Properties

Label 1805.2.a.i.1.4
Level $1805$
Weight $2$
Character 1805.1
Self dual yes
Analytic conductor $14.413$
Analytic rank $1$
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: \(1\)
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 \(2.15976\) 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)
 
\(f(q)\) \(=\) \(q+1.66454 q^{2} +1.15976 q^{3} +0.770710 q^{4} +1.00000 q^{5} +1.93047 q^{6} -2.43525 q^{7} -2.04621 q^{8} -1.65497 q^{9} +O(q^{10})\) \(q+1.66454 q^{2} +1.15976 q^{3} +0.770710 q^{4} +1.00000 q^{5} +1.93047 q^{6} -2.43525 q^{7} -2.04621 q^{8} -1.65497 q^{9} +1.66454 q^{10} -5.75477 q^{11} +0.893835 q^{12} +1.59501 q^{13} -4.05359 q^{14} +1.15976 q^{15} -4.94743 q^{16} -5.98406 q^{17} -2.75477 q^{18} +0.770710 q^{20} -2.82430 q^{21} -9.57907 q^{22} -0.940044 q^{23} -2.37310 q^{24} +1.00000 q^{25} +2.65497 q^{26} -5.39862 q^{27} -1.87687 q^{28} -2.61834 q^{29} +1.93047 q^{30} +5.26913 q^{31} -4.14280 q^{32} -6.67412 q^{33} -9.96073 q^{34} -2.43525 q^{35} -1.27550 q^{36} +2.89384 q^{37} +1.84982 q^{39} -2.04621 q^{40} +6.31534 q^{41} -4.70118 q^{42} +4.53922 q^{43} -4.43525 q^{44} -1.65497 q^{45} -1.56475 q^{46} +8.95437 q^{47} -5.73781 q^{48} -1.06953 q^{49} +1.66454 q^{50} -6.94004 q^{51} +1.22929 q^{52} +2.19639 q^{53} -8.98625 q^{54} -5.75477 q^{55} +4.98304 q^{56} -4.35834 q^{58} +10.7988 q^{59} +0.893835 q^{60} -10.5287 q^{61} +8.77071 q^{62} +4.03027 q^{63} +2.99898 q^{64} +1.59501 q^{65} -11.1094 q^{66} -1.00958 q^{67} -4.61197 q^{68} -1.09022 q^{69} -4.05359 q^{70} -8.83388 q^{71} +3.38641 q^{72} -10.2500 q^{73} +4.81692 q^{74} +1.15976 q^{75} +14.0143 q^{77} +3.07911 q^{78} -7.60459 q^{79} -4.94743 q^{80} -1.29619 q^{81} +10.5122 q^{82} +3.11355 q^{83} -2.17672 q^{84} -5.98406 q^{85} +7.55574 q^{86} -3.03663 q^{87} +11.7755 q^{88} +11.1141 q^{89} -2.75477 q^{90} -3.88426 q^{91} -0.724501 q^{92} +6.11091 q^{93} +14.9049 q^{94} -4.80463 q^{96} -4.05776 q^{97} -1.78029 q^{98} +9.52395 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} - 3 q^{3} + 5 q^{4} + 4 q^{5} + 2 q^{6} - 4 q^{7} - 12 q^{8} + q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.66454 1.17701 0.588506 0.808493i \(-0.299717\pi\)
0.588506 + 0.808493i \(0.299717\pi\)
\(3\) 1.15976 0.669585 0.334793 0.942292i \(-0.391334\pi\)
0.334793 + 0.942292i \(0.391334\pi\)
\(4\) 0.770710 0.385355
\(5\) 1.00000 0.447214
\(6\) 1.93047 0.788109
\(7\) −2.43525 −0.920440 −0.460220 0.887805i \(-0.652229\pi\)
−0.460220 + 0.887805i \(0.652229\pi\)
\(8\) −2.04621 −0.723444
\(9\) −1.65497 −0.551656
\(10\) 1.66454 0.526375
\(11\) −5.75477 −1.73513 −0.867564 0.497326i \(-0.834315\pi\)
−0.867564 + 0.497326i \(0.834315\pi\)
\(12\) 0.893835 0.258028
\(13\) 1.59501 0.442376 0.221188 0.975231i \(-0.429007\pi\)
0.221188 + 0.975231i \(0.429007\pi\)
\(14\) −4.05359 −1.08337
\(15\) 1.15976 0.299448
\(16\) −4.94743 −1.23686
\(17\) −5.98406 −1.45135 −0.725673 0.688039i \(-0.758472\pi\)
−0.725673 + 0.688039i \(0.758472\pi\)
\(18\) −2.75477 −0.649305
\(19\) 0 0
\(20\) 0.770710 0.172336
\(21\) −2.82430 −0.616313
\(22\) −9.57907 −2.04226
\(23\) −0.940044 −0.196013 −0.0980064 0.995186i \(-0.531247\pi\)
−0.0980064 + 0.995186i \(0.531247\pi\)
\(24\) −2.37310 −0.484407
\(25\) 1.00000 0.200000
\(26\) 2.65497 0.520682
\(27\) −5.39862 −1.03897
\(28\) −1.87687 −0.354696
\(29\) −2.61834 −0.486213 −0.243106 0.970000i \(-0.578166\pi\)
−0.243106 + 0.970000i \(0.578166\pi\)
\(30\) 1.93047 0.352453
\(31\) 5.26913 0.946364 0.473182 0.880965i \(-0.343105\pi\)
0.473182 + 0.880965i \(0.343105\pi\)
\(32\) −4.14280 −0.732350
\(33\) −6.67412 −1.16182
\(34\) −9.96073 −1.70825
\(35\) −2.43525 −0.411633
\(36\) −1.27550 −0.212583
\(37\) 2.89384 0.475744 0.237872 0.971297i \(-0.423550\pi\)
0.237872 + 0.971297i \(0.423550\pi\)
\(38\) 0 0
\(39\) 1.84982 0.296209
\(40\) −2.04621 −0.323534
\(41\) 6.31534 0.986291 0.493145 0.869947i \(-0.335847\pi\)
0.493145 + 0.869947i \(0.335847\pi\)
\(42\) −4.70118 −0.725407
\(43\) 4.53922 0.692225 0.346113 0.938193i \(-0.387502\pi\)
0.346113 + 0.938193i \(0.387502\pi\)
\(44\) −4.43525 −0.668640
\(45\) −1.65497 −0.246708
\(46\) −1.56475 −0.230709
\(47\) 8.95437 1.30613 0.653064 0.757303i \(-0.273483\pi\)
0.653064 + 0.757303i \(0.273483\pi\)
\(48\) −5.73781 −0.828181
\(49\) −1.06953 −0.152791
\(50\) 1.66454 0.235402
\(51\) −6.94004 −0.971801
\(52\) 1.22929 0.170472
\(53\) 2.19639 0.301697 0.150848 0.988557i \(-0.451799\pi\)
0.150848 + 0.988557i \(0.451799\pi\)
\(54\) −8.98625 −1.22287
\(55\) −5.75477 −0.775973
\(56\) 4.98304 0.665887
\(57\) 0 0
\(58\) −4.35834 −0.572278
\(59\) 10.7988 1.40588 0.702941 0.711249i \(-0.251870\pi\)
0.702941 + 0.711249i \(0.251870\pi\)
\(60\) 0.893835 0.115394
\(61\) −10.5287 −1.34806 −0.674030 0.738704i \(-0.735438\pi\)
−0.674030 + 0.738704i \(0.735438\pi\)
\(62\) 8.77071 1.11388
\(63\) 4.03027 0.507766
\(64\) 2.99898 0.374873
\(65\) 1.59501 0.197837
\(66\) −11.1094 −1.36747
\(67\) −1.00958 −0.123340 −0.0616698 0.998097i \(-0.519643\pi\)
−0.0616698 + 0.998097i \(0.519643\pi\)
\(68\) −4.61197 −0.559284
\(69\) −1.09022 −0.131247
\(70\) −4.05359 −0.484497
\(71\) −8.83388 −1.04839 −0.524194 0.851599i \(-0.675633\pi\)
−0.524194 + 0.851599i \(0.675633\pi\)
\(72\) 3.38641 0.399092
\(73\) −10.2500 −1.19967 −0.599835 0.800124i \(-0.704767\pi\)
−0.599835 + 0.800124i \(0.704767\pi\)
\(74\) 4.81692 0.559955
\(75\) 1.15976 0.133917
\(76\) 0 0
\(77\) 14.0143 1.59708
\(78\) 3.07911 0.348641
\(79\) −7.60459 −0.855583 −0.427792 0.903877i \(-0.640708\pi\)
−0.427792 + 0.903877i \(0.640708\pi\)
\(80\) −4.94743 −0.553139
\(81\) −1.29619 −0.144021
\(82\) 10.5122 1.16088
\(83\) 3.11355 0.341756 0.170878 0.985292i \(-0.445340\pi\)
0.170878 + 0.985292i \(0.445340\pi\)
\(84\) −2.17672 −0.237499
\(85\) −5.98406 −0.649062
\(86\) 7.55574 0.814757
\(87\) −3.03663 −0.325561
\(88\) 11.7755 1.25527
\(89\) 11.1141 1.17809 0.589047 0.808099i \(-0.299503\pi\)
0.589047 + 0.808099i \(0.299503\pi\)
\(90\) −2.75477 −0.290378
\(91\) −3.88426 −0.407181
\(92\) −0.724501 −0.0755345
\(93\) 6.11091 0.633672
\(94\) 14.9049 1.53733
\(95\) 0 0
\(96\) −4.80463 −0.490371
\(97\) −4.05776 −0.412003 −0.206002 0.978552i \(-0.566045\pi\)
−0.206002 + 0.978552i \(0.566045\pi\)
\(98\) −1.78029 −0.179836
\(99\) 9.52395 0.957193
\(100\) 0.770710 0.0770710
\(101\) −11.1301 −1.10748 −0.553741 0.832689i \(-0.686800\pi\)
−0.553741 + 0.832689i \(0.686800\pi\)
\(102\) −11.5520 −1.14382
\(103\) −11.5791 −1.14092 −0.570460 0.821326i \(-0.693235\pi\)
−0.570460 + 0.821326i \(0.693235\pi\)
\(104\) −3.26372 −0.320035
\(105\) −2.82430 −0.275624
\(106\) 3.65598 0.355101
\(107\) −17.9177 −1.73217 −0.866086 0.499894i \(-0.833372\pi\)
−0.866086 + 0.499894i \(0.833372\pi\)
\(108\) −4.16077 −0.400371
\(109\) −5.62470 −0.538749 −0.269374 0.963036i \(-0.586817\pi\)
−0.269374 + 0.963036i \(0.586817\pi\)
\(110\) −9.57907 −0.913328
\(111\) 3.35614 0.318551
\(112\) 12.0482 1.13845
\(113\) 15.6789 1.47494 0.737472 0.675378i \(-0.236019\pi\)
0.737472 + 0.675378i \(0.236019\pi\)
\(114\) 0 0
\(115\) −0.940044 −0.0876595
\(116\) −2.01798 −0.187364
\(117\) −2.63969 −0.244039
\(118\) 17.9751 1.65474
\(119\) 14.5727 1.33588
\(120\) −2.37310 −0.216634
\(121\) 22.1173 2.01067
\(122\) −17.5255 −1.58668
\(123\) 7.32425 0.660406
\(124\) 4.06097 0.364686
\(125\) 1.00000 0.0894427
\(126\) 6.70856 0.597646
\(127\) −6.11991 −0.543054 −0.271527 0.962431i \(-0.587529\pi\)
−0.271527 + 0.962431i \(0.587529\pi\)
\(128\) 13.2775 1.17358
\(129\) 5.26439 0.463504
\(130\) 2.65497 0.232856
\(131\) −14.8811 −1.30017 −0.650084 0.759862i \(-0.725266\pi\)
−0.650084 + 0.759862i \(0.725266\pi\)
\(132\) −5.14381 −0.447711
\(133\) 0 0
\(134\) −1.68049 −0.145172
\(135\) −5.39862 −0.464640
\(136\) 12.2446 1.04997
\(137\) 17.3504 1.48234 0.741170 0.671317i \(-0.234271\pi\)
0.741170 + 0.671317i \(0.234271\pi\)
\(138\) −1.81472 −0.154479
\(139\) −6.70534 −0.568740 −0.284370 0.958715i \(-0.591784\pi\)
−0.284370 + 0.958715i \(0.591784\pi\)
\(140\) −1.87687 −0.158625
\(141\) 10.3849 0.874564
\(142\) −14.7044 −1.23396
\(143\) −9.17891 −0.767579
\(144\) 8.18782 0.682319
\(145\) −2.61834 −0.217441
\(146\) −17.0615 −1.41202
\(147\) −1.24040 −0.102306
\(148\) 2.23031 0.183330
\(149\) 14.3928 1.17911 0.589553 0.807729i \(-0.299304\pi\)
0.589553 + 0.807729i \(0.299304\pi\)
\(150\) 1.93047 0.157622
\(151\) −12.7219 −1.03529 −0.517645 0.855595i \(-0.673191\pi\)
−0.517645 + 0.855595i \(0.673191\pi\)
\(152\) 0 0
\(153\) 9.90341 0.800644
\(154\) 23.3275 1.87978
\(155\) 5.26913 0.423227
\(156\) 1.42568 0.114145
\(157\) −3.37530 −0.269378 −0.134689 0.990888i \(-0.543004\pi\)
−0.134689 + 0.990888i \(0.543004\pi\)
\(158\) −12.6582 −1.00703
\(159\) 2.54727 0.202012
\(160\) −4.14280 −0.327517
\(161\) 2.28925 0.180418
\(162\) −2.15756 −0.169514
\(163\) 0.307960 0.0241213 0.0120607 0.999927i \(-0.496161\pi\)
0.0120607 + 0.999927i \(0.496161\pi\)
\(164\) 4.86730 0.380072
\(165\) −6.67412 −0.519580
\(166\) 5.18264 0.402251
\(167\) −14.2643 −1.10380 −0.551902 0.833909i \(-0.686098\pi\)
−0.551902 + 0.833909i \(0.686098\pi\)
\(168\) 5.77911 0.445868
\(169\) −10.4559 −0.804303
\(170\) −9.96073 −0.763953
\(171\) 0 0
\(172\) 3.49842 0.266752
\(173\) −13.3471 −1.01476 −0.507382 0.861721i \(-0.669387\pi\)
−0.507382 + 0.861721i \(0.669387\pi\)
\(174\) −5.05461 −0.383189
\(175\) −2.43525 −0.184088
\(176\) 28.4713 2.14610
\(177\) 12.5239 0.941357
\(178\) 18.5000 1.38663
\(179\) 14.2207 1.06291 0.531454 0.847087i \(-0.321646\pi\)
0.531454 + 0.847087i \(0.321646\pi\)
\(180\) −1.27550 −0.0950701
\(181\) −9.88263 −0.734570 −0.367285 0.930108i \(-0.619713\pi\)
−0.367285 + 0.930108i \(0.619713\pi\)
\(182\) −6.46552 −0.479256
\(183\) −12.2107 −0.902641
\(184\) 1.92353 0.141804
\(185\) 2.89384 0.212759
\(186\) 10.1719 0.745839
\(187\) 34.4368 2.51827
\(188\) 6.90122 0.503323
\(189\) 13.1470 0.956305
\(190\) 0 0
\(191\) −12.9942 −0.940228 −0.470114 0.882606i \(-0.655787\pi\)
−0.470114 + 0.882606i \(0.655787\pi\)
\(192\) 3.47809 0.251009
\(193\) −14.5159 −1.04488 −0.522439 0.852677i \(-0.674978\pi\)
−0.522439 + 0.852677i \(0.674978\pi\)
\(194\) −6.75432 −0.484932
\(195\) 1.84982 0.132469
\(196\) −0.824301 −0.0588786
\(197\) −25.0010 −1.78125 −0.890624 0.454740i \(-0.849732\pi\)
−0.890624 + 0.454740i \(0.849732\pi\)
\(198\) 15.8530 1.12663
\(199\) −2.25539 −0.159880 −0.0799401 0.996800i \(-0.525473\pi\)
−0.0799401 + 0.996800i \(0.525473\pi\)
\(200\) −2.04621 −0.144689
\(201\) −1.17086 −0.0825864
\(202\) −18.5265 −1.30352
\(203\) 6.37632 0.447530
\(204\) −5.34876 −0.374488
\(205\) 6.31534 0.441083
\(206\) −19.2739 −1.34287
\(207\) 1.55574 0.108131
\(208\) −7.89120 −0.547156
\(209\) 0 0
\(210\) −4.70118 −0.324412
\(211\) −22.2161 −1.52942 −0.764710 0.644374i \(-0.777118\pi\)
−0.764710 + 0.644374i \(0.777118\pi\)
\(212\) 1.69278 0.116260
\(213\) −10.2451 −0.701986
\(214\) −29.8249 −2.03879
\(215\) 4.53922 0.309572
\(216\) 11.0467 0.751634
\(217\) −12.8317 −0.871071
\(218\) −9.36257 −0.634113
\(219\) −11.8875 −0.803281
\(220\) −4.43525 −0.299025
\(221\) −9.54463 −0.642041
\(222\) 5.58645 0.374938
\(223\) −10.2160 −0.684113 −0.342056 0.939679i \(-0.611123\pi\)
−0.342056 + 0.939679i \(0.611123\pi\)
\(224\) 10.0888 0.674084
\(225\) −1.65497 −0.110331
\(226\) 26.0982 1.73602
\(227\) −4.15180 −0.275565 −0.137782 0.990463i \(-0.543997\pi\)
−0.137782 + 0.990463i \(0.543997\pi\)
\(228\) 0 0
\(229\) 6.53286 0.431703 0.215852 0.976426i \(-0.430747\pi\)
0.215852 + 0.976426i \(0.430747\pi\)
\(230\) −1.56475 −0.103176
\(231\) 16.2532 1.06938
\(232\) 5.35766 0.351748
\(233\) 5.14820 0.337270 0.168635 0.985679i \(-0.446064\pi\)
0.168635 + 0.985679i \(0.446064\pi\)
\(234\) −4.39388 −0.287237
\(235\) 8.95437 0.584118
\(236\) 8.32272 0.541763
\(237\) −8.81947 −0.572886
\(238\) 24.2569 1.57234
\(239\) 13.9962 0.905338 0.452669 0.891679i \(-0.350472\pi\)
0.452669 + 0.891679i \(0.350472\pi\)
\(240\) −5.73781 −0.370374
\(241\) −15.2257 −0.980773 −0.490387 0.871505i \(-0.663144\pi\)
−0.490387 + 0.871505i \(0.663144\pi\)
\(242\) 36.8153 2.36658
\(243\) 14.6926 0.942532
\(244\) −8.11456 −0.519482
\(245\) −1.06953 −0.0683300
\(246\) 12.1916 0.777305
\(247\) 0 0
\(248\) −10.7817 −0.684642
\(249\) 3.61095 0.228835
\(250\) 1.66454 0.105275
\(251\) −6.11259 −0.385823 −0.192912 0.981216i \(-0.561793\pi\)
−0.192912 + 0.981216i \(0.561793\pi\)
\(252\) 3.10616 0.195670
\(253\) 5.40973 0.340107
\(254\) −10.1869 −0.639181
\(255\) −6.94004 −0.434602
\(256\) 16.1031 1.00644
\(257\) −0.122683 −0.00765274 −0.00382637 0.999993i \(-0.501218\pi\)
−0.00382637 + 0.999993i \(0.501218\pi\)
\(258\) 8.76281 0.545549
\(259\) −7.04723 −0.437893
\(260\) 1.22929 0.0762373
\(261\) 4.33326 0.268222
\(262\) −24.7703 −1.53031
\(263\) −10.0605 −0.620359 −0.310179 0.950678i \(-0.600389\pi\)
−0.310179 + 0.950678i \(0.600389\pi\)
\(264\) 13.6566 0.840509
\(265\) 2.19639 0.134923
\(266\) 0 0
\(267\) 12.8897 0.788835
\(268\) −0.778092 −0.0475295
\(269\) −5.71228 −0.348284 −0.174142 0.984721i \(-0.555715\pi\)
−0.174142 + 0.984721i \(0.555715\pi\)
\(270\) −8.98625 −0.546886
\(271\) −12.7112 −0.772150 −0.386075 0.922467i \(-0.626169\pi\)
−0.386075 + 0.922467i \(0.626169\pi\)
\(272\) 29.6057 1.79511
\(273\) −4.50479 −0.272642
\(274\) 28.8804 1.74473
\(275\) −5.75477 −0.347025
\(276\) −0.840244 −0.0505768
\(277\) 17.6019 1.05760 0.528799 0.848747i \(-0.322642\pi\)
0.528799 + 0.848747i \(0.322642\pi\)
\(278\) −11.1613 −0.669413
\(279\) −8.72024 −0.522067
\(280\) 4.98304 0.297794
\(281\) 20.5004 1.22295 0.611476 0.791263i \(-0.290576\pi\)
0.611476 + 0.791263i \(0.290576\pi\)
\(282\) 17.2861 1.02937
\(283\) −11.8561 −0.704772 −0.352386 0.935855i \(-0.614630\pi\)
−0.352386 + 0.935855i \(0.614630\pi\)
\(284\) −6.80836 −0.404002
\(285\) 0 0
\(286\) −15.2787 −0.903449
\(287\) −15.3795 −0.907821
\(288\) 6.85619 0.404005
\(289\) 18.8089 1.10641
\(290\) −4.35834 −0.255930
\(291\) −4.70601 −0.275871
\(292\) −7.89976 −0.462298
\(293\) 24.9814 1.45943 0.729715 0.683751i \(-0.239653\pi\)
0.729715 + 0.683751i \(0.239653\pi\)
\(294\) −2.06470 −0.120416
\(295\) 10.7988 0.628729
\(296\) −5.92139 −0.344174
\(297\) 31.0678 1.80274
\(298\) 23.9575 1.38782
\(299\) −1.49938 −0.0867114
\(300\) 0.893835 0.0516056
\(301\) −11.0542 −0.637151
\(302\) −21.1761 −1.21855
\(303\) −12.9082 −0.741554
\(304\) 0 0
\(305\) −10.5287 −0.602871
\(306\) 16.4847 0.942366
\(307\) −16.9199 −0.965671 −0.482836 0.875711i \(-0.660393\pi\)
−0.482836 + 0.875711i \(0.660393\pi\)
\(308\) 10.8010 0.615443
\(309\) −13.4289 −0.763943
\(310\) 8.77071 0.498143
\(311\) −15.2133 −0.862670 −0.431335 0.902192i \(-0.641957\pi\)
−0.431335 + 0.902192i \(0.641957\pi\)
\(312\) −3.78512 −0.214290
\(313\) 24.9273 1.40898 0.704488 0.709716i \(-0.251177\pi\)
0.704488 + 0.709716i \(0.251177\pi\)
\(314\) −5.61834 −0.317061
\(315\) 4.03027 0.227080
\(316\) −5.86093 −0.329703
\(317\) 25.2304 1.41708 0.708541 0.705669i \(-0.249354\pi\)
0.708541 + 0.705669i \(0.249354\pi\)
\(318\) 4.24005 0.237770
\(319\) 15.0679 0.843641
\(320\) 2.99898 0.167648
\(321\) −20.7802 −1.15984
\(322\) 3.81055 0.212354
\(323\) 0 0
\(324\) −0.998983 −0.0554991
\(325\) 1.59501 0.0884753
\(326\) 0.512614 0.0283911
\(327\) −6.52328 −0.360738
\(328\) −12.9225 −0.713526
\(329\) −21.8062 −1.20221
\(330\) −11.1094 −0.611551
\(331\) 20.2063 1.11064 0.555320 0.831637i \(-0.312596\pi\)
0.555320 + 0.831637i \(0.312596\pi\)
\(332\) 2.39964 0.131697
\(333\) −4.78920 −0.262447
\(334\) −23.7436 −1.29919
\(335\) −1.00958 −0.0551592
\(336\) 13.9730 0.762291
\(337\) 31.8247 1.73360 0.866800 0.498657i \(-0.166173\pi\)
0.866800 + 0.498657i \(0.166173\pi\)
\(338\) −17.4044 −0.946674
\(339\) 18.1837 0.987601
\(340\) −4.61197 −0.250119
\(341\) −30.3226 −1.64206
\(342\) 0 0
\(343\) 19.6514 1.06107
\(344\) −9.28820 −0.500786
\(345\) −1.09022 −0.0586955
\(346\) −22.2169 −1.19439
\(347\) −3.30255 −0.177290 −0.0886451 0.996063i \(-0.528254\pi\)
−0.0886451 + 0.996063i \(0.528254\pi\)
\(348\) −2.34036 −0.125457
\(349\) 17.8486 0.955416 0.477708 0.878519i \(-0.341468\pi\)
0.477708 + 0.878519i \(0.341468\pi\)
\(350\) −4.05359 −0.216674
\(351\) −8.61086 −0.459614
\(352\) 23.8408 1.27072
\(353\) −8.29523 −0.441511 −0.220755 0.975329i \(-0.570852\pi\)
−0.220755 + 0.975329i \(0.570852\pi\)
\(354\) 20.8467 1.10799
\(355\) −8.83388 −0.468854
\(356\) 8.56576 0.453984
\(357\) 16.9008 0.894484
\(358\) 23.6710 1.25105
\(359\) 8.35022 0.440708 0.220354 0.975420i \(-0.429279\pi\)
0.220354 + 0.975420i \(0.429279\pi\)
\(360\) 3.38641 0.178479
\(361\) 0 0
\(362\) −16.4501 −0.864598
\(363\) 25.6507 1.34631
\(364\) −2.99363 −0.156909
\(365\) −10.2500 −0.536508
\(366\) −20.3253 −1.06242
\(367\) 14.4198 0.752705 0.376353 0.926477i \(-0.377178\pi\)
0.376353 + 0.926477i \(0.377178\pi\)
\(368\) 4.65080 0.242440
\(369\) −10.4517 −0.544093
\(370\) 4.81692 0.250420
\(371\) −5.34876 −0.277694
\(372\) 4.70974 0.244188
\(373\) 24.1157 1.24866 0.624332 0.781159i \(-0.285371\pi\)
0.624332 + 0.781159i \(0.285371\pi\)
\(374\) 57.3217 2.96403
\(375\) 1.15976 0.0598895
\(376\) −18.3225 −0.944911
\(377\) −4.17627 −0.215089
\(378\) 21.8838 1.12558
\(379\) −16.6757 −0.856571 −0.428285 0.903644i \(-0.640882\pi\)
−0.428285 + 0.903644i \(0.640882\pi\)
\(380\) 0 0
\(381\) −7.09760 −0.363621
\(382\) −21.6294 −1.10666
\(383\) 10.8779 0.555834 0.277917 0.960605i \(-0.410356\pi\)
0.277917 + 0.960605i \(0.410356\pi\)
\(384\) 15.3987 0.785811
\(385\) 14.0143 0.714236
\(386\) −24.1624 −1.22983
\(387\) −7.51226 −0.381870
\(388\) −3.12736 −0.158767
\(389\) 36.4543 1.84831 0.924154 0.382021i \(-0.124772\pi\)
0.924154 + 0.382021i \(0.124772\pi\)
\(390\) 3.07911 0.155917
\(391\) 5.62528 0.284482
\(392\) 2.18849 0.110535
\(393\) −17.2584 −0.870573
\(394\) −41.6153 −2.09655
\(395\) −7.60459 −0.382628
\(396\) 7.34020 0.368859
\(397\) 8.58381 0.430809 0.215405 0.976525i \(-0.430893\pi\)
0.215405 + 0.976525i \(0.430893\pi\)
\(398\) −3.75419 −0.188181
\(399\) 0 0
\(400\) −4.94743 −0.247371
\(401\) 17.0557 0.851721 0.425860 0.904789i \(-0.359971\pi\)
0.425860 + 0.904789i \(0.359971\pi\)
\(402\) −1.94896 −0.0972051
\(403\) 8.40432 0.418649
\(404\) −8.57805 −0.426774
\(405\) −1.29619 −0.0644080
\(406\) 10.6137 0.526747
\(407\) −16.6533 −0.825476
\(408\) 14.2008 0.703043
\(409\) 11.7940 0.583178 0.291589 0.956544i \(-0.405816\pi\)
0.291589 + 0.956544i \(0.405816\pi\)
\(410\) 10.5122 0.519159
\(411\) 20.1222 0.992553
\(412\) −8.92410 −0.439659
\(413\) −26.2978 −1.29403
\(414\) 2.58960 0.127272
\(415\) 3.11355 0.152838
\(416\) −6.60780 −0.323974
\(417\) −7.77656 −0.380820
\(418\) 0 0
\(419\) 1.14280 0.0558292 0.0279146 0.999610i \(-0.491113\pi\)
0.0279146 + 0.999610i \(0.491113\pi\)
\(420\) −2.17672 −0.106213
\(421\) −19.5189 −0.951292 −0.475646 0.879637i \(-0.657786\pi\)
−0.475646 + 0.879637i \(0.657786\pi\)
\(422\) −36.9797 −1.80014
\(423\) −14.8192 −0.720533
\(424\) −4.49426 −0.218261
\(425\) −5.98406 −0.290269
\(426\) −17.0535 −0.826245
\(427\) 25.6400 1.24081
\(428\) −13.8094 −0.667501
\(429\) −10.6453 −0.513960
\(430\) 7.55574 0.364370
\(431\) 36.8785 1.77637 0.888187 0.459483i \(-0.151965\pi\)
0.888187 + 0.459483i \(0.151965\pi\)
\(432\) 26.7093 1.28505
\(433\) −0.368933 −0.0177298 −0.00886490 0.999961i \(-0.502822\pi\)
−0.00886490 + 0.999961i \(0.502822\pi\)
\(434\) −21.3589 −1.02526
\(435\) −3.03663 −0.145595
\(436\) −4.33501 −0.207609
\(437\) 0 0
\(438\) −19.7872 −0.945470
\(439\) 2.26439 0.108073 0.0540367 0.998539i \(-0.482791\pi\)
0.0540367 + 0.998539i \(0.482791\pi\)
\(440\) 11.7755 0.561373
\(441\) 1.77004 0.0842878
\(442\) −15.8875 −0.755690
\(443\) −16.4627 −0.782169 −0.391084 0.920355i \(-0.627900\pi\)
−0.391084 + 0.920355i \(0.627900\pi\)
\(444\) 2.58661 0.122755
\(445\) 11.1141 0.526860
\(446\) −17.0050 −0.805209
\(447\) 16.6922 0.789513
\(448\) −7.30329 −0.345048
\(449\) −14.1613 −0.668315 −0.334158 0.942517i \(-0.608452\pi\)
−0.334158 + 0.942517i \(0.608452\pi\)
\(450\) −2.75477 −0.129861
\(451\) −36.3433 −1.71134
\(452\) 12.0839 0.568377
\(453\) −14.7543 −0.693215
\(454\) −6.91086 −0.324343
\(455\) −3.88426 −0.182097
\(456\) 0 0
\(457\) 1.22073 0.0571033 0.0285516 0.999592i \(-0.490910\pi\)
0.0285516 + 0.999592i \(0.490910\pi\)
\(458\) 10.8742 0.508120
\(459\) 32.3057 1.50790
\(460\) −0.724501 −0.0337800
\(461\) −8.69160 −0.404808 −0.202404 0.979302i \(-0.564875\pi\)
−0.202404 + 0.979302i \(0.564875\pi\)
\(462\) 27.0542 1.25867
\(463\) −19.7149 −0.916229 −0.458114 0.888893i \(-0.651475\pi\)
−0.458114 + 0.888893i \(0.651475\pi\)
\(464\) 12.9540 0.601375
\(465\) 6.11091 0.283387
\(466\) 8.56942 0.396971
\(467\) −11.4795 −0.531207 −0.265604 0.964082i \(-0.585571\pi\)
−0.265604 + 0.964082i \(0.585571\pi\)
\(468\) −2.03443 −0.0940418
\(469\) 2.45858 0.113527
\(470\) 14.9049 0.687514
\(471\) −3.91452 −0.180372
\(472\) −22.0966 −1.01708
\(473\) −26.1222 −1.20110
\(474\) −14.6804 −0.674293
\(475\) 0 0
\(476\) 11.2313 0.514787
\(477\) −3.63495 −0.166433
\(478\) 23.2973 1.06559
\(479\) 39.2631 1.79398 0.896989 0.442053i \(-0.145750\pi\)
0.896989 + 0.442053i \(0.145750\pi\)
\(480\) −4.80463 −0.219300
\(481\) 4.61570 0.210458
\(482\) −25.3439 −1.15438
\(483\) 2.65497 0.120805
\(484\) 17.0460 0.774820
\(485\) −4.05776 −0.184253
\(486\) 24.4565 1.10937
\(487\) −31.5943 −1.43168 −0.715838 0.698266i \(-0.753955\pi\)
−0.715838 + 0.698266i \(0.753955\pi\)
\(488\) 21.5439 0.975246
\(489\) 0.357159 0.0161513
\(490\) −1.78029 −0.0804252
\(491\) −11.0637 −0.499300 −0.249650 0.968336i \(-0.580316\pi\)
−0.249650 + 0.968336i \(0.580316\pi\)
\(492\) 5.64487 0.254491
\(493\) 15.6683 0.705663
\(494\) 0 0
\(495\) 9.52395 0.428070
\(496\) −26.0686 −1.17052
\(497\) 21.5127 0.964979
\(498\) 6.01060 0.269341
\(499\) −20.3735 −0.912045 −0.456023 0.889968i \(-0.650726\pi\)
−0.456023 + 0.889968i \(0.650726\pi\)
\(500\) 0.770710 0.0344672
\(501\) −16.5431 −0.739091
\(502\) −10.1747 −0.454118
\(503\) −13.6724 −0.609624 −0.304812 0.952413i \(-0.598594\pi\)
−0.304812 + 0.952413i \(0.598594\pi\)
\(504\) −8.24676 −0.367340
\(505\) −11.1301 −0.495281
\(506\) 9.00474 0.400310
\(507\) −12.1263 −0.538550
\(508\) −4.71668 −0.209269
\(509\) 7.72393 0.342357 0.171179 0.985240i \(-0.445243\pi\)
0.171179 + 0.985240i \(0.445243\pi\)
\(510\) −11.5520 −0.511532
\(511\) 24.9613 1.10422
\(512\) 0.249240 0.0110150
\(513\) 0 0
\(514\) −0.204211 −0.00900736
\(515\) −11.5791 −0.510235
\(516\) 4.05732 0.178613
\(517\) −51.5303 −2.26630
\(518\) −11.7304 −0.515405
\(519\) −15.4794 −0.679471
\(520\) −3.26372 −0.143124
\(521\) 2.16876 0.0950151 0.0475075 0.998871i \(-0.484872\pi\)
0.0475075 + 0.998871i \(0.484872\pi\)
\(522\) 7.21290 0.315700
\(523\) 23.8932 1.04478 0.522389 0.852707i \(-0.325041\pi\)
0.522389 + 0.852707i \(0.325041\pi\)
\(524\) −11.4690 −0.501026
\(525\) −2.82430 −0.123263
\(526\) −16.7462 −0.730169
\(527\) −31.5308 −1.37350
\(528\) 33.0197 1.43700
\(529\) −22.1163 −0.961579
\(530\) 3.65598 0.158806
\(531\) −17.8716 −0.775562
\(532\) 0 0
\(533\) 10.0730 0.436312
\(534\) 21.4554 0.928467
\(535\) −17.9177 −0.774651
\(536\) 2.06581 0.0892293
\(537\) 16.4926 0.711707
\(538\) −9.50835 −0.409934
\(539\) 6.15492 0.265111
\(540\) −4.16077 −0.179051
\(541\) 42.4550 1.82528 0.912641 0.408762i \(-0.134039\pi\)
0.912641 + 0.408762i \(0.134039\pi\)
\(542\) −21.1584 −0.908829
\(543\) −11.4614 −0.491858
\(544\) 24.7907 1.06289
\(545\) −5.62470 −0.240936
\(546\) −7.49842 −0.320903
\(547\) −12.0307 −0.514396 −0.257198 0.966359i \(-0.582799\pi\)
−0.257198 + 0.966359i \(0.582799\pi\)
\(548\) 13.3721 0.571227
\(549\) 17.4246 0.743665
\(550\) −9.57907 −0.408453
\(551\) 0 0
\(552\) 2.23082 0.0949500
\(553\) 18.5191 0.787513
\(554\) 29.2992 1.24481
\(555\) 3.35614 0.142460
\(556\) −5.16787 −0.219167
\(557\) −8.75270 −0.370864 −0.185432 0.982657i \(-0.559368\pi\)
−0.185432 + 0.982657i \(0.559368\pi\)
\(558\) −14.5152 −0.614479
\(559\) 7.24011 0.306224
\(560\) 12.0482 0.509131
\(561\) 39.9383 1.68620
\(562\) 34.1238 1.43943
\(563\) 35.9707 1.51598 0.757991 0.652265i \(-0.226181\pi\)
0.757991 + 0.652265i \(0.226181\pi\)
\(564\) 8.00373 0.337018
\(565\) 15.6789 0.659615
\(566\) −19.7350 −0.829524
\(567\) 3.15654 0.132562
\(568\) 18.0760 0.758450
\(569\) 20.3125 0.851543 0.425772 0.904831i \(-0.360003\pi\)
0.425772 + 0.904831i \(0.360003\pi\)
\(570\) 0 0
\(571\) 10.1773 0.425906 0.212953 0.977062i \(-0.431692\pi\)
0.212953 + 0.977062i \(0.431692\pi\)
\(572\) −7.07428 −0.295790
\(573\) −15.0701 −0.629563
\(574\) −25.5998 −1.06852
\(575\) −0.940044 −0.0392025
\(576\) −4.96322 −0.206801
\(577\) −32.7441 −1.36316 −0.681578 0.731745i \(-0.738706\pi\)
−0.681578 + 0.731745i \(0.738706\pi\)
\(578\) 31.3083 1.30225
\(579\) −16.8349 −0.699634
\(580\) −2.01798 −0.0837919
\(581\) −7.58228 −0.314566
\(582\) −7.83337 −0.324703
\(583\) −12.6397 −0.523482
\(584\) 20.9736 0.867893
\(585\) −2.63969 −0.109138
\(586\) 41.5827 1.71777
\(587\) 8.77326 0.362111 0.181056 0.983473i \(-0.442049\pi\)
0.181056 + 0.983473i \(0.442049\pi\)
\(588\) −0.955987 −0.0394243
\(589\) 0 0
\(590\) 17.9751 0.740021
\(591\) −28.9951 −1.19270
\(592\) −14.3170 −0.588427
\(593\) −32.3207 −1.32725 −0.663625 0.748065i \(-0.730983\pi\)
−0.663625 + 0.748065i \(0.730983\pi\)
\(594\) 51.7138 2.12184
\(595\) 14.5727 0.597423
\(596\) 11.0927 0.454375
\(597\) −2.61570 −0.107053
\(598\) −2.49579 −0.102060
\(599\) 19.5504 0.798807 0.399404 0.916775i \(-0.369217\pi\)
0.399404 + 0.916775i \(0.369217\pi\)
\(600\) −2.37310 −0.0968815
\(601\) 0.401837 0.0163913 0.00819564 0.999966i \(-0.497391\pi\)
0.00819564 + 0.999966i \(0.497391\pi\)
\(602\) −18.4002 −0.749934
\(603\) 1.67082 0.0680410
\(604\) −9.80486 −0.398954
\(605\) 22.1173 0.899198
\(606\) −21.4862 −0.872817
\(607\) −13.4453 −0.545727 −0.272863 0.962053i \(-0.587971\pi\)
−0.272863 + 0.962053i \(0.587971\pi\)
\(608\) 0 0
\(609\) 7.39497 0.299659
\(610\) −17.5255 −0.709586
\(611\) 14.2823 0.577800
\(612\) 7.63266 0.308532
\(613\) −20.7053 −0.836280 −0.418140 0.908383i \(-0.637318\pi\)
−0.418140 + 0.908383i \(0.637318\pi\)
\(614\) −28.1640 −1.13661
\(615\) 7.32425 0.295342
\(616\) −28.6762 −1.15540
\(617\) −9.27871 −0.373547 −0.186773 0.982403i \(-0.559803\pi\)
−0.186773 + 0.982403i \(0.559803\pi\)
\(618\) −22.3530 −0.899169
\(619\) −2.89129 −0.116211 −0.0581053 0.998310i \(-0.518506\pi\)
−0.0581053 + 0.998310i \(0.518506\pi\)
\(620\) 4.06097 0.163093
\(621\) 5.07494 0.203650
\(622\) −25.3233 −1.01537
\(623\) −27.0657 −1.08437
\(624\) −9.15186 −0.366368
\(625\) 1.00000 0.0400000
\(626\) 41.4926 1.65838
\(627\) 0 0
\(628\) −2.60138 −0.103806
\(629\) −17.3169 −0.690469
\(630\) 6.70856 0.267275
\(631\) 30.4540 1.21236 0.606178 0.795329i \(-0.292702\pi\)
0.606178 + 0.795329i \(0.292702\pi\)
\(632\) 15.5606 0.618966
\(633\) −25.7653 −1.02408
\(634\) 41.9972 1.66792
\(635\) −6.11991 −0.242861
\(636\) 1.96321 0.0778462
\(637\) −1.70592 −0.0675910
\(638\) 25.0812 0.992975
\(639\) 14.6198 0.578349
\(640\) 13.2775 0.524841
\(641\) 20.0738 0.792869 0.396434 0.918063i \(-0.370247\pi\)
0.396434 + 0.918063i \(0.370247\pi\)
\(642\) −34.5896 −1.36514
\(643\) −2.08913 −0.0823874 −0.0411937 0.999151i \(-0.513116\pi\)
−0.0411937 + 0.999151i \(0.513116\pi\)
\(644\) 1.76434 0.0695249
\(645\) 5.26439 0.207285
\(646\) 0 0
\(647\) −2.10623 −0.0828043 −0.0414021 0.999143i \(-0.513182\pi\)
−0.0414021 + 0.999143i \(0.513182\pi\)
\(648\) 2.65227 0.104191
\(649\) −62.1444 −2.43938
\(650\) 2.65497 0.104136
\(651\) −14.8816 −0.583257
\(652\) 0.237348 0.00929527
\(653\) 1.83067 0.0716395 0.0358197 0.999358i \(-0.488596\pi\)
0.0358197 + 0.999358i \(0.488596\pi\)
\(654\) −10.8583 −0.424593
\(655\) −14.8811 −0.581453
\(656\) −31.2447 −1.21990
\(657\) 16.9634 0.661804
\(658\) −36.2973 −1.41502
\(659\) −24.0536 −0.936994 −0.468497 0.883465i \(-0.655204\pi\)
−0.468497 + 0.883465i \(0.655204\pi\)
\(660\) −5.14381 −0.200223
\(661\) 17.4422 0.678423 0.339211 0.940710i \(-0.389840\pi\)
0.339211 + 0.940710i \(0.389840\pi\)
\(662\) 33.6343 1.30723
\(663\) −11.0694 −0.429902
\(664\) −6.37097 −0.247241
\(665\) 0 0
\(666\) −7.97184 −0.308902
\(667\) 2.46135 0.0953039
\(668\) −10.9936 −0.425356
\(669\) −11.8480 −0.458072
\(670\) −1.68049 −0.0649229
\(671\) 60.5901 2.33906
\(672\) 11.7005 0.451357
\(673\) −47.5187 −1.83171 −0.915856 0.401506i \(-0.868487\pi\)
−0.915856 + 0.401506i \(0.868487\pi\)
\(674\) 52.9736 2.04047
\(675\) −5.39862 −0.207793
\(676\) −8.05850 −0.309942
\(677\) −14.5531 −0.559321 −0.279661 0.960099i \(-0.590222\pi\)
−0.279661 + 0.960099i \(0.590222\pi\)
\(678\) 30.2675 1.16242
\(679\) 9.88168 0.379224
\(680\) 12.2446 0.469560
\(681\) −4.81507 −0.184514
\(682\) −50.4734 −1.93273
\(683\) −3.33714 −0.127692 −0.0638460 0.997960i \(-0.520337\pi\)
−0.0638460 + 0.997960i \(0.520337\pi\)
\(684\) 0 0
\(685\) 17.3504 0.662923
\(686\) 32.7106 1.24890
\(687\) 7.57652 0.289062
\(688\) −22.4575 −0.856183
\(689\) 3.50326 0.133464
\(690\) −1.81472 −0.0690853
\(691\) −19.3318 −0.735415 −0.367708 0.929941i \(-0.619857\pi\)
−0.367708 + 0.929941i \(0.619857\pi\)
\(692\) −10.2868 −0.391044
\(693\) −23.1932 −0.881038
\(694\) −5.49724 −0.208673
\(695\) −6.70534 −0.254348
\(696\) 6.21358 0.235525
\(697\) −37.7914 −1.43145
\(698\) 29.7099 1.12454
\(699\) 5.97066 0.225831
\(700\) −1.87687 −0.0709392
\(701\) 9.93785 0.375347 0.187674 0.982231i \(-0.439905\pi\)
0.187674 + 0.982231i \(0.439905\pi\)
\(702\) −14.3332 −0.540971
\(703\) 0 0
\(704\) −17.2584 −0.650452
\(705\) 10.3849 0.391117
\(706\) −13.8078 −0.519663
\(707\) 27.1045 1.01937
\(708\) 9.65233 0.362757
\(709\) 37.2119 1.39752 0.698760 0.715356i \(-0.253735\pi\)
0.698760 + 0.715356i \(0.253735\pi\)
\(710\) −14.7044 −0.551846
\(711\) 12.5853 0.471987
\(712\) −22.7418 −0.852285
\(713\) −4.95322 −0.185499
\(714\) 28.1321 1.05282
\(715\) −9.17891 −0.343272
\(716\) 10.9601 0.409596
\(717\) 16.2322 0.606201
\(718\) 13.8993 0.518718
\(719\) −2.64218 −0.0985365 −0.0492683 0.998786i \(-0.515689\pi\)
−0.0492683 + 0.998786i \(0.515689\pi\)
\(720\) 8.18782 0.305142
\(721\) 28.1980 1.05015
\(722\) 0 0
\(723\) −17.6581 −0.656711
\(724\) −7.61664 −0.283070
\(725\) −2.61834 −0.0972426
\(726\) 42.6968 1.58463
\(727\) −10.1731 −0.377298 −0.188649 0.982045i \(-0.560411\pi\)
−0.188649 + 0.982045i \(0.560411\pi\)
\(728\) 7.94800 0.294572
\(729\) 20.9284 0.775126
\(730\) −17.0615 −0.631476
\(731\) −27.1630 −1.00466
\(732\) −9.41091 −0.347837
\(733\) 14.8222 0.547472 0.273736 0.961805i \(-0.411741\pi\)
0.273736 + 0.961805i \(0.411741\pi\)
\(734\) 24.0023 0.885942
\(735\) −1.24040 −0.0457528
\(736\) 3.89441 0.143550
\(737\) 5.80989 0.214010
\(738\) −17.3973 −0.640403
\(739\) 35.4866 1.30539 0.652697 0.757619i \(-0.273637\pi\)
0.652697 + 0.757619i \(0.273637\pi\)
\(740\) 2.23031 0.0819877
\(741\) 0 0
\(742\) −8.90325 −0.326849
\(743\) 8.73882 0.320596 0.160298 0.987069i \(-0.448754\pi\)
0.160298 + 0.987069i \(0.448754\pi\)
\(744\) −12.5042 −0.458426
\(745\) 14.3928 0.527313
\(746\) 40.1417 1.46969
\(747\) −5.15282 −0.188532
\(748\) 26.5408 0.970428
\(749\) 43.6342 1.59436
\(750\) 1.93047 0.0704906
\(751\) −13.0991 −0.477994 −0.238997 0.971020i \(-0.576819\pi\)
−0.238997 + 0.971020i \(0.576819\pi\)
\(752\) −44.3011 −1.61549
\(753\) −7.08911 −0.258342
\(754\) −6.95160 −0.253162
\(755\) −12.7219 −0.462996
\(756\) 10.1325 0.368517
\(757\) −16.4380 −0.597450 −0.298725 0.954339i \(-0.596561\pi\)
−0.298725 + 0.954339i \(0.596561\pi\)
\(758\) −27.7574 −1.00819
\(759\) 6.27397 0.227731
\(760\) 0 0
\(761\) 16.3918 0.594203 0.297101 0.954846i \(-0.403980\pi\)
0.297101 + 0.954846i \(0.403980\pi\)
\(762\) −11.8143 −0.427986
\(763\) 13.6976 0.495886
\(764\) −10.0148 −0.362321
\(765\) 9.90341 0.358059
\(766\) 18.1067 0.654223
\(767\) 17.2242 0.621929
\(768\) 18.6756 0.673899
\(769\) −50.0421 −1.80456 −0.902282 0.431147i \(-0.858109\pi\)
−0.902282 + 0.431147i \(0.858109\pi\)
\(770\) 23.3275 0.840664
\(771\) −0.142282 −0.00512416
\(772\) −11.1875 −0.402649
\(773\) −48.6873 −1.75116 −0.875580 0.483074i \(-0.839520\pi\)
−0.875580 + 0.483074i \(0.839520\pi\)
\(774\) −12.5045 −0.449465
\(775\) 5.26913 0.189273
\(776\) 8.30302 0.298061
\(777\) −8.17306 −0.293207
\(778\) 60.6799 2.17548
\(779\) 0 0
\(780\) 1.42568 0.0510474
\(781\) 50.8369 1.81909
\(782\) 9.36352 0.334839
\(783\) 14.1354 0.505158
\(784\) 5.29144 0.188980
\(785\) −3.37530 −0.120470
\(786\) −28.7275 −1.02467
\(787\) −6.51678 −0.232298 −0.116149 0.993232i \(-0.537055\pi\)
−0.116149 + 0.993232i \(0.537055\pi\)
\(788\) −19.2685 −0.686413
\(789\) −11.6678 −0.415383
\(790\) −12.6582 −0.450358
\(791\) −38.1820 −1.35760
\(792\) −19.4880 −0.692475
\(793\) −16.7934 −0.596350
\(794\) 14.2881 0.507067
\(795\) 2.54727 0.0903424
\(796\) −1.73825 −0.0616106
\(797\) −38.3796 −1.35947 −0.679737 0.733456i \(-0.737906\pi\)
−0.679737 + 0.733456i \(0.737906\pi\)
\(798\) 0 0
\(799\) −53.5834 −1.89565
\(800\) −4.14280 −0.146470
\(801\) −18.3935 −0.649902
\(802\) 28.3900 1.00248
\(803\) 58.9862 2.08158
\(804\) −0.902397 −0.0318251
\(805\) 2.28925 0.0806853
\(806\) 13.9894 0.492755
\(807\) −6.62485 −0.233206
\(808\) 22.7744 0.801202
\(809\) 25.1409 0.883906 0.441953 0.897038i \(-0.354286\pi\)
0.441953 + 0.897038i \(0.354286\pi\)
\(810\) −2.15756 −0.0758089
\(811\) 47.0106 1.65077 0.825383 0.564573i \(-0.190959\pi\)
0.825383 + 0.564573i \(0.190959\pi\)
\(812\) 4.91429 0.172458
\(813\) −14.7419 −0.517020
\(814\) −27.7202 −0.971594
\(815\) 0.307960 0.0107874
\(816\) 34.3354 1.20198
\(817\) 0 0
\(818\) 19.6317 0.686406
\(819\) 6.42832 0.224624
\(820\) 4.86730 0.169973
\(821\) −19.8204 −0.691737 −0.345869 0.938283i \(-0.612416\pi\)
−0.345869 + 0.938283i \(0.612416\pi\)
\(822\) 33.4943 1.16825
\(823\) −39.2168 −1.36701 −0.683505 0.729946i \(-0.739545\pi\)
−0.683505 + 0.729946i \(0.739545\pi\)
\(824\) 23.6932 0.825391
\(825\) −6.67412 −0.232363
\(826\) −43.7738 −1.52309
\(827\) 11.8435 0.411839 0.205920 0.978569i \(-0.433981\pi\)
0.205920 + 0.978569i \(0.433981\pi\)
\(828\) 1.19903 0.0416690
\(829\) −46.9321 −1.63002 −0.815010 0.579447i \(-0.803269\pi\)
−0.815010 + 0.579447i \(0.803269\pi\)
\(830\) 5.18264 0.179892
\(831\) 20.4140 0.708153
\(832\) 4.78341 0.165835
\(833\) 6.40015 0.221752
\(834\) −12.9444 −0.448229
\(835\) −14.2643 −0.493636
\(836\) 0 0
\(837\) −28.4461 −0.983240
\(838\) 1.90223 0.0657116
\(839\) 53.1339 1.83438 0.917192 0.398445i \(-0.130450\pi\)
0.917192 + 0.398445i \(0.130450\pi\)
\(840\) 5.77911 0.199398
\(841\) −22.1443 −0.763597
\(842\) −32.4901 −1.11968
\(843\) 23.7755 0.818870
\(844\) −17.1222 −0.589370
\(845\) −10.4559 −0.359695
\(846\) −24.6672 −0.848075
\(847\) −53.8613 −1.85070
\(848\) −10.8665 −0.373156
\(849\) −13.7502 −0.471905
\(850\) −9.96073 −0.341650
\(851\) −2.72033 −0.0932518
\(852\) −7.89603 −0.270514
\(853\) 42.2998 1.44832 0.724159 0.689633i \(-0.242228\pi\)
0.724159 + 0.689633i \(0.242228\pi\)
\(854\) 42.6790 1.46045
\(855\) 0 0
\(856\) 36.6634 1.25313
\(857\) −23.5384 −0.804056 −0.402028 0.915627i \(-0.631695\pi\)
−0.402028 + 0.915627i \(0.631695\pi\)
\(858\) −17.7196 −0.604936
\(859\) 9.48124 0.323496 0.161748 0.986832i \(-0.448287\pi\)
0.161748 + 0.986832i \(0.448287\pi\)
\(860\) 3.49842 0.119295
\(861\) −17.8364 −0.607864
\(862\) 61.3859 2.09081
\(863\) −22.8204 −0.776816 −0.388408 0.921487i \(-0.626975\pi\)
−0.388408 + 0.921487i \(0.626975\pi\)
\(864\) 22.3654 0.760886
\(865\) −13.3471 −0.453816
\(866\) −0.614106 −0.0208682
\(867\) 21.8138 0.740834
\(868\) −9.88950 −0.335672
\(869\) 43.7626 1.48455
\(870\) −5.05461 −0.171367
\(871\) −1.61029 −0.0545625
\(872\) 11.5093 0.389755
\(873\) 6.71546 0.227284
\(874\) 0 0
\(875\) −2.43525 −0.0823266
\(876\) −9.16179 −0.309548
\(877\) 25.2942 0.854124 0.427062 0.904222i \(-0.359549\pi\)
0.427062 + 0.904222i \(0.359549\pi\)
\(878\) 3.76918 0.127204
\(879\) 28.9723 0.977213
\(880\) 28.4713 0.959767
\(881\) −33.2871 −1.12147 −0.560736 0.827995i \(-0.689482\pi\)
−0.560736 + 0.827995i \(0.689482\pi\)
\(882\) 2.94632 0.0992077
\(883\) −15.3109 −0.515252 −0.257626 0.966245i \(-0.582940\pi\)
−0.257626 + 0.966245i \(0.582940\pi\)
\(884\) −7.35614 −0.247414
\(885\) 12.5239 0.420988
\(886\) −27.4030 −0.920621
\(887\) −43.2055 −1.45070 −0.725349 0.688381i \(-0.758322\pi\)
−0.725349 + 0.688381i \(0.758322\pi\)
\(888\) −6.86737 −0.230454
\(889\) 14.9035 0.499849
\(890\) 18.5000 0.620120
\(891\) 7.45925 0.249894
\(892\) −7.87356 −0.263626
\(893\) 0 0
\(894\) 27.7849 0.929265
\(895\) 14.2207 0.475347
\(896\) −32.3342 −1.08021
\(897\) −1.73891 −0.0580607
\(898\) −23.5722 −0.786614
\(899\) −13.7964 −0.460134
\(900\) −1.27550 −0.0425166
\(901\) −13.1433 −0.437867
\(902\) −60.4951 −2.01427
\(903\) −12.8201 −0.426627
\(904\) −32.0822 −1.06704
\(905\) −9.88263 −0.328510
\(906\) −24.5591 −0.815922
\(907\) −14.3278 −0.475748 −0.237874 0.971296i \(-0.576451\pi\)
−0.237874 + 0.971296i \(0.576451\pi\)
\(908\) −3.19983 −0.106190
\(909\) 18.4199 0.610949
\(910\) −6.46552 −0.214330
\(911\) 4.61162 0.152790 0.0763949 0.997078i \(-0.475659\pi\)
0.0763949 + 0.997078i \(0.475659\pi\)
\(912\) 0 0
\(913\) −17.9177 −0.592990
\(914\) 2.03196 0.0672112
\(915\) −12.2107 −0.403674
\(916\) 5.03494 0.166359
\(917\) 36.2393 1.19673
\(918\) 53.7742 1.77481
\(919\) 26.4921 0.873892 0.436946 0.899488i \(-0.356060\pi\)
0.436946 + 0.899488i \(0.356060\pi\)
\(920\) 1.92353 0.0634168
\(921\) −19.6230 −0.646599
\(922\) −14.4676 −0.476464
\(923\) −14.0901 −0.463782
\(924\) 12.5265 0.412091
\(925\) 2.89384 0.0951487
\(926\) −32.8163 −1.07841
\(927\) 19.1630 0.629394
\(928\) 10.8472 0.356078
\(929\) −4.75725 −0.156080 −0.0780402 0.996950i \(-0.524866\pi\)
−0.0780402 + 0.996950i \(0.524866\pi\)
\(930\) 10.1719 0.333549
\(931\) 0 0
\(932\) 3.96777 0.129969
\(933\) −17.6438 −0.577631
\(934\) −19.1081 −0.625237
\(935\) 34.4368 1.12621
\(936\) 5.40135 0.176549
\(937\) −11.9367 −0.389954 −0.194977 0.980808i \(-0.562463\pi\)
−0.194977 + 0.980808i \(0.562463\pi\)
\(938\) 4.09242 0.133622
\(939\) 28.9096 0.943429
\(940\) 6.90122 0.225093
\(941\) 17.9943 0.586597 0.293299 0.956021i \(-0.405247\pi\)
0.293299 + 0.956021i \(0.405247\pi\)
\(942\) −6.51590 −0.212299
\(943\) −5.93670 −0.193326
\(944\) −53.4262 −1.73887
\(945\) 13.1470 0.427673
\(946\) −43.4815 −1.41371
\(947\) −25.2191 −0.819511 −0.409755 0.912195i \(-0.634386\pi\)
−0.409755 + 0.912195i \(0.634386\pi\)
\(948\) −6.79725 −0.220764
\(949\) −16.3488 −0.530705
\(950\) 0 0
\(951\) 29.2611 0.948858
\(952\) −29.8188 −0.966433
\(953\) −42.3179 −1.37081 −0.685405 0.728162i \(-0.740375\pi\)
−0.685405 + 0.728162i \(0.740375\pi\)
\(954\) −6.05053 −0.195893
\(955\) −12.9942 −0.420483
\(956\) 10.7870 0.348876
\(957\) 17.4751 0.564890
\(958\) 65.3552 2.11153
\(959\) −42.2525 −1.36441
\(960\) 3.47809 0.112255
\(961\) −3.23623 −0.104395
\(962\) 7.68304 0.247711
\(963\) 29.6533 0.955563
\(964\) −11.7346 −0.377946
\(965\) −14.5159 −0.467283
\(966\) 4.41931 0.142189
\(967\) −13.6964 −0.440447 −0.220223 0.975449i \(-0.570679\pi\)
−0.220223 + 0.975449i \(0.570679\pi\)
\(968\) −45.2567 −1.45460
\(969\) 0 0
\(970\) −6.75432 −0.216868
\(971\) 38.5812 1.23813 0.619065 0.785339i \(-0.287512\pi\)
0.619065 + 0.785339i \(0.287512\pi\)
\(972\) 11.3237 0.363209
\(973\) 16.3292 0.523491
\(974\) −52.5902 −1.68510
\(975\) 1.84982 0.0592417
\(976\) 52.0899 1.66736
\(977\) 17.4592 0.558568 0.279284 0.960209i \(-0.409903\pi\)
0.279284 + 0.960209i \(0.409903\pi\)
\(978\) 0.594507 0.0190102
\(979\) −63.9592 −2.04414
\(980\) −0.824301 −0.0263313
\(981\) 9.30869 0.297204
\(982\) −18.4161 −0.587681
\(983\) −43.9161 −1.40071 −0.700353 0.713796i \(-0.746974\pi\)
−0.700353 + 0.713796i \(0.746974\pi\)
\(984\) −14.9870 −0.477767
\(985\) −25.0010 −0.796599
\(986\) 26.0805 0.830574
\(987\) −25.2898 −0.804984
\(988\) 0 0
\(989\) −4.26707 −0.135685
\(990\) 15.8530 0.503843
\(991\) −46.3462 −1.47224 −0.736118 0.676853i \(-0.763343\pi\)
−0.736118 + 0.676853i \(0.763343\pi\)
\(992\) −21.8289 −0.693070
\(993\) 23.4344 0.743668
\(994\) 35.8089 1.13579
\(995\) −2.25539 −0.0715006
\(996\) 2.78300 0.0881827
\(997\) 11.7371 0.371719 0.185859 0.982576i \(-0.440493\pi\)
0.185859 + 0.982576i \(0.440493\pi\)
\(998\) −33.9127 −1.07349
\(999\) −15.6227 −0.494281
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.i.1.4 4
5.4 even 2 9025.2.a.bp.1.1 4
19.8 odd 6 95.2.e.c.26.4 yes 8
19.12 odd 6 95.2.e.c.11.4 8
19.18 odd 2 1805.2.a.o.1.1 4
57.8 even 6 855.2.k.h.406.1 8
57.50 even 6 855.2.k.h.676.1 8
76.27 even 6 1520.2.q.o.881.1 8
76.31 even 6 1520.2.q.o.961.1 8
95.8 even 12 475.2.j.c.349.7 16
95.12 even 12 475.2.j.c.49.7 16
95.27 even 12 475.2.j.c.349.2 16
95.69 odd 6 475.2.e.e.201.1 8
95.84 odd 6 475.2.e.e.26.1 8
95.88 even 12 475.2.j.c.49.2 16
95.94 odd 2 9025.2.a.bg.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.e.c.11.4 8 19.12 odd 6
95.2.e.c.26.4 yes 8 19.8 odd 6
475.2.e.e.26.1 8 95.84 odd 6
475.2.e.e.201.1 8 95.69 odd 6
475.2.j.c.49.2 16 95.88 even 12
475.2.j.c.49.7 16 95.12 even 12
475.2.j.c.349.2 16 95.27 even 12
475.2.j.c.349.7 16 95.8 even 12
855.2.k.h.406.1 8 57.8 even 6
855.2.k.h.676.1 8 57.50 even 6
1520.2.q.o.881.1 8 76.27 even 6
1520.2.q.o.961.1 8 76.31 even 6
1805.2.a.i.1.4 4 1.1 even 1 trivial
1805.2.a.o.1.1 4 19.18 odd 2
9025.2.a.bg.1.4 4 95.94 odd 2
9025.2.a.bp.1.1 4 5.4 even 2