707\) 15.0863 0.567379
\(708\) 20.5737 0.773206
\(709\) 24.4375 0.917769 0.458885 0.888496i \(-0.348249\pi\)
0.458885 + 0.888496i \(0.348249\pi\)
\(710\) −4.91514 −0.184462
\(711\) −1.41251 −0.0529732
\(712\) 44.2241 1.65737
\(713\) 2.05954 0.0771303
\(714\) −43.8597 −1.64141
\(715\) −0.556310 −0.0208048
\(716\) 112.987 4.22252
\(717\) −35.1346 −1.31213
\(718\) −74.0516 −2.76358
\(719\) −22.1126 −0.824662 −0.412331 0.911034i \(-0.635285\pi\)
−0.412331 + 0.911034i \(0.635285\pi\)
\(720\) −12.5582 −0.468015
\(721\) 1.09711 0.0408584
\(722\) 0 0
\(723\) 12.5085 0.465195
\(724\) −95.9345 −3.56538
\(725\) −9.70523 −0.360443
\(726\) −42.1848 −1.56562
\(727\) −29.0494 −1.07738 −0.538692 0.842503i \(-0.681081\pi\)
−0.538692 + 0.842503i \(0.681081\pi\)
\(728\) −18.2280 −0.675575
\(729\) 29.9075 1.10768
\(730\) −19.6671 −0.727913
\(731\) −7.70184 −0.284863
\(732\) 52.9884 1.95851
\(733\) 14.5428 0.537151 0.268576 0.963259i \(-0.413447\pi\)
0.268576 + 0.963259i \(0.413447\pi\)
\(734\) −63.3360 −2.33777
\(735\) 1.66309 0.0613442
\(736\) 10.3714 0.382296
\(737\) 2.18984 0.0806640
\(738\) 8.57009 0.315469
\(739\) −4.75596 −0.174951 −0.0874754 0.996167i \(-0.527880\pi\)
−0.0874754 + 0.996167i \(0.527880\pi\)
\(740\) 35.6845 1.31179
\(741\) 0 0
\(742\) 86.3242 3.16906
\(743\) 5.87705 0.215608 0.107804 0.994172i \(-0.465618\pi\)
0.107804 + 0.994172i \(0.465618\pi\)
\(744\) 73.2151 2.68420
\(745\) −14.9116 −0.546317
\(746\) 81.4455 2.98193
\(747\) 5.75922 0.210719
\(748\) −18.1429 −0.663370
\(749\) 18.3263 0.669629
\(750\) 4.11474 0.150249
\(751\) 1.62096 0.0591498 0.0295749 0.999563i \(-0.490585\pi\)
0.0295749 + 0.999563i \(0.490585\pi\)
\(752\) −64.1382 −2.33888
\(753\) 27.2243 0.992107
\(754\) −17.2188 −0.627072
\(755\) 21.4589 0.780971
\(756\) 89.9308 3.27075
\(757\) −28.1135 −1.02180 −0.510901 0.859639i \(-0.670688\pi\)
−0.510901 + 0.859639i \(0.670688\pi\)
\(758\) 48.2976 1.75425
\(759\) −0.538612 −0.0195504
\(760\) 0 0
\(761\) 20.1663 0.731027 0.365514 0.930806i \(-0.380893\pi\)
0.365514 + 0.930806i \(0.380893\pi\)
\(762\) 36.3534 1.31694
\(763\) −18.7122 −0.677427
\(764\) 29.6076 1.07117
\(765\) −2.89687 −0.104736
\(766\) 22.3579 0.807825
\(767\) 1.58221 0.0571303
\(768\) 97.3257 3.51194
\(769\) 45.3047 1.63373 0.816865 0.576829i \(-0.195710\pi\)
0.816865 + 0.576829i \(0.195710\pi\)
\(770\) 6.79438 0.244853
\(771\) 21.0357 0.757583
\(772\) 100.940 3.63292
\(773\) −20.1762 −0.725686 −0.362843 0.931850i \(-0.618194\pi\)
−0.362843 + 0.931850i \(0.618194\pi\)
\(774\) 4.39487 0.157970
\(775\) 4.93349 0.177216
\(776\) −107.968 −3.87584
\(777\) −27.0494 −0.970393
\(778\) 47.7353 1.71139
\(779\) 0 0
\(780\) 5.38083 0.192665
\(781\) 1.54119 0.0551480
\(782\) 4.30823 0.154062
\(783\) 54.6479 1.95296
\(784\) 18.0824 0.645800
\(785\) −2.43118 −0.0867724
\(786\) 86.0505 3.06932
\(787\) −46.1385 −1.64466 −0.822331 0.569010i \(-0.807327\pi\)
−0.822331 + 0.569010i \(0.807327\pi\)
\(788\) 45.2897 1.61338
\(789\) −9.56426 −0.340497
\(790\) 5.03207 0.179033
\(791\) −0.838961 −0.0298300
\(792\) 6.65976 0.236644
\(793\) 4.07505 0.144709
\(794\) −31.4122 −1.11478
\(795\) −16.3924 −0.581380
\(796\) 7.87031 0.278956
\(797\) −1.86497 −0.0660606 −0.0330303 0.999454i \(-0.510516\pi\)
−0.0330303 + 0.999454i \(0.510516\pi\)
\(798\) 0 0
\(799\) −14.7951 −0.523414
\(800\) 24.8441 0.878371
\(801\) −3.44191 −0.121614
\(802\) −24.6528 −0.870522
\(803\) 6.16681 0.217622
\(804\) −21.1810 −0.746996
\(805\) −1.18919 −0.0419136
\(806\) 8.75290 0.308308
\(807\) 26.8413 0.944859
\(808\) −52.6802 −1.85328
\(809\) 18.2267 0.640816 0.320408 0.947280i \(-0.396180\pi\)
0.320408 + 0.947280i \(0.396180\pi\)
\(810\) −16.7635 −0.589010
\(811\) −20.9779 −0.736634 −0.368317 0.929700i \(-0.620066\pi\)
−0.368317 + 0.929700i \(0.620066\pi\)
\(812\) 155.005 5.43962
\(813\) 17.7268 0.621707
\(814\) −15.1806 −0.532079
\(815\) 17.8175 0.624120
\(816\) 90.5558 3.17009
\(817\) 0 0
\(818\) −18.0504 −0.631118
\(819\) 1.41866 0.0495721
\(820\) −22.5035 −0.785857
\(821\) −22.0867 −0.770829 −0.385415 0.922743i \(-0.625942\pi\)
−0.385415 + 0.922743i \(0.625942\pi\)
\(822\) 21.4574 0.748413
\(823\) 15.9769 0.556921 0.278461 0.960448i \(-0.410176\pi\)
0.278461 + 0.960448i \(0.410176\pi\)
\(824\) −3.83101 −0.133460
\(825\) −1.29021 −0.0449194
\(826\) −19.3240 −0.672368
\(827\) 49.2009 1.71088 0.855441 0.517901i \(-0.173286\pi\)
0.855441 + 0.517901i \(0.173286\pi\)
\(828\) −1.81201 −0.0629717
\(829\) −35.8564 −1.24534 −0.622672 0.782483i \(-0.713953\pi\)
−0.622672 + 0.782483i \(0.713953\pi\)
\(830\) −20.5172 −0.712164
\(831\) 35.2451 1.22264
\(832\) 23.2082 0.804599
\(833\) 4.17118 0.144523
\(834\) −44.1257 −1.52795
\(835\) −0.405598 −0.0140363
\(836\) 0 0
\(837\) −27.7794 −0.960195
\(838\) 60.2463 2.08117
\(839\) −25.4830 −0.879770 −0.439885 0.898054i \(-0.644981\pi\)
−0.439885 + 0.898054i \(0.644981\pi\)
\(840\) −42.2750 −1.45863
\(841\) 65.1914 2.24798
\(842\) −80.9294 −2.78901
\(843\) 20.6024 0.709583
\(844\) 106.057 3.65065
\(845\) −12.5862 −0.432978
\(846\) 8.44249 0.290259
\(847\) 29.2046 1.00348
\(848\) −178.231 −6.12047
\(849\) −17.5410 −0.602007
\(850\) 10.3201 0.353976
\(851\) 2.65700 0.0910807
\(852\) −14.9069 −0.510703
\(853\) −57.1622 −1.95720 −0.978598 0.205782i \(-0.934026\pi\)
−0.978598 + 0.205782i \(0.934026\pi\)
\(854\) −49.7698 −1.70309
\(855\) 0 0
\(856\) −63.9940 −2.18727
\(857\) 26.1974 0.894885 0.447442 0.894313i \(-0.352335\pi\)
0.447442 + 0.894313i \(0.352335\pi\)
\(858\) −2.28907 −0.0781475
\(859\) −17.7449 −0.605449 −0.302724 0.953078i \(-0.597896\pi\)
−0.302724 + 0.953078i \(0.597896\pi\)
\(860\) −11.5401 −0.393516
\(861\) 17.0580 0.581336
\(862\) −35.5418 −1.21056
\(863\) 25.0867 0.853960 0.426980 0.904261i \(-0.359578\pi\)
0.426980 + 0.904261i \(0.359578\pi\)
\(864\) −139.891 −4.75920
\(865\) 18.0210 0.612733
\(866\) −38.1789 −1.29737
\(867\) −4.47357 −0.151930
\(868\) −78.7944 −2.67446
\(869\) −1.57785 −0.0535249
\(870\) −39.9344 −1.35390
\(871\) −1.62891 −0.0551936
\(872\) 65.3415 2.21274
\(873\) 8.40305 0.284400
\(874\) 0 0
\(875\) −2.84864 −0.0963015
\(876\) −59.6476 −2.01531
\(877\) 10.0157 0.338205 0.169103 0.985598i \(-0.445913\pi\)
0.169103 + 0.985598i \(0.445913\pi\)
\(878\) −0.195290 −0.00659071
\(879\) −40.3678 −1.36157
\(880\) −14.0282 −0.472889
\(881\) 33.3473 1.12350 0.561750 0.827307i \(-0.310128\pi\)
0.561750 + 0.827307i \(0.310128\pi\)
\(882\) −2.38018 −0.0801449
\(883\) −27.3570 −0.920638 −0.460319 0.887754i \(-0.652265\pi\)
−0.460319 + 0.887754i \(0.652265\pi\)
\(884\) 13.4956 0.453905
\(885\) 3.66951 0.123349
\(886\) −10.4505 −0.351092
\(887\) 17.1634 0.576292 0.288146 0.957586i \(-0.406961\pi\)
0.288146 + 0.957586i \(0.406961\pi\)
\(888\) 94.4544 3.16968
\(889\) −25.1675 −0.844090
\(890\) 12.2618 0.411016
\(891\) 5.25635 0.176094
\(892\) −90.5363 −3.03138
\(893\) 0 0
\(894\) −61.3571 −2.05209
\(895\) 20.1523 0.673617
\(896\) −141.905 −4.74072
\(897\) 0.400646 0.0133772
\(898\) −73.2985 −2.44600
\(899\) −47.8807 −1.59691
\(900\) −4.34056 −0.144685
\(901\) −41.1136 −1.36969
\(902\) 9.57325 0.318754
\(903\) 8.74762 0.291103
\(904\) 2.92958 0.0974364
\(905\) −17.1108 −0.568783
\(906\) 88.2979 2.93350
\(907\) 3.02106 0.100313 0.0501563 0.998741i \(-0.484028\pi\)
0.0501563 + 0.998741i \(0.484028\pi\)
\(908\) 147.559 4.89693
\(909\) 4.10004 0.135990
\(910\) −5.05399 −0.167538
\(911\) −19.5682 −0.648324 −0.324162 0.946002i \(-0.605082\pi\)
−0.324162 + 0.946002i \(0.605082\pi\)
\(912\) 0 0
\(913\) 6.43336 0.212913
\(914\) −91.4346 −3.02439
\(915\) 9.45099 0.312440
\(916\) −74.7498 −2.46980
\(917\) −59.5729 −1.96727
\(918\) −58.1100 −1.91792
\(919\) −1.81420 −0.0598448 −0.0299224 0.999552i \(-0.509526\pi\)
−0.0299224 + 0.999552i \(0.509526\pi\)
\(920\) 4.15257 0.136906
\(921\) 13.1859 0.434491
\(922\) −53.1017 −1.74881
\(923\) −1.14641 −0.0377346
\(924\) 20.6064 0.677901
\(925\) 6.36467 0.209269
\(926\) 108.035 3.55025
\(927\) 0.298163 0.00979295
\(928\) −241.117 −7.91507
\(929\) −22.4754 −0.737395 −0.368698 0.929549i \(-0.620196\pi\)
−0.368698 + 0.929549i \(0.620196\pi\)
\(930\) 20.3000 0.665664
\(931\) 0 0
\(932\) 141.901 4.64813
\(933\) −0.971975 −0.0318210
\(934\) −107.742 −3.52542
\(935\) −3.23596 −0.105827
\(936\) −4.95386 −0.161922
\(937\) 55.0385 1.79803 0.899015 0.437918i \(-0.144284\pi\)
0.899015 + 0.437918i \(0.144284\pi\)
\(938\) 19.8944 0.649576
\(939\) −4.42437 −0.144384
\(940\) −22.1685 −0.723056
\(941\) −12.1956 −0.397566 −0.198783 0.980044i \(-0.563699\pi\)
−0.198783 + 0.980044i \(0.563699\pi\)
\(942\) −10.0036 −0.325937
\(943\) −1.67557 −0.0545640
\(944\) 39.8977 1.29856
\(945\) 16.0400 0.521782
\(946\) 4.90931 0.159615
\(947\) 30.4307 0.988863 0.494432 0.869216i \(-0.335376\pi\)
0.494432 + 0.869216i \(0.335376\pi\)
\(948\) 15.2615 0.495672
\(949\) −4.58717 −0.148906
\(950\) 0 0
\(951\) −15.4859 −0.502164
\(952\) −106.029 −3.43642
\(953\) 15.7378 0.509798 0.254899 0.966968i \(-0.417958\pi\)
0.254899 + 0.966968i \(0.417958\pi\)
\(954\) 23.4605 0.759561
\(955\) 5.28080 0.170883
\(956\) −132.036 −4.27036
\(957\) 12.5218 0.404772
\(958\) −68.2748 −2.20586
\(959\) −14.8550 −0.479693
\(960\) 53.8252 1.73720
\(961\) −6.66065 −0.214860
\(962\) 11.2921 0.364071
\(963\) 4.98058 0.160497
\(964\) 47.0070 1.51399
\(965\) 18.0036 0.579558
\(966\) −4.89322 −0.157437
\(967\) 30.6373 0.985228 0.492614 0.870248i \(-0.336042\pi\)
0.492614 + 0.870248i \(0.336042\pi\)
\(968\) −101.980 −3.27776
\(969\) 0 0
\(970\) −29.9359 −0.961184
\(971\) 16.4765 0.528755 0.264378 0.964419i \(-0.414833\pi\)
0.264378 + 0.964419i \(0.414833\pi\)
\(972\) 43.8679 1.40706
\(973\) 30.5483 0.979334
\(974\) 60.3649 1.93422
\(975\) 0.959723 0.0307357
\(976\) 102.758 3.28921
\(977\) −2.77995 −0.0889383 −0.0444692 0.999011i \(-0.514160\pi\)
−0.0444692 + 0.999011i \(0.514160\pi\)
\(978\) 73.3144 2.34433
\(979\) −3.84480 −0.122880
\(980\) 6.24993 0.199647
\(981\) −5.08545 −0.162366
\(982\) 25.9130 0.826918
\(983\) −1.71171 −0.0545951 −0.0272976 0.999627i \(-0.508690\pi\)
−0.0272976 + 0.999627i \(0.508690\pi\)
\(984\) −59.5653 −1.89887
\(985\) 8.07785 0.257382
\(986\) −100.159 −3.18971
\(987\) 16.8041 0.534879
\(988\) 0 0
\(989\) −0.859257 −0.0273228
\(990\) 1.84652 0.0586863
\(991\) −9.67421 −0.307311 −0.153656 0.988124i \(-0.549105\pi\)
−0.153656 + 0.988124i \(0.549105\pi\)
\(992\) 122.568 3.89154
\(993\) 22.5164 0.714535
\(994\) 14.0015 0.444099
\(995\) 1.40374 0.0445017
\(996\) −62.2259 −1.97170
\(997\) 22.7311 0.719902 0.359951 0.932971i \(-0.382793\pi\)
0.359951 + 0.932971i \(0.382793\pi\)
\(998\) 68.7064 2.17486
\(999\) −35.8380 −1.13386
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 1805.2.a.o.1.4 4
5.4 even 2 9025.2.a.bg.1.1 4
19.7 even 3 95.2.e.c.11.1 8
19.11 even 3 95.2.e.c.26.1 yes 8
19.18 odd 2 1805.2.a.i.1.1 4
57.11 odd 6 855.2.k.h.406.4 8
57.26 odd 6 855.2.k.h.676.4 8
76.7 odd 6 1520.2.q.o.961.3 8
76.11 odd 6 1520.2.q.o.881.3 8
95.7 odd 12 475.2.j.c.49.1 16
95.49 even 6 475.2.e.e.26.4 8
95.64 even 6 475.2.e.e.201.4 8
95.68 odd 12 475.2.j.c.349.1 16
95.83 odd 12 475.2.j.c.49.8 16
95.87 odd 12 475.2.j.c.349.8 16
95.94 odd 2 9025.2.a.bp.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.e.c.11.1 8 19.7 even 3
95.2.e.c.26.1 yes 8 19.11 even 3
475.2.e.e.26.4 8 95.49 even 6
475.2.e.e.201.4 8 95.64 even 6
475.2.j.c.49.1 16 95.7 odd 12
475.2.j.c.49.8 16 95.83 odd 12
475.2.j.c.349.1 16 95.68 odd 12
475.2.j.c.349.8 16 95.87 odd 12
855.2.k.h.406.4 8 57.11 odd 6
855.2.k.h.676.4 8 57.26 odd 6
1520.2.q.o.881.3 8 76.11 odd 6
1520.2.q.o.961.3 8 76.7 odd 6
1805.2.a.i.1.1 4 19.18 odd 2
1805.2.a.o.1.4 4 1.1 even 1 trivial
9025.2.a.bg.1.1 4 5.4 even 2
9025.2.a.bp.1.4 4 95.94 odd 2