Properties

Label 103.2.a.a.1.1
Level $103$
Weight $2$
Character 103.1
Self dual yes
Analytic conductor $0.822$
Analytic rank $1$
Dimension $2$
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] = [103,2,Mod(1,103)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(103, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("103.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 103.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: \(0.822459140819\)
Analytic rank: \(1\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 103.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.61803 q^{2} -1.00000 q^{3} +4.85410 q^{4} -0.381966 q^{5} +2.61803 q^{6} -1.00000 q^{7} -7.47214 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q-2.61803 q^{2} -1.00000 q^{3} +4.85410 q^{4} -0.381966 q^{5} +2.61803 q^{6} -1.00000 q^{7} -7.47214 q^{8} -2.00000 q^{9} +1.00000 q^{10} -2.61803 q^{11} -4.85410 q^{12} -4.85410 q^{13} +2.61803 q^{14} +0.381966 q^{15} +9.85410 q^{16} -5.61803 q^{17} +5.23607 q^{18} +5.85410 q^{19} -1.85410 q^{20} +1.00000 q^{21} +6.85410 q^{22} +4.47214 q^{23} +7.47214 q^{24} -4.85410 q^{25} +12.7082 q^{26} +5.00000 q^{27} -4.85410 q^{28} -5.23607 q^{29} -1.00000 q^{30} -6.70820 q^{31} -10.8541 q^{32} +2.61803 q^{33} +14.7082 q^{34} +0.381966 q^{35} -9.70820 q^{36} +6.70820 q^{37} -15.3262 q^{38} +4.85410 q^{39} +2.85410 q^{40} +8.94427 q^{41} -2.61803 q^{42} -8.70820 q^{43} -12.7082 q^{44} +0.763932 q^{45} -11.7082 q^{46} +4.09017 q^{47} -9.85410 q^{48} -6.00000 q^{49} +12.7082 q^{50} +5.61803 q^{51} -23.5623 q^{52} +1.09017 q^{53} -13.0902 q^{54} +1.00000 q^{55} +7.47214 q^{56} -5.85410 q^{57} +13.7082 q^{58} +6.38197 q^{59} +1.85410 q^{60} +4.14590 q^{61} +17.5623 q^{62} +2.00000 q^{63} +8.70820 q^{64} +1.85410 q^{65} -6.85410 q^{66} +14.4164 q^{67} -27.2705 q^{68} -4.47214 q^{69} -1.00000 q^{70} -4.09017 q^{71} +14.9443 q^{72} -10.8541 q^{73} -17.5623 q^{74} +4.85410 q^{75} +28.4164 q^{76} +2.61803 q^{77} -12.7082 q^{78} -6.56231 q^{79} -3.76393 q^{80} +1.00000 q^{81} -23.4164 q^{82} -6.32624 q^{83} +4.85410 q^{84} +2.14590 q^{85} +22.7984 q^{86} +5.23607 q^{87} +19.5623 q^{88} -2.29180 q^{89} -2.00000 q^{90} +4.85410 q^{91} +21.7082 q^{92} +6.70820 q^{93} -10.7082 q^{94} -2.23607 q^{95} +10.8541 q^{96} -1.70820 q^{97} +15.7082 q^{98} +5.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} - 2 q^{3} + 3 q^{4} - 3 q^{5} + 3 q^{6} - 2 q^{7} - 6 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} - 2 q^{3} + 3 q^{4} - 3 q^{5} + 3 q^{6} - 2 q^{7} - 6 q^{8} - 4 q^{9} + 2 q^{10} - 3 q^{11} - 3 q^{12} - 3 q^{13} + 3 q^{14} + 3 q^{15} + 13 q^{16} - 9 q^{17} + 6 q^{18} + 5 q^{19} + 3 q^{20} + 2 q^{21} + 7 q^{22} + 6 q^{24} - 3 q^{25} + 12 q^{26} + 10 q^{27} - 3 q^{28} - 6 q^{29} - 2 q^{30} - 15 q^{32} + 3 q^{33} + 16 q^{34} + 3 q^{35} - 6 q^{36} - 15 q^{38} + 3 q^{39} - q^{40} - 3 q^{42} - 4 q^{43} - 12 q^{44} + 6 q^{45} - 10 q^{46} - 3 q^{47} - 13 q^{48} - 12 q^{49} + 12 q^{50} + 9 q^{51} - 27 q^{52} - 9 q^{53} - 15 q^{54} + 2 q^{55} + 6 q^{56} - 5 q^{57} + 14 q^{58} + 15 q^{59} - 3 q^{60} + 15 q^{61} + 15 q^{62} + 4 q^{63} + 4 q^{64} - 3 q^{65} - 7 q^{66} + 2 q^{67} - 21 q^{68} - 2 q^{70} + 3 q^{71} + 12 q^{72} - 15 q^{73} - 15 q^{74} + 3 q^{75} + 30 q^{76} + 3 q^{77} - 12 q^{78} + 7 q^{79} - 12 q^{80} + 2 q^{81} - 20 q^{82} + 3 q^{83} + 3 q^{84} + 11 q^{85} + 21 q^{86} + 6 q^{87} + 19 q^{88} - 18 q^{89} - 4 q^{90} + 3 q^{91} + 30 q^{92} - 8 q^{94} + 15 q^{96} + 10 q^{97} + 18 q^{98} + 6 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.61803 −1.85123 −0.925615 0.378467i \(-0.876451\pi\)
−0.925615 + 0.378467i \(0.876451\pi\)
\(3\) −1.00000 −0.577350 −0.288675 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 4.85410 2.42705
\(5\) −0.381966 −0.170820 −0.0854102 0.996346i \(-0.527220\pi\)
−0.0854102 + 0.996346i \(0.527220\pi\)
\(6\) 2.61803 1.06881
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) −7.47214 −2.64180
\(9\) −2.00000 −0.666667
\(10\) 1.00000 0.316228
\(11\) −2.61803 −0.789367 −0.394683 0.918817i \(-0.629146\pi\)
−0.394683 + 0.918817i \(0.629146\pi\)
\(12\) −4.85410 −1.40126
\(13\) −4.85410 −1.34629 −0.673143 0.739512i \(-0.735056\pi\)
−0.673143 + 0.739512i \(0.735056\pi\)
\(14\) 2.61803 0.699699
\(15\) 0.381966 0.0986232
\(16\) 9.85410 2.46353
\(17\) −5.61803 −1.36257 −0.681287 0.732017i \(-0.738579\pi\)
−0.681287 + 0.732017i \(0.738579\pi\)
\(18\) 5.23607 1.23415
\(19\) 5.85410 1.34302 0.671512 0.740994i \(-0.265645\pi\)
0.671512 + 0.740994i \(0.265645\pi\)
\(20\) −1.85410 −0.414590
\(21\) 1.00000 0.218218
\(22\) 6.85410 1.46130
\(23\) 4.47214 0.932505 0.466252 0.884652i \(-0.345604\pi\)
0.466252 + 0.884652i \(0.345604\pi\)
\(24\) 7.47214 1.52524
\(25\) −4.85410 −0.970820
\(26\) 12.7082 2.49228
\(27\) 5.00000 0.962250
\(28\) −4.85410 −0.917339
\(29\) −5.23607 −0.972313 −0.486157 0.873872i \(-0.661602\pi\)
−0.486157 + 0.873872i \(0.661602\pi\)
\(30\) −1.00000 −0.182574
\(31\) −6.70820 −1.20483 −0.602414 0.798183i \(-0.705795\pi\)
−0.602414 + 0.798183i \(0.705795\pi\)
\(32\) −10.8541 −1.91875
\(33\) 2.61803 0.455741
\(34\) 14.7082 2.52244
\(35\) 0.381966 0.0645640
\(36\) −9.70820 −1.61803
\(37\) 6.70820 1.10282 0.551411 0.834234i \(-0.314090\pi\)
0.551411 + 0.834234i \(0.314090\pi\)
\(38\) −15.3262 −2.48624
\(39\) 4.85410 0.777278
\(40\) 2.85410 0.451273
\(41\) 8.94427 1.39686 0.698430 0.715678i \(-0.253882\pi\)
0.698430 + 0.715678i \(0.253882\pi\)
\(42\) −2.61803 −0.403971
\(43\) −8.70820 −1.32799 −0.663994 0.747738i \(-0.731140\pi\)
−0.663994 + 0.747738i \(0.731140\pi\)
\(44\) −12.7082 −1.91583
\(45\) 0.763932 0.113880
\(46\) −11.7082 −1.72628
\(47\) 4.09017 0.596613 0.298306 0.954470i \(-0.403578\pi\)
0.298306 + 0.954470i \(0.403578\pi\)
\(48\) −9.85410 −1.42232
\(49\) −6.00000 −0.857143
\(50\) 12.7082 1.79721
\(51\) 5.61803 0.786682
\(52\) −23.5623 −3.26750
\(53\) 1.09017 0.149746 0.0748732 0.997193i \(-0.476145\pi\)
0.0748732 + 0.997193i \(0.476145\pi\)
\(54\) −13.0902 −1.78135
\(55\) 1.00000 0.134840
\(56\) 7.47214 0.998506
\(57\) −5.85410 −0.775395
\(58\) 13.7082 1.79998
\(59\) 6.38197 0.830861 0.415431 0.909625i \(-0.363631\pi\)
0.415431 + 0.909625i \(0.363631\pi\)
\(60\) 1.85410 0.239364
\(61\) 4.14590 0.530828 0.265414 0.964135i \(-0.414491\pi\)
0.265414 + 0.964135i \(0.414491\pi\)
\(62\) 17.5623 2.23042
\(63\) 2.00000 0.251976
\(64\) 8.70820 1.08853
\(65\) 1.85410 0.229973
\(66\) −6.85410 −0.843682
\(67\) 14.4164 1.76124 0.880622 0.473819i \(-0.157125\pi\)
0.880622 + 0.473819i \(0.157125\pi\)
\(68\) −27.2705 −3.30704
\(69\) −4.47214 −0.538382
\(70\) −1.00000 −0.119523
\(71\) −4.09017 −0.485414 −0.242707 0.970100i \(-0.578035\pi\)
−0.242707 + 0.970100i \(0.578035\pi\)
\(72\) 14.9443 1.76120
\(73\) −10.8541 −1.27038 −0.635188 0.772357i \(-0.719077\pi\)
−0.635188 + 0.772357i \(0.719077\pi\)
\(74\) −17.5623 −2.04158
\(75\) 4.85410 0.560503
\(76\) 28.4164 3.25959
\(77\) 2.61803 0.298353
\(78\) −12.7082 −1.43892
\(79\) −6.56231 −0.738317 −0.369159 0.929366i \(-0.620354\pi\)
−0.369159 + 0.929366i \(0.620354\pi\)
\(80\) −3.76393 −0.420820
\(81\) 1.00000 0.111111
\(82\) −23.4164 −2.58591
\(83\) −6.32624 −0.694395 −0.347197 0.937792i \(-0.612867\pi\)
−0.347197 + 0.937792i \(0.612867\pi\)
\(84\) 4.85410 0.529626
\(85\) 2.14590 0.232755
\(86\) 22.7984 2.45841
\(87\) 5.23607 0.561365
\(88\) 19.5623 2.08535
\(89\) −2.29180 −0.242930 −0.121465 0.992596i \(-0.538759\pi\)
−0.121465 + 0.992596i \(0.538759\pi\)
\(90\) −2.00000 −0.210819
\(91\) 4.85410 0.508848
\(92\) 21.7082 2.26324
\(93\) 6.70820 0.695608
\(94\) −10.7082 −1.10447
\(95\) −2.23607 −0.229416
\(96\) 10.8541 1.10779
\(97\) −1.70820 −0.173442 −0.0867209 0.996233i \(-0.527639\pi\)
−0.0867209 + 0.996233i \(0.527639\pi\)
\(98\) 15.7082 1.58677
\(99\) 5.23607 0.526245
\(100\) −23.5623 −2.35623
\(101\) 1.90983 0.190035 0.0950176 0.995476i \(-0.469709\pi\)
0.0950176 + 0.995476i \(0.469709\pi\)
\(102\) −14.7082 −1.45633
\(103\) −1.00000 −0.0985329
\(104\) 36.2705 3.55662
\(105\) −0.381966 −0.0372761
\(106\) −2.85410 −0.277215
\(107\) −7.09017 −0.685433 −0.342716 0.939439i \(-0.611347\pi\)
−0.342716 + 0.939439i \(0.611347\pi\)
\(108\) 24.2705 2.33543
\(109\) −9.56231 −0.915903 −0.457951 0.888977i \(-0.651417\pi\)
−0.457951 + 0.888977i \(0.651417\pi\)
\(110\) −2.61803 −0.249620
\(111\) −6.70820 −0.636715
\(112\) −9.85410 −0.931125
\(113\) −15.0000 −1.41108 −0.705541 0.708669i \(-0.749296\pi\)
−0.705541 + 0.708669i \(0.749296\pi\)
\(114\) 15.3262 1.43543
\(115\) −1.70820 −0.159291
\(116\) −25.4164 −2.35985
\(117\) 9.70820 0.897524
\(118\) −16.7082 −1.53811
\(119\) 5.61803 0.515004
\(120\) −2.85410 −0.260543
\(121\) −4.14590 −0.376900
\(122\) −10.8541 −0.982684
\(123\) −8.94427 −0.806478
\(124\) −32.5623 −2.92418
\(125\) 3.76393 0.336656
\(126\) −5.23607 −0.466466
\(127\) 15.2705 1.35504 0.677519 0.735505i \(-0.263055\pi\)
0.677519 + 0.735505i \(0.263055\pi\)
\(128\) −1.09017 −0.0963583
\(129\) 8.70820 0.766715
\(130\) −4.85410 −0.425733
\(131\) −2.23607 −0.195366 −0.0976831 0.995218i \(-0.531143\pi\)
−0.0976831 + 0.995218i \(0.531143\pi\)
\(132\) 12.7082 1.10611
\(133\) −5.85410 −0.507615
\(134\) −37.7426 −3.26047
\(135\) −1.90983 −0.164372
\(136\) 41.9787 3.59965
\(137\) −12.7082 −1.08574 −0.542868 0.839818i \(-0.682661\pi\)
−0.542868 + 0.839818i \(0.682661\pi\)
\(138\) 11.7082 0.996669
\(139\) −11.1459 −0.945383 −0.472691 0.881228i \(-0.656717\pi\)
−0.472691 + 0.881228i \(0.656717\pi\)
\(140\) 1.85410 0.156700
\(141\) −4.09017 −0.344454
\(142\) 10.7082 0.898613
\(143\) 12.7082 1.06271
\(144\) −19.7082 −1.64235
\(145\) 2.00000 0.166091
\(146\) 28.4164 2.35176
\(147\) 6.00000 0.494872
\(148\) 32.5623 2.67661
\(149\) −7.47214 −0.612141 −0.306071 0.952009i \(-0.599014\pi\)
−0.306071 + 0.952009i \(0.599014\pi\)
\(150\) −12.7082 −1.03762
\(151\) −19.0000 −1.54620 −0.773099 0.634285i \(-0.781294\pi\)
−0.773099 + 0.634285i \(0.781294\pi\)
\(152\) −43.7426 −3.54800
\(153\) 11.2361 0.908382
\(154\) −6.85410 −0.552319
\(155\) 2.56231 0.205809
\(156\) 23.5623 1.88649
\(157\) 16.7082 1.33346 0.666730 0.745299i \(-0.267693\pi\)
0.666730 + 0.745299i \(0.267693\pi\)
\(158\) 17.1803 1.36679
\(159\) −1.09017 −0.0864561
\(160\) 4.14590 0.327762
\(161\) −4.47214 −0.352454
\(162\) −2.61803 −0.205692
\(163\) 2.70820 0.212123 0.106061 0.994360i \(-0.466176\pi\)
0.106061 + 0.994360i \(0.466176\pi\)
\(164\) 43.4164 3.39025
\(165\) −1.00000 −0.0778499
\(166\) 16.5623 1.28548
\(167\) 9.00000 0.696441 0.348220 0.937413i \(-0.386786\pi\)
0.348220 + 0.937413i \(0.386786\pi\)
\(168\) −7.47214 −0.576488
\(169\) 10.5623 0.812485
\(170\) −5.61803 −0.430884
\(171\) −11.7082 −0.895349
\(172\) −42.2705 −3.22310
\(173\) 16.0344 1.21908 0.609538 0.792757i \(-0.291355\pi\)
0.609538 + 0.792757i \(0.291355\pi\)
\(174\) −13.7082 −1.03922
\(175\) 4.85410 0.366936
\(176\) −25.7984 −1.94463
\(177\) −6.38197 −0.479698
\(178\) 6.00000 0.449719
\(179\) −7.85410 −0.587043 −0.293522 0.955952i \(-0.594827\pi\)
−0.293522 + 0.955952i \(0.594827\pi\)
\(180\) 3.70820 0.276393
\(181\) 3.85410 0.286473 0.143237 0.989688i \(-0.454249\pi\)
0.143237 + 0.989688i \(0.454249\pi\)
\(182\) −12.7082 −0.941995
\(183\) −4.14590 −0.306474
\(184\) −33.4164 −2.46349
\(185\) −2.56231 −0.188384
\(186\) −17.5623 −1.28773
\(187\) 14.7082 1.07557
\(188\) 19.8541 1.44801
\(189\) −5.00000 −0.363696
\(190\) 5.85410 0.424701
\(191\) 5.61803 0.406507 0.203253 0.979126i \(-0.434848\pi\)
0.203253 + 0.979126i \(0.434848\pi\)
\(192\) −8.70820 −0.628460
\(193\) 20.1246 1.44860 0.724301 0.689484i \(-0.242163\pi\)
0.724301 + 0.689484i \(0.242163\pi\)
\(194\) 4.47214 0.321081
\(195\) −1.85410 −0.132775
\(196\) −29.1246 −2.08033
\(197\) −16.4164 −1.16962 −0.584810 0.811170i \(-0.698831\pi\)
−0.584810 + 0.811170i \(0.698831\pi\)
\(198\) −13.7082 −0.974200
\(199\) −23.4164 −1.65995 −0.829973 0.557804i \(-0.811644\pi\)
−0.829973 + 0.557804i \(0.811644\pi\)
\(200\) 36.2705 2.56471
\(201\) −14.4164 −1.01686
\(202\) −5.00000 −0.351799
\(203\) 5.23607 0.367500
\(204\) 27.2705 1.90932
\(205\) −3.41641 −0.238612
\(206\) 2.61803 0.182407
\(207\) −8.94427 −0.621670
\(208\) −47.8328 −3.31661
\(209\) −15.3262 −1.06014
\(210\) 1.00000 0.0690066
\(211\) 14.8541 1.02260 0.511299 0.859403i \(-0.329164\pi\)
0.511299 + 0.859403i \(0.329164\pi\)
\(212\) 5.29180 0.363442
\(213\) 4.09017 0.280254
\(214\) 18.5623 1.26889
\(215\) 3.32624 0.226848
\(216\) −37.3607 −2.54207
\(217\) 6.70820 0.455383
\(218\) 25.0344 1.69555
\(219\) 10.8541 0.733452
\(220\) 4.85410 0.327263
\(221\) 27.2705 1.83441
\(222\) 17.5623 1.17870
\(223\) 7.70820 0.516180 0.258090 0.966121i \(-0.416907\pi\)
0.258090 + 0.966121i \(0.416907\pi\)
\(224\) 10.8541 0.725220
\(225\) 9.70820 0.647214
\(226\) 39.2705 2.61224
\(227\) 2.94427 0.195418 0.0977091 0.995215i \(-0.468849\pi\)
0.0977091 + 0.995215i \(0.468849\pi\)
\(228\) −28.4164 −1.88192
\(229\) 6.70820 0.443291 0.221645 0.975127i \(-0.428857\pi\)
0.221645 + 0.975127i \(0.428857\pi\)
\(230\) 4.47214 0.294884
\(231\) −2.61803 −0.172254
\(232\) 39.1246 2.56866
\(233\) −26.8885 −1.76153 −0.880764 0.473556i \(-0.842970\pi\)
−0.880764 + 0.473556i \(0.842970\pi\)
\(234\) −25.4164 −1.66152
\(235\) −1.56231 −0.101914
\(236\) 30.9787 2.01654
\(237\) 6.56231 0.426268
\(238\) −14.7082 −0.953391
\(239\) 8.67376 0.561059 0.280530 0.959845i \(-0.409490\pi\)
0.280530 + 0.959845i \(0.409490\pi\)
\(240\) 3.76393 0.242961
\(241\) −8.27051 −0.532750 −0.266375 0.963869i \(-0.585826\pi\)
−0.266375 + 0.963869i \(0.585826\pi\)
\(242\) 10.8541 0.697728
\(243\) −16.0000 −1.02640
\(244\) 20.1246 1.28835
\(245\) 2.29180 0.146417
\(246\) 23.4164 1.49298
\(247\) −28.4164 −1.80809
\(248\) 50.1246 3.18292
\(249\) 6.32624 0.400909
\(250\) −9.85410 −0.623228
\(251\) 11.2361 0.709214 0.354607 0.935015i \(-0.384615\pi\)
0.354607 + 0.935015i \(0.384615\pi\)
\(252\) 9.70820 0.611559
\(253\) −11.7082 −0.736088
\(254\) −39.9787 −2.50849
\(255\) −2.14590 −0.134381
\(256\) −14.5623 −0.910144
\(257\) −13.4721 −0.840369 −0.420184 0.907439i \(-0.638035\pi\)
−0.420184 + 0.907439i \(0.638035\pi\)
\(258\) −22.7984 −1.41936
\(259\) −6.70820 −0.416828
\(260\) 9.00000 0.558156
\(261\) 10.4721 0.648209
\(262\) 5.85410 0.361668
\(263\) −20.6180 −1.27136 −0.635681 0.771952i \(-0.719281\pi\)
−0.635681 + 0.771952i \(0.719281\pi\)
\(264\) −19.5623 −1.20398
\(265\) −0.416408 −0.0255797
\(266\) 15.3262 0.939712
\(267\) 2.29180 0.140256
\(268\) 69.9787 4.27463
\(269\) 12.3262 0.751544 0.375772 0.926712i \(-0.377378\pi\)
0.375772 + 0.926712i \(0.377378\pi\)
\(270\) 5.00000 0.304290
\(271\) 1.00000 0.0607457 0.0303728 0.999539i \(-0.490331\pi\)
0.0303728 + 0.999539i \(0.490331\pi\)
\(272\) −55.3607 −3.35673
\(273\) −4.85410 −0.293784
\(274\) 33.2705 2.00995
\(275\) 12.7082 0.766334
\(276\) −21.7082 −1.30668
\(277\) 4.70820 0.282889 0.141444 0.989946i \(-0.454825\pi\)
0.141444 + 0.989946i \(0.454825\pi\)
\(278\) 29.1803 1.75012
\(279\) 13.4164 0.803219
\(280\) −2.85410 −0.170565
\(281\) −31.4721 −1.87747 −0.938735 0.344640i \(-0.888001\pi\)
−0.938735 + 0.344640i \(0.888001\pi\)
\(282\) 10.7082 0.637664
\(283\) −19.7082 −1.17153 −0.585766 0.810481i \(-0.699206\pi\)
−0.585766 + 0.810481i \(0.699206\pi\)
\(284\) −19.8541 −1.17812
\(285\) 2.23607 0.132453
\(286\) −33.2705 −1.96733
\(287\) −8.94427 −0.527964
\(288\) 21.7082 1.27917
\(289\) 14.5623 0.856606
\(290\) −5.23607 −0.307472
\(291\) 1.70820 0.100137
\(292\) −52.6869 −3.08327
\(293\) −21.6525 −1.26495 −0.632476 0.774580i \(-0.717961\pi\)
−0.632476 + 0.774580i \(0.717961\pi\)
\(294\) −15.7082 −0.916121
\(295\) −2.43769 −0.141928
\(296\) −50.1246 −2.91343
\(297\) −13.0902 −0.759569
\(298\) 19.5623 1.13321
\(299\) −21.7082 −1.25542
\(300\) 23.5623 1.36037
\(301\) 8.70820 0.501933
\(302\) 49.7426 2.86237
\(303\) −1.90983 −0.109717
\(304\) 57.6869 3.30857
\(305\) −1.58359 −0.0906762
\(306\) −29.4164 −1.68162
\(307\) −2.85410 −0.162892 −0.0814461 0.996678i \(-0.525954\pi\)
−0.0814461 + 0.996678i \(0.525954\pi\)
\(308\) 12.7082 0.724117
\(309\) 1.00000 0.0568880
\(310\) −6.70820 −0.381000
\(311\) 2.88854 0.163794 0.0818971 0.996641i \(-0.473902\pi\)
0.0818971 + 0.996641i \(0.473902\pi\)
\(312\) −36.2705 −2.05341
\(313\) 16.2918 0.920867 0.460433 0.887694i \(-0.347694\pi\)
0.460433 + 0.887694i \(0.347694\pi\)
\(314\) −43.7426 −2.46854
\(315\) −0.763932 −0.0430427
\(316\) −31.8541 −1.79193
\(317\) 28.4164 1.59602 0.798012 0.602641i \(-0.205885\pi\)
0.798012 + 0.602641i \(0.205885\pi\)
\(318\) 2.85410 0.160050
\(319\) 13.7082 0.767512
\(320\) −3.32624 −0.185942
\(321\) 7.09017 0.395735
\(322\) 11.7082 0.652473
\(323\) −32.8885 −1.82997
\(324\) 4.85410 0.269672
\(325\) 23.5623 1.30700
\(326\) −7.09017 −0.392688
\(327\) 9.56231 0.528797
\(328\) −66.8328 −3.69022
\(329\) −4.09017 −0.225498
\(330\) 2.61803 0.144118
\(331\) −16.1459 −0.887459 −0.443729 0.896161i \(-0.646345\pi\)
−0.443729 + 0.896161i \(0.646345\pi\)
\(332\) −30.7082 −1.68533
\(333\) −13.4164 −0.735215
\(334\) −23.5623 −1.28927
\(335\) −5.50658 −0.300856
\(336\) 9.85410 0.537585
\(337\) −2.43769 −0.132790 −0.0663948 0.997793i \(-0.521150\pi\)
−0.0663948 + 0.997793i \(0.521150\pi\)
\(338\) −27.6525 −1.50410
\(339\) 15.0000 0.814688
\(340\) 10.4164 0.564909
\(341\) 17.5623 0.951052
\(342\) 30.6525 1.65750
\(343\) 13.0000 0.701934
\(344\) 65.0689 3.50828
\(345\) 1.70820 0.0919666
\(346\) −41.9787 −2.25679
\(347\) −1.47214 −0.0790284 −0.0395142 0.999219i \(-0.512581\pi\)
−0.0395142 + 0.999219i \(0.512581\pi\)
\(348\) 25.4164 1.36246
\(349\) 15.4164 0.825221 0.412611 0.910907i \(-0.364617\pi\)
0.412611 + 0.910907i \(0.364617\pi\)
\(350\) −12.7082 −0.679282
\(351\) −24.2705 −1.29546
\(352\) 28.4164 1.51460
\(353\) −25.0344 −1.33245 −0.666224 0.745751i \(-0.732091\pi\)
−0.666224 + 0.745751i \(0.732091\pi\)
\(354\) 16.7082 0.888031
\(355\) 1.56231 0.0829186
\(356\) −11.1246 −0.589603
\(357\) −5.61803 −0.297338
\(358\) 20.5623 1.08675
\(359\) 14.6738 0.774452 0.387226 0.921985i \(-0.373433\pi\)
0.387226 + 0.921985i \(0.373433\pi\)
\(360\) −5.70820 −0.300849
\(361\) 15.2705 0.803711
\(362\) −10.0902 −0.530328
\(363\) 4.14590 0.217603
\(364\) 23.5623 1.23500
\(365\) 4.14590 0.217006
\(366\) 10.8541 0.567353
\(367\) 16.4377 0.858041 0.429020 0.903295i \(-0.358859\pi\)
0.429020 + 0.903295i \(0.358859\pi\)
\(368\) 44.0689 2.29725
\(369\) −17.8885 −0.931240
\(370\) 6.70820 0.348743
\(371\) −1.09017 −0.0565988
\(372\) 32.5623 1.68828
\(373\) −22.6869 −1.17468 −0.587342 0.809339i \(-0.699826\pi\)
−0.587342 + 0.809339i \(0.699826\pi\)
\(374\) −38.5066 −1.99113
\(375\) −3.76393 −0.194369
\(376\) −30.5623 −1.57613
\(377\) 25.4164 1.30901
\(378\) 13.0902 0.673286
\(379\) 5.00000 0.256833 0.128416 0.991720i \(-0.459011\pi\)
0.128416 + 0.991720i \(0.459011\pi\)
\(380\) −10.8541 −0.556804
\(381\) −15.2705 −0.782332
\(382\) −14.7082 −0.752537
\(383\) 0.819660 0.0418827 0.0209413 0.999781i \(-0.493334\pi\)
0.0209413 + 0.999781i \(0.493334\pi\)
\(384\) 1.09017 0.0556325
\(385\) −1.00000 −0.0509647
\(386\) −52.6869 −2.68169
\(387\) 17.4164 0.885326
\(388\) −8.29180 −0.420952
\(389\) 7.41641 0.376027 0.188013 0.982166i \(-0.439795\pi\)
0.188013 + 0.982166i \(0.439795\pi\)
\(390\) 4.85410 0.245797
\(391\) −25.1246 −1.27061
\(392\) 44.8328 2.26440
\(393\) 2.23607 0.112795
\(394\) 42.9787 2.16524
\(395\) 2.50658 0.126120
\(396\) 25.4164 1.27722
\(397\) 20.0000 1.00377 0.501886 0.864934i \(-0.332640\pi\)
0.501886 + 0.864934i \(0.332640\pi\)
\(398\) 61.3050 3.07294
\(399\) 5.85410 0.293072
\(400\) −47.8328 −2.39164
\(401\) −23.8885 −1.19294 −0.596468 0.802637i \(-0.703430\pi\)
−0.596468 + 0.802637i \(0.703430\pi\)
\(402\) 37.7426 1.88243
\(403\) 32.5623 1.62204
\(404\) 9.27051 0.461225
\(405\) −0.381966 −0.0189800
\(406\) −13.7082 −0.680327
\(407\) −17.5623 −0.870531
\(408\) −41.9787 −2.07826
\(409\) 36.7082 1.81510 0.907552 0.419940i \(-0.137949\pi\)
0.907552 + 0.419940i \(0.137949\pi\)
\(410\) 8.94427 0.441726
\(411\) 12.7082 0.626849
\(412\) −4.85410 −0.239144
\(413\) −6.38197 −0.314036
\(414\) 23.4164 1.15085
\(415\) 2.41641 0.118617
\(416\) 52.6869 2.58319
\(417\) 11.1459 0.545817
\(418\) 40.1246 1.96256
\(419\) −4.09017 −0.199818 −0.0999089 0.994997i \(-0.531855\pi\)
−0.0999089 + 0.994997i \(0.531855\pi\)
\(420\) −1.85410 −0.0904709
\(421\) 3.00000 0.146211 0.0731055 0.997324i \(-0.476709\pi\)
0.0731055 + 0.997324i \(0.476709\pi\)
\(422\) −38.8885 −1.89306
\(423\) −8.18034 −0.397742
\(424\) −8.14590 −0.395600
\(425\) 27.2705 1.32281
\(426\) −10.7082 −0.518814
\(427\) −4.14590 −0.200634
\(428\) −34.4164 −1.66358
\(429\) −12.7082 −0.613558
\(430\) −8.70820 −0.419947
\(431\) 34.3607 1.65510 0.827548 0.561395i \(-0.189735\pi\)
0.827548 + 0.561395i \(0.189735\pi\)
\(432\) 49.2705 2.37053
\(433\) 14.4164 0.692808 0.346404 0.938085i \(-0.387403\pi\)
0.346404 + 0.938085i \(0.387403\pi\)
\(434\) −17.5623 −0.843018
\(435\) −2.00000 −0.0958927
\(436\) −46.4164 −2.22294
\(437\) 26.1803 1.25238
\(438\) −28.4164 −1.35779
\(439\) 29.5623 1.41093 0.705466 0.708744i \(-0.250738\pi\)
0.705466 + 0.708744i \(0.250738\pi\)
\(440\) −7.47214 −0.356220
\(441\) 12.0000 0.571429
\(442\) −71.3951 −3.39592
\(443\) −15.4377 −0.733467 −0.366733 0.930326i \(-0.619524\pi\)
−0.366733 + 0.930326i \(0.619524\pi\)
\(444\) −32.5623 −1.54534
\(445\) 0.875388 0.0414974
\(446\) −20.1803 −0.955567
\(447\) 7.47214 0.353420
\(448\) −8.70820 −0.411424
\(449\) −13.3607 −0.630529 −0.315265 0.949004i \(-0.602093\pi\)
−0.315265 + 0.949004i \(0.602093\pi\)
\(450\) −25.4164 −1.19814
\(451\) −23.4164 −1.10264
\(452\) −72.8115 −3.42477
\(453\) 19.0000 0.892698
\(454\) −7.70820 −0.361764
\(455\) −1.85410 −0.0869216
\(456\) 43.7426 2.04844
\(457\) −9.85410 −0.460955 −0.230478 0.973078i \(-0.574029\pi\)
−0.230478 + 0.973078i \(0.574029\pi\)
\(458\) −17.5623 −0.820633
\(459\) −28.0902 −1.31114
\(460\) −8.29180 −0.386607
\(461\) 12.2148 0.568899 0.284450 0.958691i \(-0.408189\pi\)
0.284450 + 0.958691i \(0.408189\pi\)
\(462\) 6.85410 0.318882
\(463\) −21.4164 −0.995305 −0.497652 0.867377i \(-0.665804\pi\)
−0.497652 + 0.867377i \(0.665804\pi\)
\(464\) −51.5967 −2.39532
\(465\) −2.56231 −0.118824
\(466\) 70.3951 3.26099
\(467\) −3.65248 −0.169016 −0.0845082 0.996423i \(-0.526932\pi\)
−0.0845082 + 0.996423i \(0.526932\pi\)
\(468\) 47.1246 2.17834
\(469\) −14.4164 −0.665688
\(470\) 4.09017 0.188665
\(471\) −16.7082 −0.769873
\(472\) −47.6869 −2.19497
\(473\) 22.7984 1.04827
\(474\) −17.1803 −0.789119
\(475\) −28.4164 −1.30383
\(476\) 27.2705 1.24994
\(477\) −2.18034 −0.0998309
\(478\) −22.7082 −1.03865
\(479\) 8.18034 0.373769 0.186885 0.982382i \(-0.440161\pi\)
0.186885 + 0.982382i \(0.440161\pi\)
\(480\) −4.14590 −0.189233
\(481\) −32.5623 −1.48471
\(482\) 21.6525 0.986243
\(483\) 4.47214 0.203489
\(484\) −20.1246 −0.914755
\(485\) 0.652476 0.0296274
\(486\) 41.8885 1.90010
\(487\) −23.0000 −1.04223 −0.521115 0.853487i \(-0.674484\pi\)
−0.521115 + 0.853487i \(0.674484\pi\)
\(488\) −30.9787 −1.40234
\(489\) −2.70820 −0.122469
\(490\) −6.00000 −0.271052
\(491\) 35.7771 1.61460 0.807299 0.590143i \(-0.200929\pi\)
0.807299 + 0.590143i \(0.200929\pi\)
\(492\) −43.4164 −1.95736
\(493\) 29.4164 1.32485
\(494\) 74.3951 3.34719
\(495\) −2.00000 −0.0898933
\(496\) −66.1033 −2.96813
\(497\) 4.09017 0.183469
\(498\) −16.5623 −0.742175
\(499\) 13.2705 0.594070 0.297035 0.954867i \(-0.404002\pi\)
0.297035 + 0.954867i \(0.404002\pi\)
\(500\) 18.2705 0.817082
\(501\) −9.00000 −0.402090
\(502\) −29.4164 −1.31292
\(503\) 13.3607 0.595723 0.297862 0.954609i \(-0.403727\pi\)
0.297862 + 0.954609i \(0.403727\pi\)
\(504\) −14.9443 −0.665671
\(505\) −0.729490 −0.0324619
\(506\) 30.6525 1.36267
\(507\) −10.5623 −0.469088
\(508\) 74.1246 3.28875
\(509\) 26.6180 1.17982 0.589912 0.807468i \(-0.299163\pi\)
0.589912 + 0.807468i \(0.299163\pi\)
\(510\) 5.61803 0.248771
\(511\) 10.8541 0.480157
\(512\) 40.3050 1.78124
\(513\) 29.2705 1.29232
\(514\) 35.2705 1.55572
\(515\) 0.381966 0.0168314
\(516\) 42.2705 1.86086
\(517\) −10.7082 −0.470946
\(518\) 17.5623 0.771643
\(519\) −16.0344 −0.703834
\(520\) −13.8541 −0.607543
\(521\) 17.1803 0.752684 0.376342 0.926481i \(-0.377182\pi\)
0.376342 + 0.926481i \(0.377182\pi\)
\(522\) −27.4164 −1.19998
\(523\) −10.5836 −0.462788 −0.231394 0.972860i \(-0.574329\pi\)
−0.231394 + 0.972860i \(0.574329\pi\)
\(524\) −10.8541 −0.474164
\(525\) −4.85410 −0.211850
\(526\) 53.9787 2.35358
\(527\) 37.6869 1.64167
\(528\) 25.7984 1.12273
\(529\) −3.00000 −0.130435
\(530\) 1.09017 0.0473540
\(531\) −12.7639 −0.553907
\(532\) −28.4164 −1.23201
\(533\) −43.4164 −1.88057
\(534\) −6.00000 −0.259645
\(535\) 2.70820 0.117086
\(536\) −107.721 −4.65285
\(537\) 7.85410 0.338930
\(538\) −32.2705 −1.39128
\(539\) 15.7082 0.676600
\(540\) −9.27051 −0.398939
\(541\) 9.14590 0.393213 0.196606 0.980482i \(-0.437008\pi\)
0.196606 + 0.980482i \(0.437008\pi\)
\(542\) −2.61803 −0.112454
\(543\) −3.85410 −0.165395
\(544\) 60.9787 2.61444
\(545\) 3.65248 0.156455
\(546\) 12.7082 0.543861
\(547\) −2.27051 −0.0970800 −0.0485400 0.998821i \(-0.515457\pi\)
−0.0485400 + 0.998821i \(0.515457\pi\)
\(548\) −61.6869 −2.63513
\(549\) −8.29180 −0.353885
\(550\) −33.2705 −1.41866
\(551\) −30.6525 −1.30584
\(552\) 33.4164 1.42230
\(553\) 6.56231 0.279058
\(554\) −12.3262 −0.523692
\(555\) 2.56231 0.108764
\(556\) −54.1033 −2.29449
\(557\) −31.6869 −1.34262 −0.671309 0.741178i \(-0.734268\pi\)
−0.671309 + 0.741178i \(0.734268\pi\)
\(558\) −35.1246 −1.48694
\(559\) 42.2705 1.78785
\(560\) 3.76393 0.159055
\(561\) −14.7082 −0.620981
\(562\) 82.3951 3.47563
\(563\) −1.20163 −0.0506425 −0.0253213 0.999679i \(-0.508061\pi\)
−0.0253213 + 0.999679i \(0.508061\pi\)
\(564\) −19.8541 −0.836009
\(565\) 5.72949 0.241041
\(566\) 51.5967 2.16877
\(567\) −1.00000 −0.0419961
\(568\) 30.5623 1.28237
\(569\) −27.1591 −1.13857 −0.569283 0.822141i \(-0.692779\pi\)
−0.569283 + 0.822141i \(0.692779\pi\)
\(570\) −5.85410 −0.245201
\(571\) −22.5623 −0.944203 −0.472102 0.881544i \(-0.656504\pi\)
−0.472102 + 0.881544i \(0.656504\pi\)
\(572\) 61.6869 2.57926
\(573\) −5.61803 −0.234697
\(574\) 23.4164 0.977382
\(575\) −21.7082 −0.905295
\(576\) −17.4164 −0.725684
\(577\) −36.8328 −1.53337 −0.766685 0.642023i \(-0.778095\pi\)
−0.766685 + 0.642023i \(0.778095\pi\)
\(578\) −38.1246 −1.58577
\(579\) −20.1246 −0.836350
\(580\) 9.70820 0.403111
\(581\) 6.32624 0.262457
\(582\) −4.47214 −0.185376
\(583\) −2.85410 −0.118205
\(584\) 81.1033 3.35608
\(585\) −3.70820 −0.153315
\(586\) 56.6869 2.34171
\(587\) −47.0689 −1.94274 −0.971370 0.237570i \(-0.923649\pi\)
−0.971370 + 0.237570i \(0.923649\pi\)
\(588\) 29.1246 1.20108
\(589\) −39.2705 −1.61811
\(590\) 6.38197 0.262741
\(591\) 16.4164 0.675281
\(592\) 66.1033 2.71683
\(593\) 2.18034 0.0895358 0.0447679 0.998997i \(-0.485745\pi\)
0.0447679 + 0.998997i \(0.485745\pi\)
\(594\) 34.2705 1.40614
\(595\) −2.14590 −0.0879732
\(596\) −36.2705 −1.48570
\(597\) 23.4164 0.958370
\(598\) 56.8328 2.32407
\(599\) −35.4508 −1.44848 −0.724241 0.689547i \(-0.757810\pi\)
−0.724241 + 0.689547i \(0.757810\pi\)
\(600\) −36.2705 −1.48074
\(601\) 3.56231 0.145309 0.0726547 0.997357i \(-0.476853\pi\)
0.0726547 + 0.997357i \(0.476853\pi\)
\(602\) −22.7984 −0.929192
\(603\) −28.8328 −1.17416
\(604\) −92.2279 −3.75270
\(605\) 1.58359 0.0643822
\(606\) 5.00000 0.203111
\(607\) 5.70820 0.231689 0.115844 0.993267i \(-0.463043\pi\)
0.115844 + 0.993267i \(0.463043\pi\)
\(608\) −63.5410 −2.57693
\(609\) −5.23607 −0.212176
\(610\) 4.14590 0.167863
\(611\) −19.8541 −0.803211
\(612\) 54.5410 2.20469
\(613\) 24.4164 0.986169 0.493085 0.869981i \(-0.335869\pi\)
0.493085 + 0.869981i \(0.335869\pi\)
\(614\) 7.47214 0.301551
\(615\) 3.41641 0.137763
\(616\) −19.5623 −0.788188
\(617\) 3.27051 0.131666 0.0658329 0.997831i \(-0.479030\pi\)
0.0658329 + 0.997831i \(0.479030\pi\)
\(618\) −2.61803 −0.105313
\(619\) −28.6869 −1.15302 −0.576512 0.817088i \(-0.695587\pi\)
−0.576512 + 0.817088i \(0.695587\pi\)
\(620\) 12.4377 0.499510
\(621\) 22.3607 0.897303
\(622\) −7.56231 −0.303221
\(623\) 2.29180 0.0918189
\(624\) 47.8328 1.91485
\(625\) 22.8328 0.913313
\(626\) −42.6525 −1.70474
\(627\) 15.3262 0.612071
\(628\) 81.1033 3.23638
\(629\) −37.6869 −1.50268
\(630\) 2.00000 0.0796819
\(631\) 42.2705 1.68276 0.841381 0.540442i \(-0.181743\pi\)
0.841381 + 0.540442i \(0.181743\pi\)
\(632\) 49.0344 1.95049
\(633\) −14.8541 −0.590398
\(634\) −74.3951 −2.95461
\(635\) −5.83282 −0.231468
\(636\) −5.29180 −0.209833
\(637\) 29.1246 1.15396
\(638\) −35.8885 −1.42084
\(639\) 8.18034 0.323609
\(640\) 0.416408 0.0164600
\(641\) −15.0000 −0.592464 −0.296232 0.955116i \(-0.595730\pi\)
−0.296232 + 0.955116i \(0.595730\pi\)
\(642\) −18.5623 −0.732596
\(643\) 5.00000 0.197181 0.0985904 0.995128i \(-0.468567\pi\)
0.0985904 + 0.995128i \(0.468567\pi\)
\(644\) −21.7082 −0.855423
\(645\) −3.32624 −0.130970
\(646\) 86.1033 3.38769
\(647\) 1.25735 0.0494317 0.0247158 0.999695i \(-0.492132\pi\)
0.0247158 + 0.999695i \(0.492132\pi\)
\(648\) −7.47214 −0.293533
\(649\) −16.7082 −0.655854
\(650\) −61.6869 −2.41956
\(651\) −6.70820 −0.262915
\(652\) 13.1459 0.514833
\(653\) −5.23607 −0.204903 −0.102452 0.994738i \(-0.532669\pi\)
−0.102452 + 0.994738i \(0.532669\pi\)
\(654\) −25.0344 −0.978924
\(655\) 0.854102 0.0333725
\(656\) 88.1378 3.44120
\(657\) 21.7082 0.846918
\(658\) 10.7082 0.417449
\(659\) 36.5967 1.42561 0.712803 0.701364i \(-0.247425\pi\)
0.712803 + 0.701364i \(0.247425\pi\)
\(660\) −4.85410 −0.188946
\(661\) −31.5623 −1.22763 −0.613816 0.789449i \(-0.710366\pi\)
−0.613816 + 0.789449i \(0.710366\pi\)
\(662\) 42.2705 1.64289
\(663\) −27.2705 −1.05910
\(664\) 47.2705 1.83445
\(665\) 2.23607 0.0867110
\(666\) 35.1246 1.36105
\(667\) −23.4164 −0.906687
\(668\) 43.6869 1.69030
\(669\) −7.70820 −0.298016
\(670\) 14.4164 0.556954
\(671\) −10.8541 −0.419018
\(672\) −10.8541 −0.418706
\(673\) −24.2918 −0.936380 −0.468190 0.883628i \(-0.655094\pi\)
−0.468190 + 0.883628i \(0.655094\pi\)
\(674\) 6.38197 0.245824
\(675\) −24.2705 −0.934172
\(676\) 51.2705 1.97194
\(677\) 40.0344 1.53865 0.769324 0.638858i \(-0.220593\pi\)
0.769324 + 0.638858i \(0.220593\pi\)
\(678\) −39.2705 −1.50817
\(679\) 1.70820 0.0655549
\(680\) −16.0344 −0.614893
\(681\) −2.94427 −0.112825
\(682\) −45.9787 −1.76062
\(683\) 47.2361 1.80744 0.903719 0.428126i \(-0.140826\pi\)
0.903719 + 0.428126i \(0.140826\pi\)
\(684\) −56.8328 −2.17306
\(685\) 4.85410 0.185466
\(686\) −34.0344 −1.29944
\(687\) −6.70820 −0.255934
\(688\) −85.8115 −3.27153
\(689\) −5.29180 −0.201601
\(690\) −4.47214 −0.170251
\(691\) 7.85410 0.298784 0.149392 0.988778i \(-0.452268\pi\)
0.149392 + 0.988778i \(0.452268\pi\)
\(692\) 77.8328 2.95876
\(693\) −5.23607 −0.198902
\(694\) 3.85410 0.146300
\(695\) 4.25735 0.161491
\(696\) −39.1246 −1.48301
\(697\) −50.2492 −1.90333
\(698\) −40.3607 −1.52767
\(699\) 26.8885 1.01702
\(700\) 23.5623 0.890571
\(701\) −25.7984 −0.974391 −0.487196 0.873293i \(-0.661980\pi\)
−0.487196 + 0.873293i \(0.661980\pi\)
\(702\) 63.5410 2.39820
\(703\) 39.2705 1.48112
\(704\) −22.7984 −0.859246
\(705\) 1.56231 0.0588398
\(706\) 65.5410 2.46667
\(707\) −1.90983 −0.0718266
\(708\) −30.9787 −1.16425
\(709\) −1.02129 −0.0383552 −0.0191776 0.999816i \(-0.506105\pi\)
−0.0191776 + 0.999816i \(0.506105\pi\)
\(710\) −4.09017 −0.153501
\(711\) 13.1246 0.492211
\(712\) 17.1246 0.641772
\(713\) −30.0000 −1.12351
\(714\) 14.7082 0.550441
\(715\) −4.85410 −0.181533
\(716\) −38.1246 −1.42478
\(717\) −8.67376 −0.323928
\(718\) −38.4164 −1.43369
\(719\) 39.3262 1.46662 0.733311 0.679894i \(-0.237974\pi\)
0.733311 + 0.679894i \(0.237974\pi\)
\(720\) 7.52786 0.280547
\(721\) 1.00000 0.0372419
\(722\) −39.9787 −1.48785
\(723\) 8.27051 0.307584
\(724\) 18.7082 0.695285
\(725\) 25.4164 0.943942
\(726\) −10.8541 −0.402834
\(727\) −4.72949 −0.175407 −0.0877035 0.996147i \(-0.527953\pi\)
−0.0877035 + 0.996147i \(0.527953\pi\)
\(728\) −36.2705 −1.34427
\(729\) 13.0000 0.481481
\(730\) −10.8541 −0.401728
\(731\) 48.9230 1.80948
\(732\) −20.1246 −0.743827
\(733\) 14.7082 0.543260 0.271630 0.962402i \(-0.412437\pi\)
0.271630 + 0.962402i \(0.412437\pi\)
\(734\) −43.0344 −1.58843
\(735\) −2.29180 −0.0845342
\(736\) −48.5410 −1.78925
\(737\) −37.7426 −1.39027
\(738\) 46.8328 1.72394
\(739\) −16.8328 −0.619205 −0.309603 0.950866i \(-0.600196\pi\)
−0.309603 + 0.950866i \(0.600196\pi\)
\(740\) −12.4377 −0.457219
\(741\) 28.4164 1.04390
\(742\) 2.85410 0.104777
\(743\) 30.7082 1.12657 0.563287 0.826261i \(-0.309536\pi\)
0.563287 + 0.826261i \(0.309536\pi\)
\(744\) −50.1246 −1.83766
\(745\) 2.85410 0.104566
\(746\) 59.3951 2.17461
\(747\) 12.6525 0.462930
\(748\) 71.3951 2.61046
\(749\) 7.09017 0.259069
\(750\) 9.85410 0.359821
\(751\) −44.1246 −1.61013 −0.805065 0.593187i \(-0.797870\pi\)
−0.805065 + 0.593187i \(0.797870\pi\)
\(752\) 40.3050 1.46977
\(753\) −11.2361 −0.409465
\(754\) −66.5410 −2.42328
\(755\) 7.25735 0.264122
\(756\) −24.2705 −0.882710
\(757\) −21.2918 −0.773863 −0.386932 0.922108i \(-0.626465\pi\)
−0.386932 + 0.922108i \(0.626465\pi\)
\(758\) −13.0902 −0.475456
\(759\) 11.7082 0.424981
\(760\) 16.7082 0.606070
\(761\) −7.47214 −0.270865 −0.135432 0.990787i \(-0.543242\pi\)
−0.135432 + 0.990787i \(0.543242\pi\)
\(762\) 39.9787 1.44828
\(763\) 9.56231 0.346179
\(764\) 27.2705 0.986612
\(765\) −4.29180 −0.155170
\(766\) −2.14590 −0.0775344
\(767\) −30.9787 −1.11858
\(768\) 14.5623 0.525472
\(769\) 31.3951 1.13214 0.566069 0.824358i \(-0.308464\pi\)
0.566069 + 0.824358i \(0.308464\pi\)
\(770\) 2.61803 0.0943474
\(771\) 13.4721 0.485187
\(772\) 97.6869 3.51583
\(773\) −16.5279 −0.594466 −0.297233 0.954805i \(-0.596064\pi\)
−0.297233 + 0.954805i \(0.596064\pi\)
\(774\) −45.5967 −1.63894
\(775\) 32.5623 1.16967
\(776\) 12.7639 0.458198
\(777\) 6.70820 0.240655
\(778\) −19.4164 −0.696112
\(779\) 52.3607 1.87602
\(780\) −9.00000 −0.322252
\(781\) 10.7082 0.383170
\(782\) 65.7771 2.35218
\(783\) −26.1803 −0.935609
\(784\) −59.1246 −2.11159
\(785\) −6.38197 −0.227782
\(786\) −5.85410 −0.208809
\(787\) 43.4164 1.54763 0.773814 0.633413i \(-0.218347\pi\)
0.773814 + 0.633413i \(0.218347\pi\)
\(788\) −79.6869 −2.83873
\(789\) 20.6180 0.734021
\(790\) −6.56231 −0.233476
\(791\) 15.0000 0.533339
\(792\) −39.1246 −1.39023
\(793\) −20.1246 −0.714646
\(794\) −52.3607 −1.85821
\(795\) 0.416408 0.0147685
\(796\) −113.666 −4.02877
\(797\) −50.1246 −1.77550 −0.887752 0.460321i \(-0.847734\pi\)
−0.887752 + 0.460321i \(0.847734\pi\)
\(798\) −15.3262 −0.542543
\(799\) −22.9787 −0.812928
\(800\) 52.6869 1.86276
\(801\) 4.58359 0.161953
\(802\) 62.5410 2.20840
\(803\) 28.4164 1.00279
\(804\) −69.9787 −2.46796
\(805\) 1.70820 0.0602063
\(806\) −85.2492 −3.00278
\(807\) −12.3262 −0.433904
\(808\) −14.2705 −0.502035
\(809\) 2.94427 0.103515 0.0517575 0.998660i \(-0.483518\pi\)
0.0517575 + 0.998660i \(0.483518\pi\)
\(810\) 1.00000 0.0351364
\(811\) 21.5410 0.756408 0.378204 0.925722i \(-0.376542\pi\)
0.378204 + 0.925722i \(0.376542\pi\)
\(812\) 25.4164 0.891941
\(813\) −1.00000 −0.0350715
\(814\) 45.9787 1.61155
\(815\) −1.03444 −0.0362349
\(816\) 55.3607 1.93801
\(817\) −50.9787 −1.78352
\(818\) −96.1033 −3.36017
\(819\) −9.70820 −0.339232
\(820\) −16.5836 −0.579124
\(821\) 33.0000 1.15171 0.575854 0.817553i \(-0.304670\pi\)
0.575854 + 0.817553i \(0.304670\pi\)
\(822\) −33.2705 −1.16044
\(823\) 3.43769 0.119830 0.0599152 0.998203i \(-0.480917\pi\)
0.0599152 + 0.998203i \(0.480917\pi\)
\(824\) 7.47214 0.260304
\(825\) −12.7082 −0.442443
\(826\) 16.7082 0.581353
\(827\) 6.70820 0.233267 0.116634 0.993175i \(-0.462790\pi\)
0.116634 + 0.993175i \(0.462790\pi\)
\(828\) −43.4164 −1.50882
\(829\) 14.2705 0.495635 0.247818 0.968807i \(-0.420287\pi\)
0.247818 + 0.968807i \(0.420287\pi\)
\(830\) −6.32624 −0.219587
\(831\) −4.70820 −0.163326
\(832\) −42.2705 −1.46547
\(833\) 33.7082 1.16792
\(834\) −29.1803 −1.01043
\(835\) −3.43769 −0.118966
\(836\) −74.3951 −2.57301
\(837\) −33.5410 −1.15935
\(838\) 10.7082 0.369909
\(839\) 23.6180 0.815385 0.407693 0.913119i \(-0.366334\pi\)
0.407693 + 0.913119i \(0.366334\pi\)
\(840\) 2.85410 0.0984759
\(841\) −1.58359 −0.0546066
\(842\) −7.85410 −0.270670
\(843\) 31.4721 1.08396
\(844\) 72.1033 2.48190
\(845\) −4.03444 −0.138789
\(846\) 21.4164 0.736311
\(847\) 4.14590 0.142455
\(848\) 10.7426 0.368904
\(849\) 19.7082 0.676384
\(850\) −71.3951 −2.44883
\(851\) 30.0000 1.02839
\(852\) 19.8541 0.680190
\(853\) −44.2705 −1.51579 −0.757897 0.652375i \(-0.773773\pi\)
−0.757897 + 0.652375i \(0.773773\pi\)
\(854\) 10.8541 0.371420
\(855\) 4.47214 0.152944
\(856\) 52.9787 1.81078
\(857\) 8.23607 0.281339 0.140669 0.990057i \(-0.455075\pi\)
0.140669 + 0.990057i \(0.455075\pi\)
\(858\) 33.2705 1.13584
\(859\) 10.5623 0.360381 0.180191 0.983632i \(-0.442329\pi\)
0.180191 + 0.983632i \(0.442329\pi\)
\(860\) 16.1459 0.550571
\(861\) 8.94427 0.304820
\(862\) −89.9574 −3.06396
\(863\) −21.7082 −0.738956 −0.369478 0.929240i \(-0.620463\pi\)
−0.369478 + 0.929240i \(0.620463\pi\)
\(864\) −54.2705 −1.84632
\(865\) −6.12461 −0.208243
\(866\) −37.7426 −1.28255
\(867\) −14.5623 −0.494562
\(868\) 32.5623 1.10524
\(869\) 17.1803 0.582803
\(870\) 5.23607 0.177519
\(871\) −69.9787 −2.37114
\(872\) 71.4508 2.41963
\(873\) 3.41641 0.115628
\(874\) −68.5410 −2.31843
\(875\) −3.76393 −0.127244
\(876\) 52.6869 1.78013
\(877\) −13.0000 −0.438979 −0.219489 0.975615i \(-0.570439\pi\)
−0.219489 + 0.975615i \(0.570439\pi\)
\(878\) −77.3951 −2.61196
\(879\) 21.6525 0.730320
\(880\) 9.85410 0.332182
\(881\) 29.8885 1.00697 0.503485 0.864004i \(-0.332051\pi\)
0.503485 + 0.864004i \(0.332051\pi\)
\(882\) −31.4164 −1.05785
\(883\) 5.87539 0.197723 0.0988613 0.995101i \(-0.468480\pi\)
0.0988613 + 0.995101i \(0.468480\pi\)
\(884\) 132.374 4.45221
\(885\) 2.43769 0.0819422
\(886\) 40.4164 1.35782
\(887\) 40.1935 1.34957 0.674783 0.738016i \(-0.264237\pi\)
0.674783 + 0.738016i \(0.264237\pi\)
\(888\) 50.1246 1.68207
\(889\) −15.2705 −0.512156
\(890\) −2.29180 −0.0768212
\(891\) −2.61803 −0.0877074
\(892\) 37.4164 1.25279
\(893\) 23.9443 0.801265
\(894\) −19.5623 −0.654261
\(895\) 3.00000 0.100279
\(896\) 1.09017 0.0364200
\(897\) 21.7082 0.724816
\(898\) 34.9787 1.16725
\(899\) 35.1246 1.17147
\(900\) 47.1246 1.57082
\(901\) −6.12461 −0.204040
\(902\) 61.3050 2.04123
\(903\) −8.70820 −0.289791
\(904\) 112.082 3.72779
\(905\) −1.47214 −0.0489355
\(906\) −49.7426 −1.65259
\(907\) −7.12461 −0.236569 −0.118284 0.992980i \(-0.537739\pi\)
−0.118284 + 0.992980i \(0.537739\pi\)
\(908\) 14.2918 0.474290
\(909\) −3.81966 −0.126690
\(910\) 4.85410 0.160912
\(911\) −10.0344 −0.332456 −0.166228 0.986087i \(-0.553159\pi\)
−0.166228 + 0.986087i \(0.553159\pi\)
\(912\) −57.6869 −1.91020
\(913\) 16.5623 0.548132
\(914\) 25.7984 0.853334
\(915\) 1.58359 0.0523519
\(916\) 32.5623 1.07589
\(917\) 2.23607 0.0738415
\(918\) 73.5410 2.42722
\(919\) −27.9787 −0.922933 −0.461466 0.887158i \(-0.652676\pi\)
−0.461466 + 0.887158i \(0.652676\pi\)
\(920\) 12.7639 0.420814
\(921\) 2.85410 0.0940459
\(922\) −31.9787 −1.05316
\(923\) 19.8541 0.653506
\(924\) −12.7082 −0.418069
\(925\) −32.5623 −1.07064
\(926\) 56.0689 1.84254
\(927\) 2.00000 0.0656886
\(928\) 56.8328 1.86563
\(929\) 14.9443 0.490306 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(930\) 6.70820 0.219971
\(931\) −35.1246 −1.15116
\(932\) −130.520 −4.27532
\(933\) −2.88854 −0.0945667
\(934\) 9.56231 0.312888
\(935\) −5.61803 −0.183729
\(936\) −72.5410 −2.37108
\(937\) −11.0000 −0.359354 −0.179677 0.983726i \(-0.557505\pi\)
−0.179677 + 0.983726i \(0.557505\pi\)
\(938\) 37.7426 1.23234
\(939\) −16.2918 −0.531663
\(940\) −7.58359 −0.247350
\(941\) −23.3951 −0.762659 −0.381330 0.924439i \(-0.624534\pi\)
−0.381330 + 0.924439i \(0.624534\pi\)
\(942\) 43.7426 1.42521
\(943\) 40.0000 1.30258
\(944\) 62.8885 2.04685
\(945\) 1.90983 0.0621268
\(946\) −59.6869 −1.94059
\(947\) −41.0132 −1.33275 −0.666374 0.745617i \(-0.732155\pi\)
−0.666374 + 0.745617i \(0.732155\pi\)
\(948\) 31.8541 1.03457
\(949\) 52.6869 1.71029
\(950\) 74.3951 2.41370
\(951\) −28.4164 −0.921465
\(952\) −41.9787 −1.36054
\(953\) 13.3607 0.432795 0.216397 0.976305i \(-0.430569\pi\)
0.216397 + 0.976305i \(0.430569\pi\)
\(954\) 5.70820 0.184810
\(955\) −2.14590 −0.0694396
\(956\) 42.1033 1.36172
\(957\) −13.7082 −0.443123
\(958\) −21.4164 −0.691933
\(959\) 12.7082 0.410369
\(960\) 3.32624 0.107354
\(961\) 14.0000 0.451613
\(962\) 85.2492 2.74855
\(963\) 14.1803 0.456955
\(964\) −40.1459 −1.29301
\(965\) −7.68692 −0.247451
\(966\) −11.7082 −0.376705
\(967\) −14.4164 −0.463600 −0.231800 0.972763i \(-0.574462\pi\)
−0.231800 + 0.972763i \(0.574462\pi\)
\(968\) 30.9787 0.995694
\(969\) 32.8885 1.05653
\(970\) −1.70820 −0.0548471
\(971\) −53.0132 −1.70127 −0.850637 0.525754i \(-0.823783\pi\)
−0.850637 + 0.525754i \(0.823783\pi\)
\(972\) −77.6656 −2.49113
\(973\) 11.1459 0.357321
\(974\) 60.2148 1.92941
\(975\) −23.5623 −0.754598
\(976\) 40.8541 1.30771
\(977\) 46.7426 1.49543 0.747715 0.664020i \(-0.231151\pi\)
0.747715 + 0.664020i \(0.231151\pi\)
\(978\) 7.09017 0.226719
\(979\) 6.00000 0.191761
\(980\) 11.1246 0.355363
\(981\) 19.1246 0.610602
\(982\) −93.6656 −2.98899
\(983\) 18.6525 0.594922 0.297461 0.954734i \(-0.403860\pi\)
0.297461 + 0.954734i \(0.403860\pi\)
\(984\) 66.8328 2.13055
\(985\) 6.27051 0.199795
\(986\) −77.0132 −2.45260
\(987\) 4.09017 0.130192
\(988\) −137.936 −4.38833
\(989\) −38.9443 −1.23836
\(990\) 5.23607 0.166413
\(991\) 24.2705 0.770978 0.385489 0.922712i \(-0.374033\pi\)
0.385489 + 0.922712i \(0.374033\pi\)
\(992\) 72.8115 2.31177
\(993\) 16.1459 0.512375
\(994\) −10.7082 −0.339644
\(995\) 8.94427 0.283552
\(996\) 30.7082 0.973027
\(997\) −39.2918 −1.24438 −0.622192 0.782865i \(-0.713758\pi\)
−0.622192 + 0.782865i \(0.713758\pi\)
\(998\) −34.7426 −1.09976
\(999\) 33.5410 1.06119
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 103.2.a.a.1.1 2
3.2 odd 2 927.2.a.b.1.2 2
4.3 odd 2 1648.2.a.f.1.2 2
5.4 even 2 2575.2.a.g.1.2 2
7.6 odd 2 5047.2.a.a.1.1 2
8.3 odd 2 6592.2.a.h.1.1 2
8.5 even 2 6592.2.a.t.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
103.2.a.a.1.1 2 1.1 even 1 trivial
927.2.a.b.1.2 2 3.2 odd 2
1648.2.a.f.1.2 2 4.3 odd 2
2575.2.a.g.1.2 2 5.4 even 2
5047.2.a.a.1.1 2 7.6 odd 2
6592.2.a.h.1.1 2 8.3 odd 2
6592.2.a.t.1.1 2 8.5 even 2