Properties

Label 2541.2.a.w.1.2
Level $2541$
Weight $2$
Character 2541.1
Self dual yes
Analytic conductor $20.290$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2541,2,Mod(1,2541)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2541, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2541.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2541 = 3 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2541.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: \(20.2899871536\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 2541.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.23607 q^{2} -1.00000 q^{3} +3.00000 q^{4} -0.618034 q^{5} -2.23607 q^{6} +1.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.23607 q^{2} -1.00000 q^{3} +3.00000 q^{4} -0.618034 q^{5} -2.23607 q^{6} +1.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} -1.38197 q^{10} -3.00000 q^{12} +3.23607 q^{13} +2.23607 q^{14} +0.618034 q^{15} -1.00000 q^{16} +4.85410 q^{17} +2.23607 q^{18} +2.85410 q^{19} -1.85410 q^{20} -1.00000 q^{21} +4.38197 q^{23} -2.23607 q^{24} -4.61803 q^{25} +7.23607 q^{26} -1.00000 q^{27} +3.00000 q^{28} -6.00000 q^{29} +1.38197 q^{30} -3.09017 q^{31} -6.70820 q^{32} +10.8541 q^{34} -0.618034 q^{35} +3.00000 q^{36} +4.61803 q^{37} +6.38197 q^{38} -3.23607 q^{39} -1.38197 q^{40} +7.38197 q^{41} -2.23607 q^{42} +9.70820 q^{43} -0.618034 q^{45} +9.79837 q^{46} +4.47214 q^{47} +1.00000 q^{48} +1.00000 q^{49} -10.3262 q^{50} -4.85410 q^{51} +9.70820 q^{52} +6.76393 q^{53} -2.23607 q^{54} +2.23607 q^{56} -2.85410 q^{57} -13.4164 q^{58} +3.23607 q^{59} +1.85410 q^{60} -6.90983 q^{62} +1.00000 q^{63} -13.0000 q^{64} -2.00000 q^{65} +14.5623 q^{68} -4.38197 q^{69} -1.38197 q^{70} -4.94427 q^{71} +2.23607 q^{72} +13.7082 q^{73} +10.3262 q^{74} +4.61803 q^{75} +8.56231 q^{76} -7.23607 q^{78} -10.0000 q^{79} +0.618034 q^{80} +1.00000 q^{81} +16.5066 q^{82} +16.4721 q^{83} -3.00000 q^{84} -3.00000 q^{85} +21.7082 q^{86} +6.00000 q^{87} +5.61803 q^{89} -1.38197 q^{90} +3.23607 q^{91} +13.1459 q^{92} +3.09017 q^{93} +10.0000 q^{94} -1.76393 q^{95} +6.70820 q^{96} -6.00000 q^{97} +2.23607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 6 q^{4} + q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 6 q^{4} + q^{5} + 2 q^{7} + 2 q^{9} - 5 q^{10} - 6 q^{12} + 2 q^{13} - q^{15} - 2 q^{16} + 3 q^{17} - q^{19} + 3 q^{20} - 2 q^{21} + 11 q^{23} - 7 q^{25} + 10 q^{26} - 2 q^{27} + 6 q^{28} - 12 q^{29} + 5 q^{30} + 5 q^{31} + 15 q^{34} + q^{35} + 6 q^{36} + 7 q^{37} + 15 q^{38} - 2 q^{39} - 5 q^{40} + 17 q^{41} + 6 q^{43} + q^{45} - 5 q^{46} + 2 q^{48} + 2 q^{49} - 5 q^{50} - 3 q^{51} + 6 q^{52} + 18 q^{53} + q^{57} + 2 q^{59} - 3 q^{60} - 25 q^{62} + 2 q^{63} - 26 q^{64} - 4 q^{65} + 9 q^{68} - 11 q^{69} - 5 q^{70} + 8 q^{71} + 14 q^{73} + 5 q^{74} + 7 q^{75} - 3 q^{76} - 10 q^{78} - 20 q^{79} - q^{80} + 2 q^{81} - 5 q^{82} + 24 q^{83} - 6 q^{84} - 6 q^{85} + 30 q^{86} + 12 q^{87} + 9 q^{89} - 5 q^{90} + 2 q^{91} + 33 q^{92} - 5 q^{93} + 20 q^{94} - 8 q^{95} - 12 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.23607 1.58114 0.790569 0.612372i \(-0.209785\pi\)
0.790569 + 0.612372i \(0.209785\pi\)
\(3\) −1.00000 −0.577350
\(4\) 3.00000 1.50000
\(5\) −0.618034 −0.276393 −0.138197 0.990405i \(-0.544131\pi\)
−0.138197 + 0.990405i \(0.544131\pi\)
\(6\) −2.23607 −0.912871
\(7\) 1.00000 0.377964
\(8\) 2.23607 0.790569
\(9\) 1.00000 0.333333
\(10\) −1.38197 −0.437016
\(11\) 0 0
\(12\) −3.00000 −0.866025
\(13\) 3.23607 0.897524 0.448762 0.893651i \(-0.351865\pi\)
0.448762 + 0.893651i \(0.351865\pi\)
\(14\) 2.23607 0.597614
\(15\) 0.618034 0.159576
\(16\) −1.00000 −0.250000
\(17\) 4.85410 1.17729 0.588646 0.808391i \(-0.299661\pi\)
0.588646 + 0.808391i \(0.299661\pi\)
\(18\) 2.23607 0.527046
\(19\) 2.85410 0.654776 0.327388 0.944890i \(-0.393832\pi\)
0.327388 + 0.944890i \(0.393832\pi\)
\(20\) −1.85410 −0.414590
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 4.38197 0.913703 0.456852 0.889543i \(-0.348977\pi\)
0.456852 + 0.889543i \(0.348977\pi\)
\(24\) −2.23607 −0.456435
\(25\) −4.61803 −0.923607
\(26\) 7.23607 1.41911
\(27\) −1.00000 −0.192450
\(28\) 3.00000 0.566947
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 1.38197 0.252311
\(31\) −3.09017 −0.555011 −0.277505 0.960724i \(-0.589508\pi\)
−0.277505 + 0.960724i \(0.589508\pi\)
\(32\) −6.70820 −1.18585
\(33\) 0 0
\(34\) 10.8541 1.86146
\(35\) −0.618034 −0.104467
\(36\) 3.00000 0.500000
\(37\) 4.61803 0.759200 0.379600 0.925151i \(-0.376062\pi\)
0.379600 + 0.925151i \(0.376062\pi\)
\(38\) 6.38197 1.03529
\(39\) −3.23607 −0.518186
\(40\) −1.38197 −0.218508
\(41\) 7.38197 1.15287 0.576435 0.817143i \(-0.304444\pi\)
0.576435 + 0.817143i \(0.304444\pi\)
\(42\) −2.23607 −0.345033
\(43\) 9.70820 1.48049 0.740244 0.672339i \(-0.234710\pi\)
0.740244 + 0.672339i \(0.234710\pi\)
\(44\) 0 0
\(45\) −0.618034 −0.0921311
\(46\) 9.79837 1.44469
\(47\) 4.47214 0.652328 0.326164 0.945313i \(-0.394244\pi\)
0.326164 + 0.945313i \(0.394244\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −10.3262 −1.46035
\(51\) −4.85410 −0.679710
\(52\) 9.70820 1.34629
\(53\) 6.76393 0.929098 0.464549 0.885548i \(-0.346217\pi\)
0.464549 + 0.885548i \(0.346217\pi\)
\(54\) −2.23607 −0.304290
\(55\) 0 0
\(56\) 2.23607 0.298807
\(57\) −2.85410 −0.378035
\(58\) −13.4164 −1.76166
\(59\) 3.23607 0.421300 0.210650 0.977562i \(-0.432442\pi\)
0.210650 + 0.977562i \(0.432442\pi\)
\(60\) 1.85410 0.239364
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) −6.90983 −0.877549
\(63\) 1.00000 0.125988
\(64\) −13.0000 −1.62500
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 14.5623 1.76594
\(69\) −4.38197 −0.527527
\(70\) −1.38197 −0.165177
\(71\) −4.94427 −0.586777 −0.293389 0.955993i \(-0.594783\pi\)
−0.293389 + 0.955993i \(0.594783\pi\)
\(72\) 2.23607 0.263523
\(73\) 13.7082 1.60442 0.802212 0.597039i \(-0.203656\pi\)
0.802212 + 0.597039i \(0.203656\pi\)
\(74\) 10.3262 1.20040
\(75\) 4.61803 0.533245
\(76\) 8.56231 0.982164
\(77\) 0 0
\(78\) −7.23607 −0.819323
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0.618034 0.0690983
\(81\) 1.00000 0.111111
\(82\) 16.5066 1.82285
\(83\) 16.4721 1.80805 0.904026 0.427478i \(-0.140598\pi\)
0.904026 + 0.427478i \(0.140598\pi\)
\(84\) −3.00000 −0.327327
\(85\) −3.00000 −0.325396
\(86\) 21.7082 2.34086
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) 5.61803 0.595510 0.297755 0.954642i \(-0.403762\pi\)
0.297755 + 0.954642i \(0.403762\pi\)
\(90\) −1.38197 −0.145672
\(91\) 3.23607 0.339232
\(92\) 13.1459 1.37055
\(93\) 3.09017 0.320436
\(94\) 10.0000 1.03142
\(95\) −1.76393 −0.180976
\(96\) 6.70820 0.684653
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 2.23607 0.225877
\(99\) 0 0
\(100\) −13.8541 −1.38541
\(101\) −6.79837 −0.676463 −0.338232 0.941063i \(-0.609829\pi\)
−0.338232 + 0.941063i \(0.609829\pi\)
\(102\) −10.8541 −1.07472
\(103\) 1.32624 0.130678 0.0653391 0.997863i \(-0.479187\pi\)
0.0653391 + 0.997863i \(0.479187\pi\)
\(104\) 7.23607 0.709555
\(105\) 0.618034 0.0603139
\(106\) 15.1246 1.46903
\(107\) 15.7984 1.52729 0.763643 0.645638i \(-0.223409\pi\)
0.763643 + 0.645638i \(0.223409\pi\)
\(108\) −3.00000 −0.288675
\(109\) −17.5623 −1.68216 −0.841082 0.540908i \(-0.818081\pi\)
−0.841082 + 0.540908i \(0.818081\pi\)
\(110\) 0 0
\(111\) −4.61803 −0.438324
\(112\) −1.00000 −0.0944911
\(113\) −5.23607 −0.492568 −0.246284 0.969198i \(-0.579210\pi\)
−0.246284 + 0.969198i \(0.579210\pi\)
\(114\) −6.38197 −0.597726
\(115\) −2.70820 −0.252541
\(116\) −18.0000 −1.67126
\(117\) 3.23607 0.299175
\(118\) 7.23607 0.666134
\(119\) 4.85410 0.444975
\(120\) 1.38197 0.126156
\(121\) 0 0
\(122\) 0 0
\(123\) −7.38197 −0.665609
\(124\) −9.27051 −0.832516
\(125\) 5.94427 0.531672
\(126\) 2.23607 0.199205
\(127\) 0.291796 0.0258927 0.0129464 0.999916i \(-0.495879\pi\)
0.0129464 + 0.999916i \(0.495879\pi\)
\(128\) −15.6525 −1.38350
\(129\) −9.70820 −0.854760
\(130\) −4.47214 −0.392232
\(131\) −16.6525 −1.45493 −0.727467 0.686143i \(-0.759302\pi\)
−0.727467 + 0.686143i \(0.759302\pi\)
\(132\) 0 0
\(133\) 2.85410 0.247482
\(134\) 0 0
\(135\) 0.618034 0.0531919
\(136\) 10.8541 0.930732
\(137\) 8.18034 0.698894 0.349447 0.936956i \(-0.386370\pi\)
0.349447 + 0.936956i \(0.386370\pi\)
\(138\) −9.79837 −0.834093
\(139\) 13.1459 1.11502 0.557510 0.830170i \(-0.311757\pi\)
0.557510 + 0.830170i \(0.311757\pi\)
\(140\) −1.85410 −0.156700
\(141\) −4.47214 −0.376622
\(142\) −11.0557 −0.927776
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 3.70820 0.307950
\(146\) 30.6525 2.53682
\(147\) −1.00000 −0.0824786
\(148\) 13.8541 1.13880
\(149\) −4.94427 −0.405051 −0.202525 0.979277i \(-0.564915\pi\)
−0.202525 + 0.979277i \(0.564915\pi\)
\(150\) 10.3262 0.843134
\(151\) −15.5279 −1.26364 −0.631820 0.775115i \(-0.717692\pi\)
−0.631820 + 0.775115i \(0.717692\pi\)
\(152\) 6.38197 0.517646
\(153\) 4.85410 0.392431
\(154\) 0 0
\(155\) 1.90983 0.153401
\(156\) −9.70820 −0.777278
\(157\) −23.1246 −1.84554 −0.922772 0.385345i \(-0.874082\pi\)
−0.922772 + 0.385345i \(0.874082\pi\)
\(158\) −22.3607 −1.77892
\(159\) −6.76393 −0.536415
\(160\) 4.14590 0.327762
\(161\) 4.38197 0.345347
\(162\) 2.23607 0.175682
\(163\) −10.0000 −0.783260 −0.391630 0.920123i \(-0.628089\pi\)
−0.391630 + 0.920123i \(0.628089\pi\)
\(164\) 22.1459 1.72930
\(165\) 0 0
\(166\) 36.8328 2.85878
\(167\) −21.2361 −1.64330 −0.821648 0.569995i \(-0.806945\pi\)
−0.821648 + 0.569995i \(0.806945\pi\)
\(168\) −2.23607 −0.172516
\(169\) −2.52786 −0.194451
\(170\) −6.70820 −0.514496
\(171\) 2.85410 0.218259
\(172\) 29.1246 2.22073
\(173\) −13.6180 −1.03536 −0.517680 0.855574i \(-0.673204\pi\)
−0.517680 + 0.855574i \(0.673204\pi\)
\(174\) 13.4164 1.01710
\(175\) −4.61803 −0.349091
\(176\) 0 0
\(177\) −3.23607 −0.243238
\(178\) 12.5623 0.941585
\(179\) −14.7984 −1.10608 −0.553041 0.833154i \(-0.686533\pi\)
−0.553041 + 0.833154i \(0.686533\pi\)
\(180\) −1.85410 −0.138197
\(181\) −19.8885 −1.47830 −0.739152 0.673539i \(-0.764773\pi\)
−0.739152 + 0.673539i \(0.764773\pi\)
\(182\) 7.23607 0.536373
\(183\) 0 0
\(184\) 9.79837 0.722346
\(185\) −2.85410 −0.209838
\(186\) 6.90983 0.506653
\(187\) 0 0
\(188\) 13.4164 0.978492
\(189\) −1.00000 −0.0727393
\(190\) −3.94427 −0.286148
\(191\) 4.90983 0.355263 0.177631 0.984097i \(-0.443157\pi\)
0.177631 + 0.984097i \(0.443157\pi\)
\(192\) 13.0000 0.938194
\(193\) −21.8541 −1.57309 −0.786546 0.617531i \(-0.788133\pi\)
−0.786546 + 0.617531i \(0.788133\pi\)
\(194\) −13.4164 −0.963242
\(195\) 2.00000 0.143223
\(196\) 3.00000 0.214286
\(197\) 22.4721 1.60107 0.800537 0.599284i \(-0.204548\pi\)
0.800537 + 0.599284i \(0.204548\pi\)
\(198\) 0 0
\(199\) 13.1459 0.931888 0.465944 0.884814i \(-0.345715\pi\)
0.465944 + 0.884814i \(0.345715\pi\)
\(200\) −10.3262 −0.730175
\(201\) 0 0
\(202\) −15.2016 −1.06958
\(203\) −6.00000 −0.421117
\(204\) −14.5623 −1.01957
\(205\) −4.56231 −0.318645
\(206\) 2.96556 0.206620
\(207\) 4.38197 0.304568
\(208\) −3.23607 −0.224381
\(209\) 0 0
\(210\) 1.38197 0.0953647
\(211\) −18.3607 −1.26400 −0.632001 0.774968i \(-0.717766\pi\)
−0.632001 + 0.774968i \(0.717766\pi\)
\(212\) 20.2918 1.39365
\(213\) 4.94427 0.338776
\(214\) 35.3262 2.41485
\(215\) −6.00000 −0.409197
\(216\) −2.23607 −0.152145
\(217\) −3.09017 −0.209774
\(218\) −39.2705 −2.65973
\(219\) −13.7082 −0.926315
\(220\) 0 0
\(221\) 15.7082 1.05665
\(222\) −10.3262 −0.693052
\(223\) 19.6180 1.31372 0.656860 0.754012i \(-0.271884\pi\)
0.656860 + 0.754012i \(0.271884\pi\)
\(224\) −6.70820 −0.448211
\(225\) −4.61803 −0.307869
\(226\) −11.7082 −0.778818
\(227\) 12.4721 0.827805 0.413902 0.910321i \(-0.364165\pi\)
0.413902 + 0.910321i \(0.364165\pi\)
\(228\) −8.56231 −0.567053
\(229\) 16.1803 1.06923 0.534613 0.845097i \(-0.320457\pi\)
0.534613 + 0.845097i \(0.320457\pi\)
\(230\) −6.05573 −0.399303
\(231\) 0 0
\(232\) −13.4164 −0.880830
\(233\) −15.7082 −1.02908 −0.514539 0.857467i \(-0.672037\pi\)
−0.514539 + 0.857467i \(0.672037\pi\)
\(234\) 7.23607 0.473037
\(235\) −2.76393 −0.180299
\(236\) 9.70820 0.631950
\(237\) 10.0000 0.649570
\(238\) 10.8541 0.703567
\(239\) −19.0344 −1.23124 −0.615618 0.788045i \(-0.711093\pi\)
−0.615618 + 0.788045i \(0.711093\pi\)
\(240\) −0.618034 −0.0398939
\(241\) 12.0000 0.772988 0.386494 0.922292i \(-0.373686\pi\)
0.386494 + 0.922292i \(0.373686\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −0.618034 −0.0394847
\(246\) −16.5066 −1.05242
\(247\) 9.23607 0.587677
\(248\) −6.90983 −0.438775
\(249\) −16.4721 −1.04388
\(250\) 13.2918 0.840647
\(251\) 27.7082 1.74893 0.874463 0.485092i \(-0.161214\pi\)
0.874463 + 0.485092i \(0.161214\pi\)
\(252\) 3.00000 0.188982
\(253\) 0 0
\(254\) 0.652476 0.0409400
\(255\) 3.00000 0.187867
\(256\) −9.00000 −0.562500
\(257\) 24.4508 1.52520 0.762601 0.646869i \(-0.223922\pi\)
0.762601 + 0.646869i \(0.223922\pi\)
\(258\) −21.7082 −1.35149
\(259\) 4.61803 0.286951
\(260\) −6.00000 −0.372104
\(261\) −6.00000 −0.371391
\(262\) −37.2361 −2.30045
\(263\) 10.8541 0.669293 0.334646 0.942344i \(-0.391383\pi\)
0.334646 + 0.942344i \(0.391383\pi\)
\(264\) 0 0
\(265\) −4.18034 −0.256796
\(266\) 6.38197 0.391303
\(267\) −5.61803 −0.343818
\(268\) 0 0
\(269\) −13.4164 −0.818013 −0.409006 0.912532i \(-0.634125\pi\)
−0.409006 + 0.912532i \(0.634125\pi\)
\(270\) 1.38197 0.0841038
\(271\) −22.5066 −1.36718 −0.683589 0.729868i \(-0.739582\pi\)
−0.683589 + 0.729868i \(0.739582\pi\)
\(272\) −4.85410 −0.294323
\(273\) −3.23607 −0.195856
\(274\) 18.2918 1.10505
\(275\) 0 0
\(276\) −13.1459 −0.791290
\(277\) −0.0901699 −0.00541779 −0.00270889 0.999996i \(-0.500862\pi\)
−0.00270889 + 0.999996i \(0.500862\pi\)
\(278\) 29.3951 1.76300
\(279\) −3.09017 −0.185004
\(280\) −1.38197 −0.0825883
\(281\) −26.9443 −1.60736 −0.803680 0.595061i \(-0.797128\pi\)
−0.803680 + 0.595061i \(0.797128\pi\)
\(282\) −10.0000 −0.595491
\(283\) −17.2705 −1.02663 −0.513313 0.858202i \(-0.671582\pi\)
−0.513313 + 0.858202i \(0.671582\pi\)
\(284\) −14.8328 −0.880166
\(285\) 1.76393 0.104486
\(286\) 0 0
\(287\) 7.38197 0.435744
\(288\) −6.70820 −0.395285
\(289\) 6.56231 0.386018
\(290\) 8.29180 0.486911
\(291\) 6.00000 0.351726
\(292\) 41.1246 2.40664
\(293\) −18.2148 −1.06412 −0.532059 0.846707i \(-0.678582\pi\)
−0.532059 + 0.846707i \(0.678582\pi\)
\(294\) −2.23607 −0.130410
\(295\) −2.00000 −0.116445
\(296\) 10.3262 0.600200
\(297\) 0 0
\(298\) −11.0557 −0.640441
\(299\) 14.1803 0.820070
\(300\) 13.8541 0.799867
\(301\) 9.70820 0.559572
\(302\) −34.7214 −1.99799
\(303\) 6.79837 0.390556
\(304\) −2.85410 −0.163694
\(305\) 0 0
\(306\) 10.8541 0.620488
\(307\) −17.5623 −1.00233 −0.501167 0.865351i \(-0.667096\pi\)
−0.501167 + 0.865351i \(0.667096\pi\)
\(308\) 0 0
\(309\) −1.32624 −0.0754470
\(310\) 4.27051 0.242549
\(311\) 20.6525 1.17109 0.585547 0.810638i \(-0.300880\pi\)
0.585547 + 0.810638i \(0.300880\pi\)
\(312\) −7.23607 −0.409662
\(313\) −7.81966 −0.441993 −0.220997 0.975275i \(-0.570931\pi\)
−0.220997 + 0.975275i \(0.570931\pi\)
\(314\) −51.7082 −2.91806
\(315\) −0.618034 −0.0348223
\(316\) −30.0000 −1.68763
\(317\) 6.76393 0.379900 0.189950 0.981794i \(-0.439167\pi\)
0.189950 + 0.981794i \(0.439167\pi\)
\(318\) −15.1246 −0.848146
\(319\) 0 0
\(320\) 8.03444 0.449139
\(321\) −15.7984 −0.881779
\(322\) 9.79837 0.546042
\(323\) 13.8541 0.770863
\(324\) 3.00000 0.166667
\(325\) −14.9443 −0.828959
\(326\) −22.3607 −1.23844
\(327\) 17.5623 0.971198
\(328\) 16.5066 0.911423
\(329\) 4.47214 0.246557
\(330\) 0 0
\(331\) 17.4164 0.957292 0.478646 0.878008i \(-0.341128\pi\)
0.478646 + 0.878008i \(0.341128\pi\)
\(332\) 49.4164 2.71208
\(333\) 4.61803 0.253067
\(334\) −47.4853 −2.59828
\(335\) 0 0
\(336\) 1.00000 0.0545545
\(337\) 27.0902 1.47570 0.737848 0.674967i \(-0.235842\pi\)
0.737848 + 0.674967i \(0.235842\pi\)
\(338\) −5.65248 −0.307454
\(339\) 5.23607 0.284384
\(340\) −9.00000 −0.488094
\(341\) 0 0
\(342\) 6.38197 0.345097
\(343\) 1.00000 0.0539949
\(344\) 21.7082 1.17043
\(345\) 2.70820 0.145805
\(346\) −30.4508 −1.63705
\(347\) −14.5066 −0.778754 −0.389377 0.921078i \(-0.627310\pi\)
−0.389377 + 0.921078i \(0.627310\pi\)
\(348\) 18.0000 0.964901
\(349\) −21.7082 −1.16201 −0.581007 0.813899i \(-0.697341\pi\)
−0.581007 + 0.813899i \(0.697341\pi\)
\(350\) −10.3262 −0.551961
\(351\) −3.23607 −0.172729
\(352\) 0 0
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) −7.23607 −0.384593
\(355\) 3.05573 0.162181
\(356\) 16.8541 0.893266
\(357\) −4.85410 −0.256906
\(358\) −33.0902 −1.74887
\(359\) 6.43769 0.339768 0.169884 0.985464i \(-0.445661\pi\)
0.169884 + 0.985464i \(0.445661\pi\)
\(360\) −1.38197 −0.0728360
\(361\) −10.8541 −0.571269
\(362\) −44.4721 −2.33740
\(363\) 0 0
\(364\) 9.70820 0.508848
\(365\) −8.47214 −0.443452
\(366\) 0 0
\(367\) 2.72949 0.142478 0.0712391 0.997459i \(-0.477305\pi\)
0.0712391 + 0.997459i \(0.477305\pi\)
\(368\) −4.38197 −0.228426
\(369\) 7.38197 0.384290
\(370\) −6.38197 −0.331783
\(371\) 6.76393 0.351166
\(372\) 9.27051 0.480654
\(373\) 16.3262 0.845341 0.422670 0.906284i \(-0.361093\pi\)
0.422670 + 0.906284i \(0.361093\pi\)
\(374\) 0 0
\(375\) −5.94427 −0.306961
\(376\) 10.0000 0.515711
\(377\) −19.4164 −0.999996
\(378\) −2.23607 −0.115011
\(379\) 21.5279 1.10581 0.552906 0.833244i \(-0.313519\pi\)
0.552906 + 0.833244i \(0.313519\pi\)
\(380\) −5.29180 −0.271463
\(381\) −0.291796 −0.0149492
\(382\) 10.9787 0.561720
\(383\) −20.8328 −1.06451 −0.532254 0.846585i \(-0.678655\pi\)
−0.532254 + 0.846585i \(0.678655\pi\)
\(384\) 15.6525 0.798762
\(385\) 0 0
\(386\) −48.8673 −2.48728
\(387\) 9.70820 0.493496
\(388\) −18.0000 −0.913812
\(389\) 15.2361 0.772499 0.386250 0.922394i \(-0.373770\pi\)
0.386250 + 0.922394i \(0.373770\pi\)
\(390\) 4.47214 0.226455
\(391\) 21.2705 1.07570
\(392\) 2.23607 0.112938
\(393\) 16.6525 0.840006
\(394\) 50.2492 2.53152
\(395\) 6.18034 0.310967
\(396\) 0 0
\(397\) 30.5410 1.53281 0.766405 0.642358i \(-0.222044\pi\)
0.766405 + 0.642358i \(0.222044\pi\)
\(398\) 29.3951 1.47344
\(399\) −2.85410 −0.142884
\(400\) 4.61803 0.230902
\(401\) 7.41641 0.370358 0.185179 0.982705i \(-0.440714\pi\)
0.185179 + 0.982705i \(0.440714\pi\)
\(402\) 0 0
\(403\) −10.0000 −0.498135
\(404\) −20.3951 −1.01470
\(405\) −0.618034 −0.0307104
\(406\) −13.4164 −0.665845
\(407\) 0 0
\(408\) −10.8541 −0.537358
\(409\) −22.1803 −1.09675 −0.548374 0.836233i \(-0.684753\pi\)
−0.548374 + 0.836233i \(0.684753\pi\)
\(410\) −10.2016 −0.503822
\(411\) −8.18034 −0.403506
\(412\) 3.97871 0.196017
\(413\) 3.23607 0.159236
\(414\) 9.79837 0.481564
\(415\) −10.1803 −0.499733
\(416\) −21.7082 −1.06433
\(417\) −13.1459 −0.643757
\(418\) 0 0
\(419\) −3.23607 −0.158092 −0.0790461 0.996871i \(-0.525187\pi\)
−0.0790461 + 0.996871i \(0.525187\pi\)
\(420\) 1.85410 0.0904709
\(421\) 9.79837 0.477544 0.238772 0.971076i \(-0.423255\pi\)
0.238772 + 0.971076i \(0.423255\pi\)
\(422\) −41.0557 −1.99856
\(423\) 4.47214 0.217443
\(424\) 15.1246 0.734516
\(425\) −22.4164 −1.08736
\(426\) 11.0557 0.535652
\(427\) 0 0
\(428\) 47.3951 2.29093
\(429\) 0 0
\(430\) −13.4164 −0.646997
\(431\) 15.0902 0.726868 0.363434 0.931620i \(-0.381604\pi\)
0.363434 + 0.931620i \(0.381604\pi\)
\(432\) 1.00000 0.0481125
\(433\) −31.2361 −1.50111 −0.750555 0.660808i \(-0.770214\pi\)
−0.750555 + 0.660808i \(0.770214\pi\)
\(434\) −6.90983 −0.331682
\(435\) −3.70820 −0.177795
\(436\) −52.6869 −2.52325
\(437\) 12.5066 0.598271
\(438\) −30.6525 −1.46463
\(439\) 1.85410 0.0884915 0.0442457 0.999021i \(-0.485912\pi\)
0.0442457 + 0.999021i \(0.485912\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 35.1246 1.67071
\(443\) 20.3820 0.968376 0.484188 0.874964i \(-0.339115\pi\)
0.484188 + 0.874964i \(0.339115\pi\)
\(444\) −13.8541 −0.657487
\(445\) −3.47214 −0.164595
\(446\) 43.8673 2.07717
\(447\) 4.94427 0.233856
\(448\) −13.0000 −0.614192
\(449\) −41.8885 −1.97684 −0.988421 0.151734i \(-0.951514\pi\)
−0.988421 + 0.151734i \(0.951514\pi\)
\(450\) −10.3262 −0.486784
\(451\) 0 0
\(452\) −15.7082 −0.738852
\(453\) 15.5279 0.729563
\(454\) 27.8885 1.30887
\(455\) −2.00000 −0.0937614
\(456\) −6.38197 −0.298863
\(457\) 21.4164 1.00182 0.500909 0.865500i \(-0.332999\pi\)
0.500909 + 0.865500i \(0.332999\pi\)
\(458\) 36.1803 1.69060
\(459\) −4.85410 −0.226570
\(460\) −8.12461 −0.378812
\(461\) −6.00000 −0.279448 −0.139724 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 0 0
\(463\) −23.4164 −1.08825 −0.544126 0.839003i \(-0.683139\pi\)
−0.544126 + 0.839003i \(0.683139\pi\)
\(464\) 6.00000 0.278543
\(465\) −1.90983 −0.0885662
\(466\) −35.1246 −1.62712
\(467\) 26.1803 1.21148 0.605741 0.795662i \(-0.292877\pi\)
0.605741 + 0.795662i \(0.292877\pi\)
\(468\) 9.70820 0.448762
\(469\) 0 0
\(470\) −6.18034 −0.285078
\(471\) 23.1246 1.06553
\(472\) 7.23607 0.333067
\(473\) 0 0
\(474\) 22.3607 1.02706
\(475\) −13.1803 −0.604755
\(476\) 14.5623 0.667462
\(477\) 6.76393 0.309699
\(478\) −42.5623 −1.94675
\(479\) −0.944272 −0.0431449 −0.0215724 0.999767i \(-0.506867\pi\)
−0.0215724 + 0.999767i \(0.506867\pi\)
\(480\) −4.14590 −0.189233
\(481\) 14.9443 0.681400
\(482\) 26.8328 1.22220
\(483\) −4.38197 −0.199386
\(484\) 0 0
\(485\) 3.70820 0.168381
\(486\) −2.23607 −0.101430
\(487\) 27.7082 1.25558 0.627789 0.778383i \(-0.283960\pi\)
0.627789 + 0.778383i \(0.283960\pi\)
\(488\) 0 0
\(489\) 10.0000 0.452216
\(490\) −1.38197 −0.0624309
\(491\) 1.50658 0.0679909 0.0339955 0.999422i \(-0.489177\pi\)
0.0339955 + 0.999422i \(0.489177\pi\)
\(492\) −22.1459 −0.998414
\(493\) −29.1246 −1.31171
\(494\) 20.6525 0.929199
\(495\) 0 0
\(496\) 3.09017 0.138753
\(497\) −4.94427 −0.221781
\(498\) −36.8328 −1.65052
\(499\) 21.1246 0.945668 0.472834 0.881152i \(-0.343231\pi\)
0.472834 + 0.881152i \(0.343231\pi\)
\(500\) 17.8328 0.797508
\(501\) 21.2361 0.948758
\(502\) 61.9574 2.76530
\(503\) −14.4721 −0.645281 −0.322640 0.946522i \(-0.604570\pi\)
−0.322640 + 0.946522i \(0.604570\pi\)
\(504\) 2.23607 0.0996024
\(505\) 4.20163 0.186970
\(506\) 0 0
\(507\) 2.52786 0.112266
\(508\) 0.875388 0.0388391
\(509\) −27.0902 −1.20075 −0.600375 0.799718i \(-0.704982\pi\)
−0.600375 + 0.799718i \(0.704982\pi\)
\(510\) 6.70820 0.297044
\(511\) 13.7082 0.606415
\(512\) 11.1803 0.494106
\(513\) −2.85410 −0.126012
\(514\) 54.6738 2.41156
\(515\) −0.819660 −0.0361185
\(516\) −29.1246 −1.28214
\(517\) 0 0
\(518\) 10.3262 0.453709
\(519\) 13.6180 0.597765
\(520\) −4.47214 −0.196116
\(521\) −12.3262 −0.540022 −0.270011 0.962857i \(-0.587027\pi\)
−0.270011 + 0.962857i \(0.587027\pi\)
\(522\) −13.4164 −0.587220
\(523\) 13.1459 0.574830 0.287415 0.957806i \(-0.407204\pi\)
0.287415 + 0.957806i \(0.407204\pi\)
\(524\) −49.9574 −2.18240
\(525\) 4.61803 0.201548
\(526\) 24.2705 1.05824
\(527\) −15.0000 −0.653410
\(528\) 0 0
\(529\) −3.79837 −0.165147
\(530\) −9.34752 −0.406031
\(531\) 3.23607 0.140433
\(532\) 8.56231 0.371223
\(533\) 23.8885 1.03473
\(534\) −12.5623 −0.543624
\(535\) −9.76393 −0.422132
\(536\) 0 0
\(537\) 14.7984 0.638597
\(538\) −30.0000 −1.29339
\(539\) 0 0
\(540\) 1.85410 0.0797878
\(541\) −16.3262 −0.701920 −0.350960 0.936390i \(-0.614145\pi\)
−0.350960 + 0.936390i \(0.614145\pi\)
\(542\) −50.3262 −2.16170
\(543\) 19.8885 0.853499
\(544\) −32.5623 −1.39610
\(545\) 10.8541 0.464939
\(546\) −7.23607 −0.309675
\(547\) −36.4721 −1.55944 −0.779718 0.626131i \(-0.784638\pi\)
−0.779718 + 0.626131i \(0.784638\pi\)
\(548\) 24.5410 1.04834
\(549\) 0 0
\(550\) 0 0
\(551\) −17.1246 −0.729533
\(552\) −9.79837 −0.417046
\(553\) −10.0000 −0.425243
\(554\) −0.201626 −0.00856627
\(555\) 2.85410 0.121150
\(556\) 39.4377 1.67253
\(557\) 22.3607 0.947452 0.473726 0.880672i \(-0.342909\pi\)
0.473726 + 0.880672i \(0.342909\pi\)
\(558\) −6.90983 −0.292516
\(559\) 31.4164 1.32877
\(560\) 0.618034 0.0261167
\(561\) 0 0
\(562\) −60.2492 −2.54146
\(563\) −9.52786 −0.401552 −0.200776 0.979637i \(-0.564346\pi\)
−0.200776 + 0.979637i \(0.564346\pi\)
\(564\) −13.4164 −0.564933
\(565\) 3.23607 0.136142
\(566\) −38.6180 −1.62324
\(567\) 1.00000 0.0419961
\(568\) −11.0557 −0.463888
\(569\) 11.5967 0.486161 0.243080 0.970006i \(-0.421842\pi\)
0.243080 + 0.970006i \(0.421842\pi\)
\(570\) 3.94427 0.165207
\(571\) 15.3475 0.642274 0.321137 0.947033i \(-0.395935\pi\)
0.321137 + 0.947033i \(0.395935\pi\)
\(572\) 0 0
\(573\) −4.90983 −0.205111
\(574\) 16.5066 0.688971
\(575\) −20.2361 −0.843902
\(576\) −13.0000 −0.541667
\(577\) −35.7082 −1.48655 −0.743276 0.668985i \(-0.766729\pi\)
−0.743276 + 0.668985i \(0.766729\pi\)
\(578\) 14.6738 0.610348
\(579\) 21.8541 0.908225
\(580\) 11.1246 0.461924
\(581\) 16.4721 0.683379
\(582\) 13.4164 0.556128
\(583\) 0 0
\(584\) 30.6525 1.26841
\(585\) −2.00000 −0.0826898
\(586\) −40.7295 −1.68252
\(587\) −1.41641 −0.0584614 −0.0292307 0.999573i \(-0.509306\pi\)
−0.0292307 + 0.999573i \(0.509306\pi\)
\(588\) −3.00000 −0.123718
\(589\) −8.81966 −0.363408
\(590\) −4.47214 −0.184115
\(591\) −22.4721 −0.924380
\(592\) −4.61803 −0.189800
\(593\) 22.5066 0.924234 0.462117 0.886819i \(-0.347090\pi\)
0.462117 + 0.886819i \(0.347090\pi\)
\(594\) 0 0
\(595\) −3.00000 −0.122988
\(596\) −14.8328 −0.607576
\(597\) −13.1459 −0.538026
\(598\) 31.7082 1.29664
\(599\) −23.6180 −0.965007 −0.482503 0.875894i \(-0.660272\pi\)
−0.482503 + 0.875894i \(0.660272\pi\)
\(600\) 10.3262 0.421567
\(601\) 3.52786 0.143905 0.0719523 0.997408i \(-0.477077\pi\)
0.0719523 + 0.997408i \(0.477077\pi\)
\(602\) 21.7082 0.884760
\(603\) 0 0
\(604\) −46.5836 −1.89546
\(605\) 0 0
\(606\) 15.2016 0.617524
\(607\) −11.9787 −0.486201 −0.243100 0.970001i \(-0.578164\pi\)
−0.243100 + 0.970001i \(0.578164\pi\)
\(608\) −19.1459 −0.776469
\(609\) 6.00000 0.243132
\(610\) 0 0
\(611\) 14.4721 0.585480
\(612\) 14.5623 0.588646
\(613\) −21.9787 −0.887712 −0.443856 0.896098i \(-0.646390\pi\)
−0.443856 + 0.896098i \(0.646390\pi\)
\(614\) −39.2705 −1.58483
\(615\) 4.56231 0.183970
\(616\) 0 0
\(617\) −14.0689 −0.566392 −0.283196 0.959062i \(-0.591395\pi\)
−0.283196 + 0.959062i \(0.591395\pi\)
\(618\) −2.96556 −0.119292
\(619\) −36.6869 −1.47457 −0.737286 0.675581i \(-0.763893\pi\)
−0.737286 + 0.675581i \(0.763893\pi\)
\(620\) 5.72949 0.230102
\(621\) −4.38197 −0.175842
\(622\) 46.1803 1.85166
\(623\) 5.61803 0.225082
\(624\) 3.23607 0.129546
\(625\) 19.4164 0.776656
\(626\) −17.4853 −0.698853
\(627\) 0 0
\(628\) −69.3738 −2.76832
\(629\) 22.4164 0.893801
\(630\) −1.38197 −0.0550588
\(631\) −45.3050 −1.80356 −0.901781 0.432194i \(-0.857740\pi\)
−0.901781 + 0.432194i \(0.857740\pi\)
\(632\) −22.3607 −0.889460
\(633\) 18.3607 0.729772
\(634\) 15.1246 0.600675
\(635\) −0.180340 −0.00715657
\(636\) −20.2918 −0.804622
\(637\) 3.23607 0.128218
\(638\) 0 0
\(639\) −4.94427 −0.195592
\(640\) 9.67376 0.382389
\(641\) 12.7639 0.504145 0.252073 0.967708i \(-0.418888\pi\)
0.252073 + 0.967708i \(0.418888\pi\)
\(642\) −35.3262 −1.39422
\(643\) 10.2705 0.405029 0.202515 0.979279i \(-0.435089\pi\)
0.202515 + 0.979279i \(0.435089\pi\)
\(644\) 13.1459 0.518021
\(645\) 6.00000 0.236250
\(646\) 30.9787 1.21884
\(647\) 36.4721 1.43387 0.716934 0.697141i \(-0.245545\pi\)
0.716934 + 0.697141i \(0.245545\pi\)
\(648\) 2.23607 0.0878410
\(649\) 0 0
\(650\) −33.4164 −1.31070
\(651\) 3.09017 0.121113
\(652\) −30.0000 −1.17489
\(653\) 16.3607 0.640243 0.320121 0.947377i \(-0.396276\pi\)
0.320121 + 0.947377i \(0.396276\pi\)
\(654\) 39.2705 1.53560
\(655\) 10.2918 0.402134
\(656\) −7.38197 −0.288217
\(657\) 13.7082 0.534808
\(658\) 10.0000 0.389841
\(659\) −25.8541 −1.00713 −0.503566 0.863957i \(-0.667979\pi\)
−0.503566 + 0.863957i \(0.667979\pi\)
\(660\) 0 0
\(661\) −32.5410 −1.26570 −0.632849 0.774275i \(-0.718115\pi\)
−0.632849 + 0.774275i \(0.718115\pi\)
\(662\) 38.9443 1.51361
\(663\) −15.7082 −0.610056
\(664\) 36.8328 1.42939
\(665\) −1.76393 −0.0684023
\(666\) 10.3262 0.400134
\(667\) −26.2918 −1.01802
\(668\) −63.7082 −2.46494
\(669\) −19.6180 −0.758477
\(670\) 0 0
\(671\) 0 0
\(672\) 6.70820 0.258775
\(673\) −44.8328 −1.72818 −0.864089 0.503339i \(-0.832105\pi\)
−0.864089 + 0.503339i \(0.832105\pi\)
\(674\) 60.5755 2.33328
\(675\) 4.61803 0.177748
\(676\) −7.58359 −0.291677
\(677\) 34.3607 1.32059 0.660294 0.751007i \(-0.270432\pi\)
0.660294 + 0.751007i \(0.270432\pi\)
\(678\) 11.7082 0.449651
\(679\) −6.00000 −0.230259
\(680\) −6.70820 −0.257248
\(681\) −12.4721 −0.477933
\(682\) 0 0
\(683\) 26.7984 1.02541 0.512706 0.858564i \(-0.328643\pi\)
0.512706 + 0.858564i \(0.328643\pi\)
\(684\) 8.56231 0.327388
\(685\) −5.05573 −0.193169
\(686\) 2.23607 0.0853735
\(687\) −16.1803 −0.617318
\(688\) −9.70820 −0.370122
\(689\) 21.8885 0.833887
\(690\) 6.05573 0.230538
\(691\) 48.7426 1.85426 0.927129 0.374743i \(-0.122269\pi\)
0.927129 + 0.374743i \(0.122269\pi\)
\(692\) −40.8541 −1.55304
\(693\) 0 0
\(694\) −32.4377 −1.23132
\(695\) −8.12461 −0.308184
\(696\) 13.4164 0.508548
\(697\) 35.8328 1.35726
\(698\) −48.5410 −1.83730
\(699\) 15.7082 0.594139
\(700\) −13.8541 −0.523636
\(701\) 39.5967 1.49555 0.747774 0.663953i \(-0.231123\pi\)
0.747774 + 0.663953i \(0.231123\pi\)
\(702\) −7.23607 −0.273108
\(703\) 13.1803 0.497106
\(704\) 0 0
\(705\) 2.76393 0.104096
\(706\) −40.2492 −1.51480
\(707\) −6.79837 −0.255679
\(708\) −9.70820 −0.364857
\(709\) −8.03444 −0.301740 −0.150870 0.988554i \(-0.548207\pi\)
−0.150870 + 0.988554i \(0.548207\pi\)
\(710\) 6.83282 0.256431
\(711\) −10.0000 −0.375029
\(712\) 12.5623 0.470792
\(713\) −13.5410 −0.507115
\(714\) −10.8541 −0.406205
\(715\) 0 0
\(716\) −44.3951 −1.65912
\(717\) 19.0344 0.710854
\(718\) 14.3951 0.537221
\(719\) 39.0132 1.45495 0.727473 0.686137i \(-0.240695\pi\)
0.727473 + 0.686137i \(0.240695\pi\)
\(720\) 0.618034 0.0230328
\(721\) 1.32624 0.0493917
\(722\) −24.2705 −0.903255
\(723\) −12.0000 −0.446285
\(724\) −59.6656 −2.21746
\(725\) 27.7082 1.02906
\(726\) 0 0
\(727\) −18.8541 −0.699260 −0.349630 0.936888i \(-0.613693\pi\)
−0.349630 + 0.936888i \(0.613693\pi\)
\(728\) 7.23607 0.268187
\(729\) 1.00000 0.0370370
\(730\) −18.9443 −0.701159
\(731\) 47.1246 1.74297
\(732\) 0 0
\(733\) −18.0689 −0.667389 −0.333695 0.942681i \(-0.608295\pi\)
−0.333695 + 0.942681i \(0.608295\pi\)
\(734\) 6.10333 0.225278
\(735\) 0.618034 0.0227965
\(736\) −29.3951 −1.08352
\(737\) 0 0
\(738\) 16.5066 0.607616
\(739\) 41.1246 1.51279 0.756397 0.654113i \(-0.226958\pi\)
0.756397 + 0.654113i \(0.226958\pi\)
\(740\) −8.56231 −0.314757
\(741\) −9.23607 −0.339295
\(742\) 15.1246 0.555242
\(743\) −16.3262 −0.598952 −0.299476 0.954104i \(-0.596812\pi\)
−0.299476 + 0.954104i \(0.596812\pi\)
\(744\) 6.90983 0.253327
\(745\) 3.05573 0.111953
\(746\) 36.5066 1.33660
\(747\) 16.4721 0.602684
\(748\) 0 0
\(749\) 15.7984 0.577260
\(750\) −13.2918 −0.485348
\(751\) 24.0000 0.875772 0.437886 0.899030i \(-0.355727\pi\)
0.437886 + 0.899030i \(0.355727\pi\)
\(752\) −4.47214 −0.163082
\(753\) −27.7082 −1.00974
\(754\) −43.4164 −1.58113
\(755\) 9.59675 0.349261
\(756\) −3.00000 −0.109109
\(757\) 10.5623 0.383894 0.191947 0.981405i \(-0.438520\pi\)
0.191947 + 0.981405i \(0.438520\pi\)
\(758\) 48.1378 1.74844
\(759\) 0 0
\(760\) −3.94427 −0.143074
\(761\) 1.41641 0.0513447 0.0256724 0.999670i \(-0.491827\pi\)
0.0256724 + 0.999670i \(0.491827\pi\)
\(762\) −0.652476 −0.0236367
\(763\) −17.5623 −0.635798
\(764\) 14.7295 0.532894
\(765\) −3.00000 −0.108465
\(766\) −46.5836 −1.68313
\(767\) 10.4721 0.378127
\(768\) 9.00000 0.324760
\(769\) 1.05573 0.0380705 0.0190353 0.999819i \(-0.493941\pi\)
0.0190353 + 0.999819i \(0.493941\pi\)
\(770\) 0 0
\(771\) −24.4508 −0.880576
\(772\) −65.5623 −2.35964
\(773\) 8.47214 0.304722 0.152361 0.988325i \(-0.451312\pi\)
0.152361 + 0.988325i \(0.451312\pi\)
\(774\) 21.7082 0.780285
\(775\) 14.2705 0.512612
\(776\) −13.4164 −0.481621
\(777\) −4.61803 −0.165671
\(778\) 34.0689 1.22143
\(779\) 21.0689 0.754871
\(780\) 6.00000 0.214834
\(781\) 0 0
\(782\) 47.5623 1.70082
\(783\) 6.00000 0.214423
\(784\) −1.00000 −0.0357143
\(785\) 14.2918 0.510096
\(786\) 37.2361 1.32817
\(787\) −25.8541 −0.921599 −0.460800 0.887504i \(-0.652437\pi\)
−0.460800 + 0.887504i \(0.652437\pi\)
\(788\) 67.4164 2.40161
\(789\) −10.8541 −0.386416
\(790\) 13.8197 0.491681
\(791\) −5.23607 −0.186173
\(792\) 0 0
\(793\) 0 0
\(794\) 68.2918 2.42359
\(795\) 4.18034 0.148261
\(796\) 39.4377 1.39783
\(797\) 41.5623 1.47221 0.736106 0.676866i \(-0.236662\pi\)
0.736106 + 0.676866i \(0.236662\pi\)
\(798\) −6.38197 −0.225919
\(799\) 21.7082 0.767981
\(800\) 30.9787 1.09526
\(801\) 5.61803 0.198503
\(802\) 16.5836 0.585587
\(803\) 0 0
\(804\) 0 0
\(805\) −2.70820 −0.0954516
\(806\) −22.3607 −0.787621
\(807\) 13.4164 0.472280
\(808\) −15.2016 −0.534791
\(809\) 7.23607 0.254407 0.127203 0.991877i \(-0.459400\pi\)
0.127203 + 0.991877i \(0.459400\pi\)
\(810\) −1.38197 −0.0485573
\(811\) 8.58359 0.301411 0.150705 0.988579i \(-0.451846\pi\)
0.150705 + 0.988579i \(0.451846\pi\)
\(812\) −18.0000 −0.631676
\(813\) 22.5066 0.789340
\(814\) 0 0
\(815\) 6.18034 0.216488
\(816\) 4.85410 0.169928
\(817\) 27.7082 0.969387
\(818\) −49.5967 −1.73411
\(819\) 3.23607 0.113077
\(820\) −13.6869 −0.477968
\(821\) −10.9443 −0.381958 −0.190979 0.981594i \(-0.561166\pi\)
−0.190979 + 0.981594i \(0.561166\pi\)
\(822\) −18.2918 −0.638000
\(823\) 4.83282 0.168461 0.0842307 0.996446i \(-0.473157\pi\)
0.0842307 + 0.996446i \(0.473157\pi\)
\(824\) 2.96556 0.103310
\(825\) 0 0
\(826\) 7.23607 0.251775
\(827\) 29.8673 1.03859 0.519293 0.854596i \(-0.326195\pi\)
0.519293 + 0.854596i \(0.326195\pi\)
\(828\) 13.1459 0.456852
\(829\) −30.2492 −1.05060 −0.525299 0.850917i \(-0.676047\pi\)
−0.525299 + 0.850917i \(0.676047\pi\)
\(830\) −22.7639 −0.790148
\(831\) 0.0901699 0.00312796
\(832\) −42.0689 −1.45848
\(833\) 4.85410 0.168185
\(834\) −29.3951 −1.01787
\(835\) 13.1246 0.454196
\(836\) 0 0
\(837\) 3.09017 0.106812
\(838\) −7.23607 −0.249966
\(839\) −40.2492 −1.38956 −0.694779 0.719224i \(-0.744498\pi\)
−0.694779 + 0.719224i \(0.744498\pi\)
\(840\) 1.38197 0.0476824
\(841\) 7.00000 0.241379
\(842\) 21.9098 0.755063
\(843\) 26.9443 0.928010
\(844\) −55.0820 −1.89600
\(845\) 1.56231 0.0537450
\(846\) 10.0000 0.343807
\(847\) 0 0
\(848\) −6.76393 −0.232274
\(849\) 17.2705 0.592722
\(850\) −50.1246 −1.71926
\(851\) 20.2361 0.693683
\(852\) 14.8328 0.508164
\(853\) −41.7082 −1.42806 −0.714031 0.700114i \(-0.753132\pi\)
−0.714031 + 0.700114i \(0.753132\pi\)
\(854\) 0 0
\(855\) −1.76393 −0.0603252
\(856\) 35.3262 1.20743
\(857\) −34.3607 −1.17374 −0.586869 0.809682i \(-0.699640\pi\)
−0.586869 + 0.809682i \(0.699640\pi\)
\(858\) 0 0
\(859\) 12.0000 0.409435 0.204717 0.978821i \(-0.434372\pi\)
0.204717 + 0.978821i \(0.434372\pi\)
\(860\) −18.0000 −0.613795
\(861\) −7.38197 −0.251577
\(862\) 33.7426 1.14928
\(863\) −12.3262 −0.419590 −0.209795 0.977745i \(-0.567280\pi\)
−0.209795 + 0.977745i \(0.567280\pi\)
\(864\) 6.70820 0.228218
\(865\) 8.41641 0.286166
\(866\) −69.8460 −2.37346
\(867\) −6.56231 −0.222868
\(868\) −9.27051 −0.314662
\(869\) 0 0
\(870\) −8.29180 −0.281118
\(871\) 0 0
\(872\) −39.2705 −1.32987
\(873\) −6.00000 −0.203069
\(874\) 27.9656 0.945949
\(875\) 5.94427 0.200953
\(876\) −41.1246 −1.38947
\(877\) 12.4721 0.421154 0.210577 0.977577i \(-0.432466\pi\)
0.210577 + 0.977577i \(0.432466\pi\)
\(878\) 4.14590 0.139917
\(879\) 18.2148 0.614369
\(880\) 0 0
\(881\) 11.3262 0.381591 0.190795 0.981630i \(-0.438893\pi\)
0.190795 + 0.981630i \(0.438893\pi\)
\(882\) 2.23607 0.0752923
\(883\) 19.5967 0.659483 0.329742 0.944071i \(-0.393038\pi\)
0.329742 + 0.944071i \(0.393038\pi\)
\(884\) 47.1246 1.58497
\(885\) 2.00000 0.0672293
\(886\) 45.5755 1.53114
\(887\) 27.5967 0.926608 0.463304 0.886199i \(-0.346664\pi\)
0.463304 + 0.886199i \(0.346664\pi\)
\(888\) −10.3262 −0.346526
\(889\) 0.291796 0.00978653
\(890\) −7.76393 −0.260248
\(891\) 0 0
\(892\) 58.8541 1.97058
\(893\) 12.7639 0.427129
\(894\) 11.0557 0.369759
\(895\) 9.14590 0.305714
\(896\) −15.6525 −0.522913
\(897\) −14.1803 −0.473468
\(898\) −93.6656 −3.12566
\(899\) 18.5410 0.618378
\(900\) −13.8541 −0.461803
\(901\) 32.8328 1.09382
\(902\) 0 0
\(903\) −9.70820 −0.323069
\(904\) −11.7082 −0.389409
\(905\) 12.2918 0.408593
\(906\) 34.7214 1.15354
\(907\) 16.2918 0.540960 0.270480 0.962726i \(-0.412818\pi\)
0.270480 + 0.962726i \(0.412818\pi\)
\(908\) 37.4164 1.24171
\(909\) −6.79837 −0.225488
\(910\) −4.47214 −0.148250
\(911\) −0.944272 −0.0312851 −0.0156426 0.999878i \(-0.504979\pi\)
−0.0156426 + 0.999878i \(0.504979\pi\)
\(912\) 2.85410 0.0945088
\(913\) 0 0
\(914\) 47.8885 1.58401
\(915\) 0 0
\(916\) 48.5410 1.60384
\(917\) −16.6525 −0.549913
\(918\) −10.8541 −0.358239
\(919\) −30.4721 −1.00518 −0.502592 0.864524i \(-0.667620\pi\)
−0.502592 + 0.864524i \(0.667620\pi\)
\(920\) −6.05573 −0.199651
\(921\) 17.5623 0.578698
\(922\) −13.4164 −0.441846
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) −21.3262 −0.701202
\(926\) −52.3607 −1.72068
\(927\) 1.32624 0.0435594
\(928\) 40.2492 1.32125
\(929\) 23.7295 0.778539 0.389270 0.921124i \(-0.372727\pi\)
0.389270 + 0.921124i \(0.372727\pi\)
\(930\) −4.27051 −0.140036
\(931\) 2.85410 0.0935394
\(932\) −47.1246 −1.54362
\(933\) −20.6525 −0.676132
\(934\) 58.5410 1.91552
\(935\) 0 0
\(936\) 7.23607 0.236518
\(937\) 34.0689 1.11298 0.556491 0.830854i \(-0.312147\pi\)
0.556491 + 0.830854i \(0.312147\pi\)
\(938\) 0 0
\(939\) 7.81966 0.255185
\(940\) −8.29180 −0.270449
\(941\) −0.493422 −0.0160851 −0.00804255 0.999968i \(-0.502560\pi\)
−0.00804255 + 0.999968i \(0.502560\pi\)
\(942\) 51.7082 1.68474
\(943\) 32.3475 1.05338
\(944\) −3.23607 −0.105325
\(945\) 0.618034 0.0201046
\(946\) 0 0
\(947\) 13.8541 0.450198 0.225099 0.974336i \(-0.427729\pi\)
0.225099 + 0.974336i \(0.427729\pi\)
\(948\) 30.0000 0.974355
\(949\) 44.3607 1.44001
\(950\) −29.4721 −0.956202
\(951\) −6.76393 −0.219336
\(952\) 10.8541 0.351783
\(953\) 11.8885 0.385108 0.192554 0.981286i \(-0.438323\pi\)
0.192554 + 0.981286i \(0.438323\pi\)
\(954\) 15.1246 0.489677
\(955\) −3.03444 −0.0981922
\(956\) −57.1033 −1.84685
\(957\) 0 0
\(958\) −2.11146 −0.0682181
\(959\) 8.18034 0.264157
\(960\) −8.03444 −0.259310
\(961\) −21.4508 −0.691963
\(962\) 33.4164 1.07739
\(963\) 15.7984 0.509095
\(964\) 36.0000 1.15948
\(965\) 13.5066 0.434792
\(966\) −9.79837 −0.315258
\(967\) −3.63932 −0.117033 −0.0585163 0.998286i \(-0.518637\pi\)
−0.0585163 + 0.998286i \(0.518637\pi\)
\(968\) 0 0
\(969\) −13.8541 −0.445058
\(970\) 8.29180 0.266234
\(971\) 54.5410 1.75030 0.875152 0.483848i \(-0.160761\pi\)
0.875152 + 0.483848i \(0.160761\pi\)
\(972\) −3.00000 −0.0962250
\(973\) 13.1459 0.421438
\(974\) 61.9574 1.98524
\(975\) 14.9443 0.478600
\(976\) 0 0
\(977\) −13.6393 −0.436361 −0.218180 0.975908i \(-0.570012\pi\)
−0.218180 + 0.975908i \(0.570012\pi\)
\(978\) 22.3607 0.715016
\(979\) 0 0
\(980\) −1.85410 −0.0592271
\(981\) −17.5623 −0.560721
\(982\) 3.36881 0.107503
\(983\) −10.9443 −0.349068 −0.174534 0.984651i \(-0.555842\pi\)
−0.174534 + 0.984651i \(0.555842\pi\)
\(984\) −16.5066 −0.526210
\(985\) −13.8885 −0.442526
\(986\) −65.1246 −2.07399
\(987\) −4.47214 −0.142350
\(988\) 27.7082 0.881515
\(989\) 42.5410 1.35273
\(990\) 0 0
\(991\) −33.2361 −1.05578 −0.527889 0.849313i \(-0.677016\pi\)
−0.527889 + 0.849313i \(0.677016\pi\)
\(992\) 20.7295 0.658162
\(993\) −17.4164 −0.552693
\(994\) −11.0557 −0.350666
\(995\) −8.12461 −0.257568
\(996\) −49.4164 −1.56582
\(997\) −44.4296 −1.40710 −0.703549 0.710647i \(-0.748403\pi\)
−0.703549 + 0.710647i \(0.748403\pi\)
\(998\) 47.2361 1.49523
\(999\) −4.61803 −0.146108
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 2541.2.a.w.1.2 2
3.2 odd 2 7623.2.a.bk.1.1 2
11.3 even 5 231.2.j.e.64.1 4
11.4 even 5 231.2.j.e.148.1 yes 4
11.10 odd 2 2541.2.a.v.1.1 2
33.14 odd 10 693.2.m.a.64.1 4
33.26 odd 10 693.2.m.a.379.1 4
33.32 even 2 7623.2.a.bj.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.j.e.64.1 4 11.3 even 5
231.2.j.e.148.1 yes 4 11.4 even 5
693.2.m.a.64.1 4 33.14 odd 10
693.2.m.a.379.1 4 33.26 odd 10
2541.2.a.v.1.1 2 11.10 odd 2
2541.2.a.w.1.2 2 1.1 even 1 trivial
7623.2.a.bj.1.2 2 33.32 even 2
7623.2.a.bk.1.1 2 3.2 odd 2