Properties

Label 65.2.a.b.1.1
Level $65$
Weight $2$
Character 65.1
Self dual yes
Analytic conductor $0.519$
Analytic rank $0$
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] = [65,2,Mod(1,65)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(65, 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("65.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 65 = 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 65.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: \(0.519027613138\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) 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 \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 65.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.41421 q^{2} -1.41421 q^{3} +3.82843 q^{4} +1.00000 q^{5} +3.41421 q^{6} +4.82843 q^{7} -4.41421 q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-2.41421 q^{2} -1.41421 q^{3} +3.82843 q^{4} +1.00000 q^{5} +3.41421 q^{6} +4.82843 q^{7} -4.41421 q^{8} -1.00000 q^{9} -2.41421 q^{10} +3.41421 q^{11} -5.41421 q^{12} -1.00000 q^{13} -11.6569 q^{14} -1.41421 q^{15} +3.00000 q^{16} +0.828427 q^{17} +2.41421 q^{18} +0.585786 q^{19} +3.82843 q^{20} -6.82843 q^{21} -8.24264 q^{22} +1.41421 q^{23} +6.24264 q^{24} +1.00000 q^{25} +2.41421 q^{26} +5.65685 q^{27} +18.4853 q^{28} -5.65685 q^{29} +3.41421 q^{30} +1.75736 q^{31} +1.58579 q^{32} -4.82843 q^{33} -2.00000 q^{34} +4.82843 q^{35} -3.82843 q^{36} -8.48528 q^{37} -1.41421 q^{38} +1.41421 q^{39} -4.41421 q^{40} -3.17157 q^{41} +16.4853 q^{42} -11.0711 q^{43} +13.0711 q^{44} -1.00000 q^{45} -3.41421 q^{46} -4.82843 q^{47} -4.24264 q^{48} +16.3137 q^{49} -2.41421 q^{50} -1.17157 q^{51} -3.82843 q^{52} +2.48528 q^{53} -13.6569 q^{54} +3.41421 q^{55} -21.3137 q^{56} -0.828427 q^{57} +13.6569 q^{58} +1.75736 q^{59} -5.41421 q^{60} -8.00000 q^{61} -4.24264 q^{62} -4.82843 q^{63} -9.82843 q^{64} -1.00000 q^{65} +11.6569 q^{66} -2.00000 q^{67} +3.17157 q^{68} -2.00000 q^{69} -11.6569 q^{70} +11.8995 q^{71} +4.41421 q^{72} +8.48528 q^{73} +20.4853 q^{74} -1.41421 q^{75} +2.24264 q^{76} +16.4853 q^{77} -3.41421 q^{78} -8.48528 q^{79} +3.00000 q^{80} -5.00000 q^{81} +7.65685 q^{82} -3.17157 q^{83} -26.1421 q^{84} +0.828427 q^{85} +26.7279 q^{86} +8.00000 q^{87} -15.0711 q^{88} +6.00000 q^{89} +2.41421 q^{90} -4.82843 q^{91} +5.41421 q^{92} -2.48528 q^{93} +11.6569 q^{94} +0.585786 q^{95} -2.24264 q^{96} -7.65685 q^{97} -39.3848 q^{98} -3.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} + 4 q^{6} + 4 q^{7} - 6 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} + 4 q^{6} + 4 q^{7} - 6 q^{8} - 2 q^{9} - 2 q^{10} + 4 q^{11} - 8 q^{12} - 2 q^{13} - 12 q^{14} + 6 q^{16} - 4 q^{17} + 2 q^{18} + 4 q^{19} + 2 q^{20} - 8 q^{21} - 8 q^{22} + 4 q^{24} + 2 q^{25} + 2 q^{26} + 20 q^{28} + 4 q^{30} + 12 q^{31} + 6 q^{32} - 4 q^{33} - 4 q^{34} + 4 q^{35} - 2 q^{36} - 6 q^{40} - 12 q^{41} + 16 q^{42} - 8 q^{43} + 12 q^{44} - 2 q^{45} - 4 q^{46} - 4 q^{47} + 10 q^{49} - 2 q^{50} - 8 q^{51} - 2 q^{52} - 12 q^{53} - 16 q^{54} + 4 q^{55} - 20 q^{56} + 4 q^{57} + 16 q^{58} + 12 q^{59} - 8 q^{60} - 16 q^{61} - 4 q^{63} - 14 q^{64} - 2 q^{65} + 12 q^{66} - 4 q^{67} + 12 q^{68} - 4 q^{69} - 12 q^{70} + 4 q^{71} + 6 q^{72} + 24 q^{74} - 4 q^{76} + 16 q^{77} - 4 q^{78} + 6 q^{80} - 10 q^{81} + 4 q^{82} - 12 q^{83} - 24 q^{84} - 4 q^{85} + 28 q^{86} + 16 q^{87} - 16 q^{88} + 12 q^{89} + 2 q^{90} - 4 q^{91} + 8 q^{92} + 12 q^{93} + 12 q^{94} + 4 q^{95} + 4 q^{96} - 4 q^{97} - 42 q^{98} - 4 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.41421 −1.70711 −0.853553 0.521005i \(-0.825557\pi\)
−0.853553 + 0.521005i \(0.825557\pi\)
\(3\) −1.41421 −0.816497 −0.408248 0.912871i \(-0.633860\pi\)
−0.408248 + 0.912871i \(0.633860\pi\)
\(4\) 3.82843 1.91421
\(5\) 1.00000 0.447214
\(6\) 3.41421 1.39385
\(7\) 4.82843 1.82497 0.912487 0.409106i \(-0.134159\pi\)
0.912487 + 0.409106i \(0.134159\pi\)
\(8\) −4.41421 −1.56066
\(9\) −1.00000 −0.333333
\(10\) −2.41421 −0.763441
\(11\) 3.41421 1.02942 0.514712 0.857363i \(-0.327899\pi\)
0.514712 + 0.857363i \(0.327899\pi\)
\(12\) −5.41421 −1.56295
\(13\) −1.00000 −0.277350
\(14\) −11.6569 −3.11543
\(15\) −1.41421 −0.365148
\(16\) 3.00000 0.750000
\(17\) 0.828427 0.200923 0.100462 0.994941i \(-0.467968\pi\)
0.100462 + 0.994941i \(0.467968\pi\)
\(18\) 2.41421 0.569036
\(19\) 0.585786 0.134389 0.0671943 0.997740i \(-0.478595\pi\)
0.0671943 + 0.997740i \(0.478595\pi\)
\(20\) 3.82843 0.856062
\(21\) −6.82843 −1.49008
\(22\) −8.24264 −1.75734
\(23\) 1.41421 0.294884 0.147442 0.989071i \(-0.452896\pi\)
0.147442 + 0.989071i \(0.452896\pi\)
\(24\) 6.24264 1.27427
\(25\) 1.00000 0.200000
\(26\) 2.41421 0.473466
\(27\) 5.65685 1.08866
\(28\) 18.4853 3.49339
\(29\) −5.65685 −1.05045 −0.525226 0.850963i \(-0.676019\pi\)
−0.525226 + 0.850963i \(0.676019\pi\)
\(30\) 3.41421 0.623347
\(31\) 1.75736 0.315631 0.157816 0.987469i \(-0.449555\pi\)
0.157816 + 0.987469i \(0.449555\pi\)
\(32\) 1.58579 0.280330
\(33\) −4.82843 −0.840521
\(34\) −2.00000 −0.342997
\(35\) 4.82843 0.816153
\(36\) −3.82843 −0.638071
\(37\) −8.48528 −1.39497 −0.697486 0.716599i \(-0.745698\pi\)
−0.697486 + 0.716599i \(0.745698\pi\)
\(38\) −1.41421 −0.229416
\(39\) 1.41421 0.226455
\(40\) −4.41421 −0.697948
\(41\) −3.17157 −0.495316 −0.247658 0.968847i \(-0.579661\pi\)
−0.247658 + 0.968847i \(0.579661\pi\)
\(42\) 16.4853 2.54373
\(43\) −11.0711 −1.68832 −0.844161 0.536090i \(-0.819901\pi\)
−0.844161 + 0.536090i \(0.819901\pi\)
\(44\) 13.0711 1.97054
\(45\) −1.00000 −0.149071
\(46\) −3.41421 −0.503398
\(47\) −4.82843 −0.704298 −0.352149 0.935944i \(-0.614549\pi\)
−0.352149 + 0.935944i \(0.614549\pi\)
\(48\) −4.24264 −0.612372
\(49\) 16.3137 2.33053
\(50\) −2.41421 −0.341421
\(51\) −1.17157 −0.164053
\(52\) −3.82843 −0.530907
\(53\) 2.48528 0.341380 0.170690 0.985325i \(-0.445400\pi\)
0.170690 + 0.985325i \(0.445400\pi\)
\(54\) −13.6569 −1.85846
\(55\) 3.41421 0.460372
\(56\) −21.3137 −2.84816
\(57\) −0.828427 −0.109728
\(58\) 13.6569 1.79323
\(59\) 1.75736 0.228789 0.114394 0.993435i \(-0.463507\pi\)
0.114394 + 0.993435i \(0.463507\pi\)
\(60\) −5.41421 −0.698972
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) −4.24264 −0.538816
\(63\) −4.82843 −0.608325
\(64\) −9.82843 −1.22855
\(65\) −1.00000 −0.124035
\(66\) 11.6569 1.43486
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) 3.17157 0.384610
\(69\) −2.00000 −0.240772
\(70\) −11.6569 −1.39326
\(71\) 11.8995 1.41221 0.706105 0.708107i \(-0.250451\pi\)
0.706105 + 0.708107i \(0.250451\pi\)
\(72\) 4.41421 0.520220
\(73\) 8.48528 0.993127 0.496564 0.868000i \(-0.334595\pi\)
0.496564 + 0.868000i \(0.334595\pi\)
\(74\) 20.4853 2.38137
\(75\) −1.41421 −0.163299
\(76\) 2.24264 0.257249
\(77\) 16.4853 1.87867
\(78\) −3.41421 −0.386584
\(79\) −8.48528 −0.954669 −0.477334 0.878722i \(-0.658397\pi\)
−0.477334 + 0.878722i \(0.658397\pi\)
\(80\) 3.00000 0.335410
\(81\) −5.00000 −0.555556
\(82\) 7.65685 0.845558
\(83\) −3.17157 −0.348125 −0.174063 0.984735i \(-0.555690\pi\)
−0.174063 + 0.984735i \(0.555690\pi\)
\(84\) −26.1421 −2.85234
\(85\) 0.828427 0.0898555
\(86\) 26.7279 2.88215
\(87\) 8.00000 0.857690
\(88\) −15.0711 −1.60658
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 2.41421 0.254480
\(91\) −4.82843 −0.506157
\(92\) 5.41421 0.564471
\(93\) −2.48528 −0.257712
\(94\) 11.6569 1.20231
\(95\) 0.585786 0.0601004
\(96\) −2.24264 −0.228889
\(97\) −7.65685 −0.777436 −0.388718 0.921357i \(-0.627082\pi\)
−0.388718 + 0.921357i \(0.627082\pi\)
\(98\) −39.3848 −3.97846
\(99\) −3.41421 −0.343141
\(100\) 3.82843 0.382843
\(101\) −3.65685 −0.363871 −0.181935 0.983311i \(-0.558236\pi\)
−0.181935 + 0.983311i \(0.558236\pi\)
\(102\) 2.82843 0.280056
\(103\) 14.5858 1.43718 0.718590 0.695434i \(-0.244788\pi\)
0.718590 + 0.695434i \(0.244788\pi\)
\(104\) 4.41421 0.432849
\(105\) −6.82843 −0.666386
\(106\) −6.00000 −0.582772
\(107\) −9.41421 −0.910106 −0.455053 0.890464i \(-0.650380\pi\)
−0.455053 + 0.890464i \(0.650380\pi\)
\(108\) 21.6569 2.08393
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) −8.24264 −0.785905
\(111\) 12.0000 1.13899
\(112\) 14.4853 1.36873
\(113\) −8.82843 −0.830509 −0.415254 0.909705i \(-0.636307\pi\)
−0.415254 + 0.909705i \(0.636307\pi\)
\(114\) 2.00000 0.187317
\(115\) 1.41421 0.131876
\(116\) −21.6569 −2.01079
\(117\) 1.00000 0.0924500
\(118\) −4.24264 −0.390567
\(119\) 4.00000 0.366679
\(120\) 6.24264 0.569873
\(121\) 0.656854 0.0597140
\(122\) 19.3137 1.74858
\(123\) 4.48528 0.404424
\(124\) 6.72792 0.604185
\(125\) 1.00000 0.0894427
\(126\) 11.6569 1.03848
\(127\) −6.58579 −0.584394 −0.292197 0.956358i \(-0.594386\pi\)
−0.292197 + 0.956358i \(0.594386\pi\)
\(128\) 20.5563 1.81694
\(129\) 15.6569 1.37851
\(130\) 2.41421 0.211741
\(131\) −16.9706 −1.48272 −0.741362 0.671105i \(-0.765820\pi\)
−0.741362 + 0.671105i \(0.765820\pi\)
\(132\) −18.4853 −1.60894
\(133\) 2.82843 0.245256
\(134\) 4.82843 0.417113
\(135\) 5.65685 0.486864
\(136\) −3.65685 −0.313573
\(137\) −17.3137 −1.47921 −0.739605 0.673041i \(-0.764988\pi\)
−0.739605 + 0.673041i \(0.764988\pi\)
\(138\) 4.82843 0.411023
\(139\) 4.48528 0.380437 0.190218 0.981742i \(-0.439080\pi\)
0.190218 + 0.981742i \(0.439080\pi\)
\(140\) 18.4853 1.56229
\(141\) 6.82843 0.575057
\(142\) −28.7279 −2.41079
\(143\) −3.41421 −0.285511
\(144\) −3.00000 −0.250000
\(145\) −5.65685 −0.469776
\(146\) −20.4853 −1.69537
\(147\) −23.0711 −1.90287
\(148\) −32.4853 −2.67027
\(149\) −11.6569 −0.954967 −0.477483 0.878641i \(-0.658451\pi\)
−0.477483 + 0.878641i \(0.658451\pi\)
\(150\) 3.41421 0.278769
\(151\) 9.75736 0.794043 0.397021 0.917809i \(-0.370044\pi\)
0.397021 + 0.917809i \(0.370044\pi\)
\(152\) −2.58579 −0.209735
\(153\) −0.828427 −0.0669744
\(154\) −39.7990 −3.20709
\(155\) 1.75736 0.141154
\(156\) 5.41421 0.433484
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 20.4853 1.62972
\(159\) −3.51472 −0.278735
\(160\) 1.58579 0.125367
\(161\) 6.82843 0.538155
\(162\) 12.0711 0.948393
\(163\) 18.9706 1.48589 0.742945 0.669353i \(-0.233429\pi\)
0.742945 + 0.669353i \(0.233429\pi\)
\(164\) −12.1421 −0.948141
\(165\) −4.82843 −0.375893
\(166\) 7.65685 0.594287
\(167\) −3.17157 −0.245424 −0.122712 0.992442i \(-0.539159\pi\)
−0.122712 + 0.992442i \(0.539159\pi\)
\(168\) 30.1421 2.32552
\(169\) 1.00000 0.0769231
\(170\) −2.00000 −0.153393
\(171\) −0.585786 −0.0447962
\(172\) −42.3848 −3.23181
\(173\) 16.8284 1.27944 0.639721 0.768607i \(-0.279050\pi\)
0.639721 + 0.768607i \(0.279050\pi\)
\(174\) −19.3137 −1.46417
\(175\) 4.82843 0.364995
\(176\) 10.2426 0.772068
\(177\) −2.48528 −0.186805
\(178\) −14.4853 −1.08572
\(179\) 5.65685 0.422813 0.211407 0.977398i \(-0.432196\pi\)
0.211407 + 0.977398i \(0.432196\pi\)
\(180\) −3.82843 −0.285354
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 11.6569 0.864064
\(183\) 11.3137 0.836333
\(184\) −6.24264 −0.460214
\(185\) −8.48528 −0.623850
\(186\) 6.00000 0.439941
\(187\) 2.82843 0.206835
\(188\) −18.4853 −1.34818
\(189\) 27.3137 1.98678
\(190\) −1.41421 −0.102598
\(191\) 2.34315 0.169544 0.0847720 0.996400i \(-0.472984\pi\)
0.0847720 + 0.996400i \(0.472984\pi\)
\(192\) 13.8995 1.00311
\(193\) 4.34315 0.312626 0.156313 0.987708i \(-0.450039\pi\)
0.156313 + 0.987708i \(0.450039\pi\)
\(194\) 18.4853 1.32717
\(195\) 1.41421 0.101274
\(196\) 62.4558 4.46113
\(197\) 10.9706 0.781620 0.390810 0.920471i \(-0.372195\pi\)
0.390810 + 0.920471i \(0.372195\pi\)
\(198\) 8.24264 0.585779
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) −4.41421 −0.312132
\(201\) 2.82843 0.199502
\(202\) 8.82843 0.621166
\(203\) −27.3137 −1.91705
\(204\) −4.48528 −0.314033
\(205\) −3.17157 −0.221512
\(206\) −35.2132 −2.45342
\(207\) −1.41421 −0.0982946
\(208\) −3.00000 −0.208013
\(209\) 2.00000 0.138343
\(210\) 16.4853 1.13759
\(211\) 3.31371 0.228125 0.114063 0.993474i \(-0.463614\pi\)
0.114063 + 0.993474i \(0.463614\pi\)
\(212\) 9.51472 0.653474
\(213\) −16.8284 −1.15306
\(214\) 22.7279 1.55365
\(215\) −11.0711 −0.755041
\(216\) −24.9706 −1.69903
\(217\) 8.48528 0.576018
\(218\) 4.82843 0.327022
\(219\) −12.0000 −0.810885
\(220\) 13.0711 0.881251
\(221\) −0.828427 −0.0557260
\(222\) −28.9706 −1.94438
\(223\) 9.51472 0.637153 0.318576 0.947897i \(-0.396795\pi\)
0.318576 + 0.947897i \(0.396795\pi\)
\(224\) 7.65685 0.511595
\(225\) −1.00000 −0.0666667
\(226\) 21.3137 1.41777
\(227\) −16.3431 −1.08473 −0.542366 0.840142i \(-0.682472\pi\)
−0.542366 + 0.840142i \(0.682472\pi\)
\(228\) −3.17157 −0.210043
\(229\) −4.82843 −0.319071 −0.159536 0.987192i \(-0.551000\pi\)
−0.159536 + 0.987192i \(0.551000\pi\)
\(230\) −3.41421 −0.225127
\(231\) −23.3137 −1.53393
\(232\) 24.9706 1.63940
\(233\) −20.6274 −1.35135 −0.675674 0.737201i \(-0.736147\pi\)
−0.675674 + 0.737201i \(0.736147\pi\)
\(234\) −2.41421 −0.157822
\(235\) −4.82843 −0.314972
\(236\) 6.72792 0.437950
\(237\) 12.0000 0.779484
\(238\) −9.65685 −0.625961
\(239\) −3.41421 −0.220847 −0.110424 0.993885i \(-0.535221\pi\)
−0.110424 + 0.993885i \(0.535221\pi\)
\(240\) −4.24264 −0.273861
\(241\) −14.4853 −0.933079 −0.466539 0.884500i \(-0.654499\pi\)
−0.466539 + 0.884500i \(0.654499\pi\)
\(242\) −1.58579 −0.101938
\(243\) −9.89949 −0.635053
\(244\) −30.6274 −1.96072
\(245\) 16.3137 1.04224
\(246\) −10.8284 −0.690395
\(247\) −0.585786 −0.0372727
\(248\) −7.75736 −0.492593
\(249\) 4.48528 0.284243
\(250\) −2.41421 −0.152688
\(251\) 19.7990 1.24970 0.624851 0.780744i \(-0.285160\pi\)
0.624851 + 0.780744i \(0.285160\pi\)
\(252\) −18.4853 −1.16446
\(253\) 4.82843 0.303561
\(254\) 15.8995 0.997623
\(255\) −1.17157 −0.0733667
\(256\) −29.9706 −1.87316
\(257\) 27.6569 1.72519 0.862594 0.505898i \(-0.168839\pi\)
0.862594 + 0.505898i \(0.168839\pi\)
\(258\) −37.7990 −2.35326
\(259\) −40.9706 −2.54579
\(260\) −3.82843 −0.237429
\(261\) 5.65685 0.350150
\(262\) 40.9706 2.53117
\(263\) −10.5858 −0.652748 −0.326374 0.945241i \(-0.605827\pi\)
−0.326374 + 0.945241i \(0.605827\pi\)
\(264\) 21.3137 1.31177
\(265\) 2.48528 0.152670
\(266\) −6.82843 −0.418678
\(267\) −8.48528 −0.519291
\(268\) −7.65685 −0.467717
\(269\) −25.3137 −1.54340 −0.771702 0.635984i \(-0.780594\pi\)
−0.771702 + 0.635984i \(0.780594\pi\)
\(270\) −13.6569 −0.831130
\(271\) 26.7279 1.62361 0.811803 0.583932i \(-0.198486\pi\)
0.811803 + 0.583932i \(0.198486\pi\)
\(272\) 2.48528 0.150692
\(273\) 6.82843 0.413275
\(274\) 41.7990 2.52517
\(275\) 3.41421 0.205885
\(276\) −7.65685 −0.460888
\(277\) −12.8284 −0.770785 −0.385393 0.922753i \(-0.625934\pi\)
−0.385393 + 0.922753i \(0.625934\pi\)
\(278\) −10.8284 −0.649446
\(279\) −1.75736 −0.105210
\(280\) −21.3137 −1.27374
\(281\) 21.7990 1.30042 0.650209 0.759755i \(-0.274681\pi\)
0.650209 + 0.759755i \(0.274681\pi\)
\(282\) −16.4853 −0.981684
\(283\) −16.7279 −0.994372 −0.497186 0.867644i \(-0.665633\pi\)
−0.497186 + 0.867644i \(0.665633\pi\)
\(284\) 45.5563 2.70327
\(285\) −0.828427 −0.0490718
\(286\) 8.24264 0.487398
\(287\) −15.3137 −0.903940
\(288\) −1.58579 −0.0934434
\(289\) −16.3137 −0.959630
\(290\) 13.6569 0.801958
\(291\) 10.8284 0.634774
\(292\) 32.4853 1.90106
\(293\) 26.1421 1.52724 0.763620 0.645666i \(-0.223420\pi\)
0.763620 + 0.645666i \(0.223420\pi\)
\(294\) 55.6985 3.24840
\(295\) 1.75736 0.102317
\(296\) 37.4558 2.17708
\(297\) 19.3137 1.12070
\(298\) 28.1421 1.63023
\(299\) −1.41421 −0.0817861
\(300\) −5.41421 −0.312590
\(301\) −53.4558 −3.08114
\(302\) −23.5563 −1.35552
\(303\) 5.17157 0.297099
\(304\) 1.75736 0.100791
\(305\) −8.00000 −0.458079
\(306\) 2.00000 0.114332
\(307\) 24.8284 1.41703 0.708517 0.705694i \(-0.249365\pi\)
0.708517 + 0.705694i \(0.249365\pi\)
\(308\) 63.1127 3.59618
\(309\) −20.6274 −1.17345
\(310\) −4.24264 −0.240966
\(311\) 8.48528 0.481156 0.240578 0.970630i \(-0.422663\pi\)
0.240578 + 0.970630i \(0.422663\pi\)
\(312\) −6.24264 −0.353420
\(313\) −4.82843 −0.272919 −0.136459 0.990646i \(-0.543572\pi\)
−0.136459 + 0.990646i \(0.543572\pi\)
\(314\) −43.4558 −2.45236
\(315\) −4.82843 −0.272051
\(316\) −32.4853 −1.82744
\(317\) −2.14214 −0.120314 −0.0601572 0.998189i \(-0.519160\pi\)
−0.0601572 + 0.998189i \(0.519160\pi\)
\(318\) 8.48528 0.475831
\(319\) −19.3137 −1.08136
\(320\) −9.82843 −0.549426
\(321\) 13.3137 0.743099
\(322\) −16.4853 −0.918689
\(323\) 0.485281 0.0270018
\(324\) −19.1421 −1.06345
\(325\) −1.00000 −0.0554700
\(326\) −45.7990 −2.53657
\(327\) 2.82843 0.156412
\(328\) 14.0000 0.773021
\(329\) −23.3137 −1.28533
\(330\) 11.6569 0.641689
\(331\) −26.0416 −1.43138 −0.715689 0.698419i \(-0.753887\pi\)
−0.715689 + 0.698419i \(0.753887\pi\)
\(332\) −12.1421 −0.666386
\(333\) 8.48528 0.464991
\(334\) 7.65685 0.418964
\(335\) −2.00000 −0.109272
\(336\) −20.4853 −1.11756
\(337\) 12.8284 0.698809 0.349404 0.936972i \(-0.386384\pi\)
0.349404 + 0.936972i \(0.386384\pi\)
\(338\) −2.41421 −0.131316
\(339\) 12.4853 0.678107
\(340\) 3.17157 0.172003
\(341\) 6.00000 0.324918
\(342\) 1.41421 0.0764719
\(343\) 44.9706 2.42818
\(344\) 48.8701 2.63490
\(345\) −2.00000 −0.107676
\(346\) −40.6274 −2.18414
\(347\) −4.24264 −0.227757 −0.113878 0.993495i \(-0.536327\pi\)
−0.113878 + 0.993495i \(0.536327\pi\)
\(348\) 30.6274 1.64180
\(349\) 18.4853 0.989494 0.494747 0.869037i \(-0.335261\pi\)
0.494747 + 0.869037i \(0.335261\pi\)
\(350\) −11.6569 −0.623085
\(351\) −5.65685 −0.301941
\(352\) 5.41421 0.288579
\(353\) 14.8284 0.789238 0.394619 0.918845i \(-0.370877\pi\)
0.394619 + 0.918845i \(0.370877\pi\)
\(354\) 6.00000 0.318896
\(355\) 11.8995 0.631560
\(356\) 22.9706 1.21744
\(357\) −5.65685 −0.299392
\(358\) −13.6569 −0.721787
\(359\) −8.10051 −0.427528 −0.213764 0.976885i \(-0.568572\pi\)
−0.213764 + 0.976885i \(0.568572\pi\)
\(360\) 4.41421 0.232649
\(361\) −18.6569 −0.981940
\(362\) 0 0
\(363\) −0.928932 −0.0487563
\(364\) −18.4853 −0.968892
\(365\) 8.48528 0.444140
\(366\) −27.3137 −1.42771
\(367\) 35.5563 1.85603 0.928013 0.372547i \(-0.121516\pi\)
0.928013 + 0.372547i \(0.121516\pi\)
\(368\) 4.24264 0.221163
\(369\) 3.17157 0.165105
\(370\) 20.4853 1.06498
\(371\) 12.0000 0.623009
\(372\) −9.51472 −0.493315
\(373\) −2.68629 −0.139091 −0.0695455 0.997579i \(-0.522155\pi\)
−0.0695455 + 0.997579i \(0.522155\pi\)
\(374\) −6.82843 −0.353090
\(375\) −1.41421 −0.0730297
\(376\) 21.3137 1.09917
\(377\) 5.65685 0.291343
\(378\) −65.9411 −3.39165
\(379\) 29.0711 1.49328 0.746640 0.665228i \(-0.231666\pi\)
0.746640 + 0.665228i \(0.231666\pi\)
\(380\) 2.24264 0.115045
\(381\) 9.31371 0.477156
\(382\) −5.65685 −0.289430
\(383\) −29.1127 −1.48759 −0.743795 0.668408i \(-0.766976\pi\)
−0.743795 + 0.668408i \(0.766976\pi\)
\(384\) −29.0711 −1.48353
\(385\) 16.4853 0.840168
\(386\) −10.4853 −0.533687
\(387\) 11.0711 0.562774
\(388\) −29.3137 −1.48818
\(389\) 28.6274 1.45147 0.725734 0.687976i \(-0.241500\pi\)
0.725734 + 0.687976i \(0.241500\pi\)
\(390\) −3.41421 −0.172885
\(391\) 1.17157 0.0592490
\(392\) −72.0122 −3.63717
\(393\) 24.0000 1.21064
\(394\) −26.4853 −1.33431
\(395\) −8.48528 −0.426941
\(396\) −13.0711 −0.656846
\(397\) 11.7990 0.592174 0.296087 0.955161i \(-0.404318\pi\)
0.296087 + 0.955161i \(0.404318\pi\)
\(398\) −9.65685 −0.484054
\(399\) −4.00000 −0.200250
\(400\) 3.00000 0.150000
\(401\) −5.31371 −0.265354 −0.132677 0.991159i \(-0.542357\pi\)
−0.132677 + 0.991159i \(0.542357\pi\)
\(402\) −6.82843 −0.340571
\(403\) −1.75736 −0.0875403
\(404\) −14.0000 −0.696526
\(405\) −5.00000 −0.248452
\(406\) 65.9411 3.27260
\(407\) −28.9706 −1.43602
\(408\) 5.17157 0.256031
\(409\) 7.17157 0.354611 0.177306 0.984156i \(-0.443262\pi\)
0.177306 + 0.984156i \(0.443262\pi\)
\(410\) 7.65685 0.378145
\(411\) 24.4853 1.20777
\(412\) 55.8406 2.75107
\(413\) 8.48528 0.417533
\(414\) 3.41421 0.167799
\(415\) −3.17157 −0.155686
\(416\) −1.58579 −0.0777496
\(417\) −6.34315 −0.310625
\(418\) −4.82843 −0.236166
\(419\) 10.8284 0.529003 0.264502 0.964385i \(-0.414793\pi\)
0.264502 + 0.964385i \(0.414793\pi\)
\(420\) −26.1421 −1.27561
\(421\) −34.9706 −1.70436 −0.852180 0.523248i \(-0.824720\pi\)
−0.852180 + 0.523248i \(0.824720\pi\)
\(422\) −8.00000 −0.389434
\(423\) 4.82843 0.234766
\(424\) −10.9706 −0.532778
\(425\) 0.828427 0.0401846
\(426\) 40.6274 1.96840
\(427\) −38.6274 −1.86931
\(428\) −36.0416 −1.74214
\(429\) 4.82843 0.233119
\(430\) 26.7279 1.28893
\(431\) 40.3848 1.94527 0.972633 0.232346i \(-0.0746403\pi\)
0.972633 + 0.232346i \(0.0746403\pi\)
\(432\) 16.9706 0.816497
\(433\) 7.65685 0.367965 0.183982 0.982930i \(-0.441101\pi\)
0.183982 + 0.982930i \(0.441101\pi\)
\(434\) −20.4853 −0.983325
\(435\) 8.00000 0.383571
\(436\) −7.65685 −0.366697
\(437\) 0.828427 0.0396290
\(438\) 28.9706 1.38427
\(439\) 0.970563 0.0463224 0.0231612 0.999732i \(-0.492627\pi\)
0.0231612 + 0.999732i \(0.492627\pi\)
\(440\) −15.0711 −0.718485
\(441\) −16.3137 −0.776843
\(442\) 2.00000 0.0951303
\(443\) −9.41421 −0.447283 −0.223641 0.974671i \(-0.571794\pi\)
−0.223641 + 0.974671i \(0.571794\pi\)
\(444\) 45.9411 2.18027
\(445\) 6.00000 0.284427
\(446\) −22.9706 −1.08769
\(447\) 16.4853 0.779727
\(448\) −47.4558 −2.24208
\(449\) −33.1127 −1.56268 −0.781342 0.624103i \(-0.785465\pi\)
−0.781342 + 0.624103i \(0.785465\pi\)
\(450\) 2.41421 0.113807
\(451\) −10.8284 −0.509891
\(452\) −33.7990 −1.58977
\(453\) −13.7990 −0.648333
\(454\) 39.4558 1.85175
\(455\) −4.82843 −0.226360
\(456\) 3.65685 0.171248
\(457\) −18.0000 −0.842004 −0.421002 0.907060i \(-0.638322\pi\)
−0.421002 + 0.907060i \(0.638322\pi\)
\(458\) 11.6569 0.544689
\(459\) 4.68629 0.218737
\(460\) 5.41421 0.252439
\(461\) 9.51472 0.443145 0.221572 0.975144i \(-0.428881\pi\)
0.221572 + 0.975144i \(0.428881\pi\)
\(462\) 56.2843 2.61858
\(463\) −4.34315 −0.201843 −0.100922 0.994894i \(-0.532179\pi\)
−0.100922 + 0.994894i \(0.532179\pi\)
\(464\) −16.9706 −0.787839
\(465\) −2.48528 −0.115252
\(466\) 49.7990 2.30689
\(467\) −13.4142 −0.620736 −0.310368 0.950617i \(-0.600452\pi\)
−0.310368 + 0.950617i \(0.600452\pi\)
\(468\) 3.82843 0.176969
\(469\) −9.65685 −0.445912
\(470\) 11.6569 0.537691
\(471\) −25.4558 −1.17294
\(472\) −7.75736 −0.357061
\(473\) −37.7990 −1.73800
\(474\) −28.9706 −1.33066
\(475\) 0.585786 0.0268777
\(476\) 15.3137 0.701903
\(477\) −2.48528 −0.113793
\(478\) 8.24264 0.377010
\(479\) −30.7279 −1.40399 −0.701997 0.712180i \(-0.747708\pi\)
−0.701997 + 0.712180i \(0.747708\pi\)
\(480\) −2.24264 −0.102362
\(481\) 8.48528 0.386896
\(482\) 34.9706 1.59287
\(483\) −9.65685 −0.439402
\(484\) 2.51472 0.114305
\(485\) −7.65685 −0.347680
\(486\) 23.8995 1.08410
\(487\) −10.9706 −0.497124 −0.248562 0.968616i \(-0.579958\pi\)
−0.248562 + 0.968616i \(0.579958\pi\)
\(488\) 35.3137 1.59858
\(489\) −26.8284 −1.21322
\(490\) −39.3848 −1.77922
\(491\) 5.17157 0.233390 0.116695 0.993168i \(-0.462770\pi\)
0.116695 + 0.993168i \(0.462770\pi\)
\(492\) 17.1716 0.774154
\(493\) −4.68629 −0.211060
\(494\) 1.41421 0.0636285
\(495\) −3.41421 −0.153457
\(496\) 5.27208 0.236723
\(497\) 57.4558 2.57725
\(498\) −10.8284 −0.485233
\(499\) 41.5563 1.86032 0.930159 0.367157i \(-0.119669\pi\)
0.930159 + 0.367157i \(0.119669\pi\)
\(500\) 3.82843 0.171212
\(501\) 4.48528 0.200388
\(502\) −47.7990 −2.13337
\(503\) 37.8995 1.68985 0.844927 0.534881i \(-0.179644\pi\)
0.844927 + 0.534881i \(0.179644\pi\)
\(504\) 21.3137 0.949388
\(505\) −3.65685 −0.162728
\(506\) −11.6569 −0.518210
\(507\) −1.41421 −0.0628074
\(508\) −25.2132 −1.11866
\(509\) 41.1127 1.82229 0.911144 0.412088i \(-0.135200\pi\)
0.911144 + 0.412088i \(0.135200\pi\)
\(510\) 2.82843 0.125245
\(511\) 40.9706 1.81243
\(512\) 31.2426 1.38074
\(513\) 3.31371 0.146304
\(514\) −66.7696 −2.94508
\(515\) 14.5858 0.642727
\(516\) 59.9411 2.63876
\(517\) −16.4853 −0.725022
\(518\) 98.9117 4.34593
\(519\) −23.7990 −1.04466
\(520\) 4.41421 0.193576
\(521\) −17.6569 −0.773561 −0.386780 0.922172i \(-0.626413\pi\)
−0.386780 + 0.922172i \(0.626413\pi\)
\(522\) −13.6569 −0.597744
\(523\) −19.7574 −0.863929 −0.431965 0.901891i \(-0.642179\pi\)
−0.431965 + 0.901891i \(0.642179\pi\)
\(524\) −64.9706 −2.83825
\(525\) −6.82843 −0.298017
\(526\) 25.5563 1.11431
\(527\) 1.45584 0.0634176
\(528\) −14.4853 −0.630391
\(529\) −21.0000 −0.913043
\(530\) −6.00000 −0.260623
\(531\) −1.75736 −0.0762629
\(532\) 10.8284 0.469472
\(533\) 3.17157 0.137376
\(534\) 20.4853 0.886485
\(535\) −9.41421 −0.407012
\(536\) 8.82843 0.381330
\(537\) −8.00000 −0.345225
\(538\) 61.1127 2.63476
\(539\) 55.6985 2.39910
\(540\) 21.6569 0.931963
\(541\) −7.17157 −0.308330 −0.154165 0.988045i \(-0.549269\pi\)
−0.154165 + 0.988045i \(0.549269\pi\)
\(542\) −64.5269 −2.77167
\(543\) 0 0
\(544\) 1.31371 0.0563248
\(545\) −2.00000 −0.0856706
\(546\) −16.4853 −0.705505
\(547\) 13.2132 0.564956 0.282478 0.959274i \(-0.408844\pi\)
0.282478 + 0.959274i \(0.408844\pi\)
\(548\) −66.2843 −2.83152
\(549\) 8.00000 0.341432
\(550\) −8.24264 −0.351467
\(551\) −3.31371 −0.141169
\(552\) 8.82843 0.375763
\(553\) −40.9706 −1.74225
\(554\) 30.9706 1.31581
\(555\) 12.0000 0.509372
\(556\) 17.1716 0.728237
\(557\) −35.7990 −1.51685 −0.758426 0.651759i \(-0.774031\pi\)
−0.758426 + 0.651759i \(0.774031\pi\)
\(558\) 4.24264 0.179605
\(559\) 11.0711 0.468256
\(560\) 14.4853 0.612115
\(561\) −4.00000 −0.168880
\(562\) −52.6274 −2.21995
\(563\) −7.75736 −0.326934 −0.163467 0.986549i \(-0.552268\pi\)
−0.163467 + 0.986549i \(0.552268\pi\)
\(564\) 26.1421 1.10078
\(565\) −8.82843 −0.371415
\(566\) 40.3848 1.69750
\(567\) −24.1421 −1.01387
\(568\) −52.5269 −2.20398
\(569\) −10.3431 −0.433607 −0.216804 0.976215i \(-0.569563\pi\)
−0.216804 + 0.976215i \(0.569563\pi\)
\(570\) 2.00000 0.0837708
\(571\) −11.5147 −0.481876 −0.240938 0.970541i \(-0.577455\pi\)
−0.240938 + 0.970541i \(0.577455\pi\)
\(572\) −13.0711 −0.546529
\(573\) −3.31371 −0.138432
\(574\) 36.9706 1.54312
\(575\) 1.41421 0.0589768
\(576\) 9.82843 0.409518
\(577\) −34.8284 −1.44993 −0.724963 0.688788i \(-0.758143\pi\)
−0.724963 + 0.688788i \(0.758143\pi\)
\(578\) 39.3848 1.63819
\(579\) −6.14214 −0.255258
\(580\) −21.6569 −0.899252
\(581\) −15.3137 −0.635320
\(582\) −26.1421 −1.08363
\(583\) 8.48528 0.351424
\(584\) −37.4558 −1.54993
\(585\) 1.00000 0.0413449
\(586\) −63.1127 −2.60716
\(587\) 20.3431 0.839651 0.419826 0.907605i \(-0.362091\pi\)
0.419826 + 0.907605i \(0.362091\pi\)
\(588\) −88.3259 −3.64250
\(589\) 1.02944 0.0424172
\(590\) −4.24264 −0.174667
\(591\) −15.5147 −0.638190
\(592\) −25.4558 −1.04623
\(593\) −24.6274 −1.01133 −0.505663 0.862731i \(-0.668752\pi\)
−0.505663 + 0.862731i \(0.668752\pi\)
\(594\) −46.6274 −1.91315
\(595\) 4.00000 0.163984
\(596\) −44.6274 −1.82801
\(597\) −5.65685 −0.231520
\(598\) 3.41421 0.139618
\(599\) 25.4558 1.04010 0.520049 0.854137i \(-0.325914\pi\)
0.520049 + 0.854137i \(0.325914\pi\)
\(600\) 6.24264 0.254855
\(601\) −44.6274 −1.82039 −0.910195 0.414180i \(-0.864069\pi\)
−0.910195 + 0.414180i \(0.864069\pi\)
\(602\) 129.054 5.25984
\(603\) 2.00000 0.0814463
\(604\) 37.3553 1.51997
\(605\) 0.656854 0.0267049
\(606\) −12.4853 −0.507180
\(607\) 31.7574 1.28899 0.644496 0.764608i \(-0.277067\pi\)
0.644496 + 0.764608i \(0.277067\pi\)
\(608\) 0.928932 0.0376732
\(609\) 38.6274 1.56526
\(610\) 19.3137 0.781989
\(611\) 4.82843 0.195337
\(612\) −3.17157 −0.128203
\(613\) −14.6863 −0.593174 −0.296587 0.955006i \(-0.595848\pi\)
−0.296587 + 0.955006i \(0.595848\pi\)
\(614\) −59.9411 −2.41903
\(615\) 4.48528 0.180864
\(616\) −72.7696 −2.93197
\(617\) 10.9706 0.441658 0.220829 0.975313i \(-0.429124\pi\)
0.220829 + 0.975313i \(0.429124\pi\)
\(618\) 49.7990 2.00321
\(619\) 1.75736 0.0706342 0.0353171 0.999376i \(-0.488756\pi\)
0.0353171 + 0.999376i \(0.488756\pi\)
\(620\) 6.72792 0.270200
\(621\) 8.00000 0.321029
\(622\) −20.4853 −0.821385
\(623\) 28.9706 1.16068
\(624\) 4.24264 0.169842
\(625\) 1.00000 0.0400000
\(626\) 11.6569 0.465902
\(627\) −2.82843 −0.112956
\(628\) 68.9117 2.74988
\(629\) −7.02944 −0.280282
\(630\) 11.6569 0.464420
\(631\) −9.75736 −0.388434 −0.194217 0.980959i \(-0.562217\pi\)
−0.194217 + 0.980959i \(0.562217\pi\)
\(632\) 37.4558 1.48991
\(633\) −4.68629 −0.186263
\(634\) 5.17157 0.205389
\(635\) −6.58579 −0.261349
\(636\) −13.4558 −0.533559
\(637\) −16.3137 −0.646373
\(638\) 46.6274 1.84600
\(639\) −11.8995 −0.470737
\(640\) 20.5563 0.812561
\(641\) 47.6569 1.88233 0.941166 0.337944i \(-0.109731\pi\)
0.941166 + 0.337944i \(0.109731\pi\)
\(642\) −32.1421 −1.26855
\(643\) 9.51472 0.375224 0.187612 0.982243i \(-0.439925\pi\)
0.187612 + 0.982243i \(0.439925\pi\)
\(644\) 26.1421 1.03014
\(645\) 15.6569 0.616488
\(646\) −1.17157 −0.0460949
\(647\) 9.41421 0.370111 0.185055 0.982728i \(-0.440754\pi\)
0.185055 + 0.982728i \(0.440754\pi\)
\(648\) 22.0711 0.867033
\(649\) 6.00000 0.235521
\(650\) 2.41421 0.0946932
\(651\) −12.0000 −0.470317
\(652\) 72.6274 2.84431
\(653\) 46.9706 1.83810 0.919050 0.394141i \(-0.128958\pi\)
0.919050 + 0.394141i \(0.128958\pi\)
\(654\) −6.82843 −0.267013
\(655\) −16.9706 −0.663095
\(656\) −9.51472 −0.371487
\(657\) −8.48528 −0.331042
\(658\) 56.2843 2.19419
\(659\) 17.8579 0.695644 0.347822 0.937561i \(-0.386921\pi\)
0.347822 + 0.937561i \(0.386921\pi\)
\(660\) −18.4853 −0.719539
\(661\) 29.5980 1.15123 0.575614 0.817722i \(-0.304763\pi\)
0.575614 + 0.817722i \(0.304763\pi\)
\(662\) 62.8701 2.44351
\(663\) 1.17157 0.0455001
\(664\) 14.0000 0.543305
\(665\) 2.82843 0.109682
\(666\) −20.4853 −0.793789
\(667\) −8.00000 −0.309761
\(668\) −12.1421 −0.469793
\(669\) −13.4558 −0.520233
\(670\) 4.82843 0.186538
\(671\) −27.3137 −1.05443
\(672\) −10.8284 −0.417716
\(673\) 6.48528 0.249989 0.124995 0.992157i \(-0.460109\pi\)
0.124995 + 0.992157i \(0.460109\pi\)
\(674\) −30.9706 −1.19294
\(675\) 5.65685 0.217732
\(676\) 3.82843 0.147247
\(677\) −20.1421 −0.774125 −0.387063 0.922053i \(-0.626510\pi\)
−0.387063 + 0.922053i \(0.626510\pi\)
\(678\) −30.1421 −1.15760
\(679\) −36.9706 −1.41880
\(680\) −3.65685 −0.140234
\(681\) 23.1127 0.885681
\(682\) −14.4853 −0.554670
\(683\) 10.6863 0.408900 0.204450 0.978877i \(-0.434459\pi\)
0.204450 + 0.978877i \(0.434459\pi\)
\(684\) −2.24264 −0.0857495
\(685\) −17.3137 −0.661523
\(686\) −108.569 −4.14517
\(687\) 6.82843 0.260521
\(688\) −33.2132 −1.26624
\(689\) −2.48528 −0.0946817
\(690\) 4.82843 0.183815
\(691\) 6.92893 0.263589 0.131795 0.991277i \(-0.457926\pi\)
0.131795 + 0.991277i \(0.457926\pi\)
\(692\) 64.4264 2.44912
\(693\) −16.4853 −0.626224
\(694\) 10.2426 0.388805
\(695\) 4.48528 0.170136
\(696\) −35.3137 −1.33856
\(697\) −2.62742 −0.0995205
\(698\) −44.6274 −1.68917
\(699\) 29.1716 1.10337
\(700\) 18.4853 0.698678
\(701\) 14.6863 0.554694 0.277347 0.960770i \(-0.410545\pi\)
0.277347 + 0.960770i \(0.410545\pi\)
\(702\) 13.6569 0.515445
\(703\) −4.97056 −0.187468
\(704\) −33.5563 −1.26470
\(705\) 6.82843 0.257173
\(706\) −35.7990 −1.34731
\(707\) −17.6569 −0.664054
\(708\) −9.51472 −0.357585
\(709\) 45.1127 1.69424 0.847121 0.531399i \(-0.178334\pi\)
0.847121 + 0.531399i \(0.178334\pi\)
\(710\) −28.7279 −1.07814
\(711\) 8.48528 0.318223
\(712\) −26.4853 −0.992578
\(713\) 2.48528 0.0930745
\(714\) 13.6569 0.511095
\(715\) −3.41421 −0.127684
\(716\) 21.6569 0.809355
\(717\) 4.82843 0.180321
\(718\) 19.5563 0.729836
\(719\) 28.9706 1.08042 0.540210 0.841530i \(-0.318345\pi\)
0.540210 + 0.841530i \(0.318345\pi\)
\(720\) −3.00000 −0.111803
\(721\) 70.4264 2.62282
\(722\) 45.0416 1.67628
\(723\) 20.4853 0.761856
\(724\) 0 0
\(725\) −5.65685 −0.210090
\(726\) 2.24264 0.0832322
\(727\) −51.3553 −1.90466 −0.952332 0.305063i \(-0.901322\pi\)
−0.952332 + 0.305063i \(0.901322\pi\)
\(728\) 21.3137 0.789939
\(729\) 29.0000 1.07407
\(730\) −20.4853 −0.758194
\(731\) −9.17157 −0.339223
\(732\) 43.3137 1.60092
\(733\) 21.3137 0.787240 0.393620 0.919273i \(-0.371223\pi\)
0.393620 + 0.919273i \(0.371223\pi\)
\(734\) −85.8406 −3.16844
\(735\) −23.0711 −0.850989
\(736\) 2.24264 0.0826648
\(737\) −6.82843 −0.251528
\(738\) −7.65685 −0.281853
\(739\) 5.27208 0.193937 0.0969683 0.995287i \(-0.469085\pi\)
0.0969683 + 0.995287i \(0.469085\pi\)
\(740\) −32.4853 −1.19418
\(741\) 0.828427 0.0304330
\(742\) −28.9706 −1.06354
\(743\) −21.5147 −0.789298 −0.394649 0.918832i \(-0.629134\pi\)
−0.394649 + 0.918832i \(0.629134\pi\)
\(744\) 10.9706 0.402200
\(745\) −11.6569 −0.427074
\(746\) 6.48528 0.237443
\(747\) 3.17157 0.116042
\(748\) 10.8284 0.395927
\(749\) −45.4558 −1.66092
\(750\) 3.41421 0.124669
\(751\) −27.5147 −1.00403 −0.502013 0.864860i \(-0.667407\pi\)
−0.502013 + 0.864860i \(0.667407\pi\)
\(752\) −14.4853 −0.528224
\(753\) −28.0000 −1.02038
\(754\) −13.6569 −0.497353
\(755\) 9.75736 0.355107
\(756\) 104.569 3.80312
\(757\) −24.1421 −0.877461 −0.438730 0.898619i \(-0.644572\pi\)
−0.438730 + 0.898619i \(0.644572\pi\)
\(758\) −70.1838 −2.54919
\(759\) −6.82843 −0.247856
\(760\) −2.58579 −0.0937963
\(761\) 8.62742 0.312744 0.156372 0.987698i \(-0.450020\pi\)
0.156372 + 0.987698i \(0.450020\pi\)
\(762\) −22.4853 −0.814556
\(763\) −9.65685 −0.349602
\(764\) 8.97056 0.324544
\(765\) −0.828427 −0.0299518
\(766\) 70.2843 2.53947
\(767\) −1.75736 −0.0634546
\(768\) 42.3848 1.52943
\(769\) −22.9706 −0.828340 −0.414170 0.910200i \(-0.635928\pi\)
−0.414170 + 0.910200i \(0.635928\pi\)
\(770\) −39.7990 −1.43426
\(771\) −39.1127 −1.40861
\(772\) 16.6274 0.598434
\(773\) −22.1421 −0.796397 −0.398199 0.917299i \(-0.630365\pi\)
−0.398199 + 0.917299i \(0.630365\pi\)
\(774\) −26.7279 −0.960715
\(775\) 1.75736 0.0631262
\(776\) 33.7990 1.21331
\(777\) 57.9411 2.07863
\(778\) −69.1127 −2.47781
\(779\) −1.85786 −0.0665649
\(780\) 5.41421 0.193860
\(781\) 40.6274 1.45376
\(782\) −2.82843 −0.101144
\(783\) −32.0000 −1.14359
\(784\) 48.9411 1.74790
\(785\) 18.0000 0.642448
\(786\) −57.9411 −2.06669
\(787\) 22.4853 0.801514 0.400757 0.916184i \(-0.368747\pi\)
0.400757 + 0.916184i \(0.368747\pi\)
\(788\) 42.0000 1.49619
\(789\) 14.9706 0.532966
\(790\) 20.4853 0.728834
\(791\) −42.6274 −1.51566
\(792\) 15.0711 0.535527
\(793\) 8.00000 0.284088
\(794\) −28.4853 −1.01090
\(795\) −3.51472 −0.124654
\(796\) 15.3137 0.542780
\(797\) −22.9706 −0.813659 −0.406830 0.913504i \(-0.633366\pi\)
−0.406830 + 0.913504i \(0.633366\pi\)
\(798\) 9.65685 0.341849
\(799\) −4.00000 −0.141510
\(800\) 1.58579 0.0560660
\(801\) −6.00000 −0.212000
\(802\) 12.8284 0.452988
\(803\) 28.9706 1.02235
\(804\) 10.8284 0.381889
\(805\) 6.82843 0.240670
\(806\) 4.24264 0.149441
\(807\) 35.7990 1.26018
\(808\) 16.1421 0.567878
\(809\) 45.2548 1.59108 0.795538 0.605904i \(-0.207189\pi\)
0.795538 + 0.605904i \(0.207189\pi\)
\(810\) 12.0711 0.424134
\(811\) −28.3848 −0.996724 −0.498362 0.866969i \(-0.666065\pi\)
−0.498362 + 0.866969i \(0.666065\pi\)
\(812\) −104.569 −3.66964
\(813\) −37.7990 −1.32567
\(814\) 69.9411 2.45144
\(815\) 18.9706 0.664510
\(816\) −3.51472 −0.123040
\(817\) −6.48528 −0.226891
\(818\) −17.3137 −0.605360
\(819\) 4.82843 0.168719
\(820\) −12.1421 −0.424022
\(821\) −51.2548 −1.78881 −0.894403 0.447262i \(-0.852399\pi\)
−0.894403 + 0.447262i \(0.852399\pi\)
\(822\) −59.1127 −2.06179
\(823\) −2.38478 −0.0831281 −0.0415640 0.999136i \(-0.513234\pi\)
−0.0415640 + 0.999136i \(0.513234\pi\)
\(824\) −64.3848 −2.24295
\(825\) −4.82843 −0.168104
\(826\) −20.4853 −0.712774
\(827\) 56.1421 1.95225 0.976127 0.217202i \(-0.0696930\pi\)
0.976127 + 0.217202i \(0.0696930\pi\)
\(828\) −5.41421 −0.188157
\(829\) 40.9706 1.42297 0.711483 0.702703i \(-0.248024\pi\)
0.711483 + 0.702703i \(0.248024\pi\)
\(830\) 7.65685 0.265773
\(831\) 18.1421 0.629344
\(832\) 9.82843 0.340739
\(833\) 13.5147 0.468257
\(834\) 15.3137 0.530270
\(835\) −3.17157 −0.109757
\(836\) 7.65685 0.264818
\(837\) 9.94113 0.343616
\(838\) −26.1421 −0.903065
\(839\) 6.72792 0.232274 0.116137 0.993233i \(-0.462949\pi\)
0.116137 + 0.993233i \(0.462949\pi\)
\(840\) 30.1421 1.04000
\(841\) 3.00000 0.103448
\(842\) 84.4264 2.90953
\(843\) −30.8284 −1.06179
\(844\) 12.6863 0.436680
\(845\) 1.00000 0.0344010
\(846\) −11.6569 −0.400771
\(847\) 3.17157 0.108977
\(848\) 7.45584 0.256035
\(849\) 23.6569 0.811901
\(850\) −2.00000 −0.0685994
\(851\) −12.0000 −0.411355
\(852\) −64.4264 −2.20721
\(853\) −13.4558 −0.460719 −0.230360 0.973106i \(-0.573990\pi\)
−0.230360 + 0.973106i \(0.573990\pi\)
\(854\) 93.2548 3.19111
\(855\) −0.585786 −0.0200335
\(856\) 41.5563 1.42037
\(857\) 11.6569 0.398191 0.199095 0.979980i \(-0.436200\pi\)
0.199095 + 0.979980i \(0.436200\pi\)
\(858\) −11.6569 −0.397958
\(859\) −27.7990 −0.948489 −0.474245 0.880393i \(-0.657279\pi\)
−0.474245 + 0.880393i \(0.657279\pi\)
\(860\) −42.3848 −1.44531
\(861\) 21.6569 0.738064
\(862\) −97.4975 −3.32078
\(863\) −31.4558 −1.07077 −0.535385 0.844608i \(-0.679833\pi\)
−0.535385 + 0.844608i \(0.679833\pi\)
\(864\) 8.97056 0.305185
\(865\) 16.8284 0.572184
\(866\) −18.4853 −0.628155
\(867\) 23.0711 0.783535
\(868\) 32.4853 1.10262
\(869\) −28.9706 −0.982759
\(870\) −19.3137 −0.654796
\(871\) 2.00000 0.0677674
\(872\) 8.82843 0.298968
\(873\) 7.65685 0.259145
\(874\) −2.00000 −0.0676510
\(875\) 4.82843 0.163231
\(876\) −45.9411 −1.55221
\(877\) 25.3137 0.854783 0.427392 0.904067i \(-0.359433\pi\)
0.427392 + 0.904067i \(0.359433\pi\)
\(878\) −2.34315 −0.0790773
\(879\) −36.9706 −1.24699
\(880\) 10.2426 0.345279
\(881\) 19.0294 0.641118 0.320559 0.947229i \(-0.396129\pi\)
0.320559 + 0.947229i \(0.396129\pi\)
\(882\) 39.3848 1.32615
\(883\) −23.7574 −0.799499 −0.399749 0.916624i \(-0.630903\pi\)
−0.399749 + 0.916624i \(0.630903\pi\)
\(884\) −3.17157 −0.106672
\(885\) −2.48528 −0.0835418
\(886\) 22.7279 0.763559
\(887\) −22.3848 −0.751607 −0.375804 0.926699i \(-0.622633\pi\)
−0.375804 + 0.926699i \(0.622633\pi\)
\(888\) −52.9706 −1.77758
\(889\) −31.7990 −1.06650
\(890\) −14.4853 −0.485548
\(891\) −17.0711 −0.571902
\(892\) 36.4264 1.21965
\(893\) −2.82843 −0.0946497
\(894\) −39.7990 −1.33108
\(895\) 5.65685 0.189088
\(896\) 99.2548 3.31587
\(897\) 2.00000 0.0667781
\(898\) 79.9411 2.66767
\(899\) −9.94113 −0.331555
\(900\) −3.82843 −0.127614
\(901\) 2.05887 0.0685911
\(902\) 26.1421 0.870438
\(903\) 75.5980 2.51574
\(904\) 38.9706 1.29614
\(905\) 0 0
\(906\) 33.3137 1.10677
\(907\) 9.21320 0.305919 0.152960 0.988232i \(-0.451120\pi\)
0.152960 + 0.988232i \(0.451120\pi\)
\(908\) −62.5685 −2.07641
\(909\) 3.65685 0.121290
\(910\) 11.6569 0.386421
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) −2.48528 −0.0822959
\(913\) −10.8284 −0.358369
\(914\) 43.4558 1.43739
\(915\) 11.3137 0.374020
\(916\) −18.4853 −0.610771
\(917\) −81.9411 −2.70593
\(918\) −11.3137 −0.373408
\(919\) 0.485281 0.0160080 0.00800398 0.999968i \(-0.497452\pi\)
0.00800398 + 0.999968i \(0.497452\pi\)
\(920\) −6.24264 −0.205814
\(921\) −35.1127 −1.15700
\(922\) −22.9706 −0.756495
\(923\) −11.8995 −0.391677
\(924\) −89.2548 −2.93627
\(925\) −8.48528 −0.278994
\(926\) 10.4853 0.344568
\(927\) −14.5858 −0.479060
\(928\) −8.97056 −0.294473
\(929\) 16.8284 0.552123 0.276061 0.961140i \(-0.410971\pi\)
0.276061 + 0.961140i \(0.410971\pi\)
\(930\) 6.00000 0.196748
\(931\) 9.55635 0.313197
\(932\) −78.9706 −2.58677
\(933\) −12.0000 −0.392862
\(934\) 32.3848 1.05966
\(935\) 2.82843 0.0924995
\(936\) −4.41421 −0.144283
\(937\) −22.9706 −0.750416 −0.375208 0.926941i \(-0.622429\pi\)
−0.375208 + 0.926941i \(0.622429\pi\)
\(938\) 23.3137 0.761220
\(939\) 6.82843 0.222837
\(940\) −18.4853 −0.602923
\(941\) 18.7696 0.611870 0.305935 0.952052i \(-0.401031\pi\)
0.305935 + 0.952052i \(0.401031\pi\)
\(942\) 61.4558 2.00234
\(943\) −4.48528 −0.146061
\(944\) 5.27208 0.171592
\(945\) 27.3137 0.888515
\(946\) 91.2548 2.96695
\(947\) 17.1127 0.556088 0.278044 0.960568i \(-0.410314\pi\)
0.278044 + 0.960568i \(0.410314\pi\)
\(948\) 45.9411 1.49210
\(949\) −8.48528 −0.275444
\(950\) −1.41421 −0.0458831
\(951\) 3.02944 0.0982362
\(952\) −17.6569 −0.572262
\(953\) 35.2548 1.14202 0.571008 0.820944i \(-0.306553\pi\)
0.571008 + 0.820944i \(0.306553\pi\)
\(954\) 6.00000 0.194257
\(955\) 2.34315 0.0758224
\(956\) −13.0711 −0.422749
\(957\) 27.3137 0.882927
\(958\) 74.1838 2.39677
\(959\) −83.5980 −2.69952
\(960\) 13.8995 0.448604
\(961\) −27.9117 −0.900377
\(962\) −20.4853 −0.660472
\(963\) 9.41421 0.303369
\(964\) −55.4558 −1.78611
\(965\) 4.34315 0.139811
\(966\) 23.3137 0.750106
\(967\) 47.9411 1.54168 0.770841 0.637027i \(-0.219836\pi\)
0.770841 + 0.637027i \(0.219836\pi\)
\(968\) −2.89949 −0.0931933
\(969\) −0.686292 −0.0220469
\(970\) 18.4853 0.593527
\(971\) 44.2843 1.42115 0.710575 0.703622i \(-0.248435\pi\)
0.710575 + 0.703622i \(0.248435\pi\)
\(972\) −37.8995 −1.21563
\(973\) 21.6569 0.694287
\(974\) 26.4853 0.848643
\(975\) 1.41421 0.0452911
\(976\) −24.0000 −0.768221
\(977\) −39.5147 −1.26419 −0.632094 0.774892i \(-0.717804\pi\)
−0.632094 + 0.774892i \(0.717804\pi\)
\(978\) 64.7696 2.07110
\(979\) 20.4853 0.654712
\(980\) 62.4558 1.99508
\(981\) 2.00000 0.0638551
\(982\) −12.4853 −0.398421
\(983\) −1.02944 −0.0328339 −0.0164170 0.999865i \(-0.505226\pi\)
−0.0164170 + 0.999865i \(0.505226\pi\)
\(984\) −19.7990 −0.631169
\(985\) 10.9706 0.349551
\(986\) 11.3137 0.360302
\(987\) 32.9706 1.04946
\(988\) −2.24264 −0.0713479
\(989\) −15.6569 −0.497859
\(990\) 8.24264 0.261968
\(991\) 48.9706 1.55560 0.777801 0.628511i \(-0.216335\pi\)
0.777801 + 0.628511i \(0.216335\pi\)
\(992\) 2.78680 0.0884809
\(993\) 36.8284 1.16871
\(994\) −138.711 −4.39964
\(995\) 4.00000 0.126809
\(996\) 17.1716 0.544102
\(997\) 28.8284 0.913005 0.456503 0.889722i \(-0.349102\pi\)
0.456503 + 0.889722i \(0.349102\pi\)
\(998\) −100.326 −3.17576
\(999\) −48.0000 −1.51865
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 65.2.a.b.1.1 2
3.2 odd 2 585.2.a.m.1.2 2
4.3 odd 2 1040.2.a.j.1.2 2
5.2 odd 4 325.2.b.f.274.1 4
5.3 odd 4 325.2.b.f.274.4 4
5.4 even 2 325.2.a.i.1.2 2
7.6 odd 2 3185.2.a.j.1.1 2
8.3 odd 2 4160.2.a.z.1.1 2
8.5 even 2 4160.2.a.bf.1.2 2
11.10 odd 2 7865.2.a.j.1.2 2
12.11 even 2 9360.2.a.cd.1.1 2
13.2 odd 12 845.2.m.f.316.1 8
13.3 even 3 845.2.e.h.191.2 4
13.4 even 6 845.2.e.c.146.1 4
13.5 odd 4 845.2.c.b.506.4 4
13.6 odd 12 845.2.m.f.361.4 8
13.7 odd 12 845.2.m.f.361.1 8
13.8 odd 4 845.2.c.b.506.1 4
13.9 even 3 845.2.e.h.146.2 4
13.10 even 6 845.2.e.c.191.1 4
13.11 odd 12 845.2.m.f.316.4 8
13.12 even 2 845.2.a.g.1.2 2
15.2 even 4 2925.2.c.r.2224.4 4
15.8 even 4 2925.2.c.r.2224.1 4
15.14 odd 2 2925.2.a.u.1.1 2
20.19 odd 2 5200.2.a.bu.1.1 2
39.38 odd 2 7605.2.a.x.1.1 2
65.64 even 2 4225.2.a.r.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.a.b.1.1 2 1.1 even 1 trivial
325.2.a.i.1.2 2 5.4 even 2
325.2.b.f.274.1 4 5.2 odd 4
325.2.b.f.274.4 4 5.3 odd 4
585.2.a.m.1.2 2 3.2 odd 2
845.2.a.g.1.2 2 13.12 even 2
845.2.c.b.506.1 4 13.8 odd 4
845.2.c.b.506.4 4 13.5 odd 4
845.2.e.c.146.1 4 13.4 even 6
845.2.e.c.191.1 4 13.10 even 6
845.2.e.h.146.2 4 13.9 even 3
845.2.e.h.191.2 4 13.3 even 3
845.2.m.f.316.1 8 13.2 odd 12
845.2.m.f.316.4 8 13.11 odd 12
845.2.m.f.361.1 8 13.7 odd 12
845.2.m.f.361.4 8 13.6 odd 12
1040.2.a.j.1.2 2 4.3 odd 2
2925.2.a.u.1.1 2 15.14 odd 2
2925.2.c.r.2224.1 4 15.8 even 4
2925.2.c.r.2224.4 4 15.2 even 4
3185.2.a.j.1.1 2 7.6 odd 2
4160.2.a.z.1.1 2 8.3 odd 2
4160.2.a.bf.1.2 2 8.5 even 2
4225.2.a.r.1.1 2 65.64 even 2
5200.2.a.bu.1.1 2 20.19 odd 2
7605.2.a.x.1.1 2 39.38 odd 2
7865.2.a.j.1.2 2 11.10 odd 2
9360.2.a.cd.1.1 2 12.11 even 2