Properties

Label 5225.2.a.s.1.5
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $1$
Dimension $15$
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] = [5225,2,Mod(1,5225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5225, 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("5225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5225.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: \(41.7218350561\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 5 x^{14} - 11 x^{13} + 85 x^{12} + 6 x^{11} - 537 x^{10} + 327 x^{9} + 1556 x^{8} - 1451 x^{7} + \cdots - 13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(1.38376\) of defining polynomial
Character \(\chi\) \(=\) 5225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.38376 q^{2} +0.179641 q^{3} -0.0852182 q^{4} -0.248579 q^{6} -4.32482 q^{7} +2.88543 q^{8} -2.96773 q^{9} +O(q^{10})\) \(q-1.38376 q^{2} +0.179641 q^{3} -0.0852182 q^{4} -0.248579 q^{6} -4.32482 q^{7} +2.88543 q^{8} -2.96773 q^{9} +1.00000 q^{11} -0.0153087 q^{12} -0.663216 q^{13} +5.98449 q^{14} -3.82230 q^{16} +5.90319 q^{17} +4.10661 q^{18} -1.00000 q^{19} -0.776914 q^{21} -1.38376 q^{22} -0.697968 q^{23} +0.518342 q^{24} +0.917729 q^{26} -1.07205 q^{27} +0.368553 q^{28} -3.32909 q^{29} +10.4922 q^{31} -0.481734 q^{32} +0.179641 q^{33} -8.16858 q^{34} +0.252904 q^{36} -10.8094 q^{37} +1.38376 q^{38} -0.119141 q^{39} +6.31607 q^{41} +1.07506 q^{42} -2.76329 q^{43} -0.0852182 q^{44} +0.965818 q^{46} -10.4716 q^{47} -0.686641 q^{48} +11.7040 q^{49} +1.06045 q^{51} +0.0565180 q^{52} +8.10587 q^{53} +1.48345 q^{54} -12.4790 q^{56} -0.179641 q^{57} +4.60665 q^{58} -0.264035 q^{59} +14.2963 q^{61} -14.5186 q^{62} +12.8349 q^{63} +8.31121 q^{64} -0.248579 q^{66} +14.6511 q^{67} -0.503059 q^{68} -0.125384 q^{69} -9.64601 q^{71} -8.56319 q^{72} +0.803845 q^{73} +14.9576 q^{74} +0.0852182 q^{76} -4.32482 q^{77} +0.164862 q^{78} +8.25623 q^{79} +8.71060 q^{81} -8.73990 q^{82} -1.39198 q^{83} +0.0662072 q^{84} +3.82372 q^{86} -0.598040 q^{87} +2.88543 q^{88} +9.69068 q^{89} +2.86829 q^{91} +0.0594796 q^{92} +1.88482 q^{93} +14.4901 q^{94} -0.0865391 q^{96} -6.48020 q^{97} -16.1955 q^{98} -2.96773 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - 5 q^{2} - 4 q^{3} + 17 q^{4} - q^{6} - 15 q^{7} - 15 q^{8} + 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - 5 q^{2} - 4 q^{3} + 17 q^{4} - q^{6} - 15 q^{7} - 15 q^{8} + 19 q^{9} + 15 q^{11} - 9 q^{12} - 13 q^{13} + 9 q^{14} + 21 q^{16} - 11 q^{17} - 16 q^{18} - 15 q^{19} - 5 q^{22} - 10 q^{23} + 17 q^{24} - 17 q^{26} - 13 q^{27} - 30 q^{28} + 5 q^{29} + 6 q^{31} - 40 q^{32} - 4 q^{33} + 17 q^{34} + 28 q^{36} - 13 q^{37} + 5 q^{38} - 22 q^{39} - 30 q^{42} - 36 q^{43} + 17 q^{44} + 13 q^{46} + 6 q^{47} - 14 q^{48} + 16 q^{49} + 4 q^{51} - 50 q^{52} - 9 q^{53} + 9 q^{54} - 18 q^{56} + 4 q^{57} - 2 q^{58} - 7 q^{59} - 2 q^{61} - 11 q^{62} - 39 q^{63} + 17 q^{64} - q^{66} - 35 q^{67} + 18 q^{68} - 9 q^{69} + 13 q^{71} - 68 q^{72} - 2 q^{73} + 13 q^{74} - 17 q^{76} - 15 q^{77} + 10 q^{78} + 6 q^{79} + 11 q^{81} - 14 q^{82} - 30 q^{83} - 6 q^{84} - 25 q^{86} - 19 q^{87} - 15 q^{88} + 55 q^{89} + 26 q^{91} + 18 q^{92} - 14 q^{93} - 22 q^{94} - 17 q^{96} - 28 q^{97} + 22 q^{98} + 19 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.38376 −0.978464 −0.489232 0.872154i \(-0.662723\pi\)
−0.489232 + 0.872154i \(0.662723\pi\)
\(3\) 0.179641 0.103716 0.0518578 0.998654i \(-0.483486\pi\)
0.0518578 + 0.998654i \(0.483486\pi\)
\(4\) −0.0852182 −0.0426091
\(5\) 0 0
\(6\) −0.248579 −0.101482
\(7\) −4.32482 −1.63463 −0.817313 0.576193i \(-0.804538\pi\)
−0.817313 + 0.576193i \(0.804538\pi\)
\(8\) 2.88543 1.02015
\(9\) −2.96773 −0.989243
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) −0.0153087 −0.00441923
\(13\) −0.663216 −0.183943 −0.0919715 0.995762i \(-0.529317\pi\)
−0.0919715 + 0.995762i \(0.529317\pi\)
\(14\) 5.98449 1.59942
\(15\) 0 0
\(16\) −3.82230 −0.955575
\(17\) 5.90319 1.43173 0.715867 0.698237i \(-0.246032\pi\)
0.715867 + 0.698237i \(0.246032\pi\)
\(18\) 4.10661 0.967938
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −0.776914 −0.169536
\(22\) −1.38376 −0.295018
\(23\) −0.697968 −0.145536 −0.0727682 0.997349i \(-0.523183\pi\)
−0.0727682 + 0.997349i \(0.523183\pi\)
\(24\) 0.518342 0.105806
\(25\) 0 0
\(26\) 0.917729 0.179982
\(27\) −1.07205 −0.206316
\(28\) 0.368553 0.0696500
\(29\) −3.32909 −0.618196 −0.309098 0.951030i \(-0.600027\pi\)
−0.309098 + 0.951030i \(0.600027\pi\)
\(30\) 0 0
\(31\) 10.4922 1.88445 0.942224 0.334984i \(-0.108731\pi\)
0.942224 + 0.334984i \(0.108731\pi\)
\(32\) −0.481734 −0.0851593
\(33\) 0.179641 0.0312715
\(34\) −8.16858 −1.40090
\(35\) 0 0
\(36\) 0.252904 0.0421507
\(37\) −10.8094 −1.77706 −0.888530 0.458819i \(-0.848273\pi\)
−0.888530 + 0.458819i \(0.848273\pi\)
\(38\) 1.38376 0.224475
\(39\) −0.119141 −0.0190778
\(40\) 0 0
\(41\) 6.31607 0.986405 0.493202 0.869915i \(-0.335826\pi\)
0.493202 + 0.869915i \(0.335826\pi\)
\(42\) 1.07506 0.165885
\(43\) −2.76329 −0.421397 −0.210699 0.977551i \(-0.567574\pi\)
−0.210699 + 0.977551i \(0.567574\pi\)
\(44\) −0.0852182 −0.0128471
\(45\) 0 0
\(46\) 0.965818 0.142402
\(47\) −10.4716 −1.52744 −0.763720 0.645548i \(-0.776629\pi\)
−0.763720 + 0.645548i \(0.776629\pi\)
\(48\) −0.686641 −0.0991082
\(49\) 11.7040 1.67201
\(50\) 0 0
\(51\) 1.06045 0.148493
\(52\) 0.0565180 0.00783764
\(53\) 8.10587 1.11343 0.556713 0.830705i \(-0.312062\pi\)
0.556713 + 0.830705i \(0.312062\pi\)
\(54\) 1.48345 0.201872
\(55\) 0 0
\(56\) −12.4790 −1.66757
\(57\) −0.179641 −0.0237940
\(58\) 4.60665 0.604883
\(59\) −0.264035 −0.0343744 −0.0171872 0.999852i \(-0.505471\pi\)
−0.0171872 + 0.999852i \(0.505471\pi\)
\(60\) 0 0
\(61\) 14.2963 1.83045 0.915227 0.402939i \(-0.132011\pi\)
0.915227 + 0.402939i \(0.132011\pi\)
\(62\) −14.5186 −1.84386
\(63\) 12.8349 1.61704
\(64\) 8.31121 1.03890
\(65\) 0 0
\(66\) −0.248579 −0.0305980
\(67\) 14.6511 1.78992 0.894959 0.446149i \(-0.147205\pi\)
0.894959 + 0.446149i \(0.147205\pi\)
\(68\) −0.503059 −0.0610049
\(69\) −0.125384 −0.0150944
\(70\) 0 0
\(71\) −9.64601 −1.14477 −0.572385 0.819985i \(-0.693982\pi\)
−0.572385 + 0.819985i \(0.693982\pi\)
\(72\) −8.56319 −1.00918
\(73\) 0.803845 0.0940830 0.0470415 0.998893i \(-0.485021\pi\)
0.0470415 + 0.998893i \(0.485021\pi\)
\(74\) 14.9576 1.73879
\(75\) 0 0
\(76\) 0.0852182 0.00977520
\(77\) −4.32482 −0.492859
\(78\) 0.164862 0.0186669
\(79\) 8.25623 0.928899 0.464449 0.885600i \(-0.346252\pi\)
0.464449 + 0.885600i \(0.346252\pi\)
\(80\) 0 0
\(81\) 8.71060 0.967845
\(82\) −8.73990 −0.965161
\(83\) −1.39198 −0.152790 −0.0763951 0.997078i \(-0.524341\pi\)
−0.0763951 + 0.997078i \(0.524341\pi\)
\(84\) 0.0662072 0.00722379
\(85\) 0 0
\(86\) 3.82372 0.412322
\(87\) −0.598040 −0.0641167
\(88\) 2.88543 0.307588
\(89\) 9.69068 1.02721 0.513605 0.858027i \(-0.328310\pi\)
0.513605 + 0.858027i \(0.328310\pi\)
\(90\) 0 0
\(91\) 2.86829 0.300678
\(92\) 0.0594796 0.00620117
\(93\) 1.88482 0.195447
\(94\) 14.4901 1.49454
\(95\) 0 0
\(96\) −0.0865391 −0.00883236
\(97\) −6.48020 −0.657964 −0.328982 0.944336i \(-0.606706\pi\)
−0.328982 + 0.944336i \(0.606706\pi\)
\(98\) −16.1955 −1.63600
\(99\) −2.96773 −0.298268
\(100\) 0 0
\(101\) −8.25408 −0.821312 −0.410656 0.911790i \(-0.634700\pi\)
−0.410656 + 0.911790i \(0.634700\pi\)
\(102\) −1.46741 −0.145295
\(103\) −11.2648 −1.10996 −0.554979 0.831864i \(-0.687274\pi\)
−0.554979 + 0.831864i \(0.687274\pi\)
\(104\) −1.91367 −0.187650
\(105\) 0 0
\(106\) −11.2165 −1.08945
\(107\) −11.9862 −1.15875 −0.579377 0.815060i \(-0.696704\pi\)
−0.579377 + 0.815060i \(0.696704\pi\)
\(108\) 0.0913580 0.00879092
\(109\) 0.188775 0.0180813 0.00904067 0.999959i \(-0.497122\pi\)
0.00904067 + 0.999959i \(0.497122\pi\)
\(110\) 0 0
\(111\) −1.94181 −0.184309
\(112\) 16.5308 1.56201
\(113\) 12.2684 1.15411 0.577056 0.816704i \(-0.304201\pi\)
0.577056 + 0.816704i \(0.304201\pi\)
\(114\) 0.248579 0.0232816
\(115\) 0 0
\(116\) 0.283699 0.0263408
\(117\) 1.96825 0.181964
\(118\) 0.365360 0.0336341
\(119\) −25.5302 −2.34035
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −19.7826 −1.79103
\(123\) 1.13462 0.102306
\(124\) −0.894123 −0.0802946
\(125\) 0 0
\(126\) −17.7604 −1.58222
\(127\) 7.42133 0.658537 0.329268 0.944236i \(-0.393198\pi\)
0.329268 + 0.944236i \(0.393198\pi\)
\(128\) −10.5372 −0.931367
\(129\) −0.496399 −0.0437055
\(130\) 0 0
\(131\) −1.15539 −0.100947 −0.0504734 0.998725i \(-0.516073\pi\)
−0.0504734 + 0.998725i \(0.516073\pi\)
\(132\) −0.0153087 −0.00133245
\(133\) 4.32482 0.375009
\(134\) −20.2736 −1.75137
\(135\) 0 0
\(136\) 17.0333 1.46059
\(137\) 14.7417 1.25947 0.629735 0.776810i \(-0.283163\pi\)
0.629735 + 0.776810i \(0.283163\pi\)
\(138\) 0.173500 0.0147693
\(139\) 6.38501 0.541570 0.270785 0.962640i \(-0.412717\pi\)
0.270785 + 0.962640i \(0.412717\pi\)
\(140\) 0 0
\(141\) −1.88113 −0.158419
\(142\) 13.3477 1.12012
\(143\) −0.663216 −0.0554609
\(144\) 11.3436 0.945296
\(145\) 0 0
\(146\) −1.11233 −0.0920568
\(147\) 2.10252 0.173413
\(148\) 0.921160 0.0757189
\(149\) −7.66886 −0.628257 −0.314129 0.949380i \(-0.601712\pi\)
−0.314129 + 0.949380i \(0.601712\pi\)
\(150\) 0 0
\(151\) 7.24054 0.589227 0.294613 0.955617i \(-0.404809\pi\)
0.294613 + 0.955617i \(0.404809\pi\)
\(152\) −2.88543 −0.234040
\(153\) −17.5191 −1.41633
\(154\) 5.98449 0.482244
\(155\) 0 0
\(156\) 0.0101529 0.000812887 0
\(157\) −6.79146 −0.542018 −0.271009 0.962577i \(-0.587357\pi\)
−0.271009 + 0.962577i \(0.587357\pi\)
\(158\) −11.4246 −0.908894
\(159\) 1.45614 0.115480
\(160\) 0 0
\(161\) 3.01858 0.237898
\(162\) −12.0534 −0.947001
\(163\) −11.6881 −0.915481 −0.457741 0.889086i \(-0.651341\pi\)
−0.457741 + 0.889086i \(0.651341\pi\)
\(164\) −0.538244 −0.0420298
\(165\) 0 0
\(166\) 1.92617 0.149500
\(167\) −4.48188 −0.346818 −0.173409 0.984850i \(-0.555478\pi\)
−0.173409 + 0.984850i \(0.555478\pi\)
\(168\) −2.24173 −0.172953
\(169\) −12.5601 −0.966165
\(170\) 0 0
\(171\) 2.96773 0.226948
\(172\) 0.235482 0.0179554
\(173\) −18.6634 −1.41895 −0.709475 0.704731i \(-0.751068\pi\)
−0.709475 + 0.704731i \(0.751068\pi\)
\(174\) 0.827542 0.0627358
\(175\) 0 0
\(176\) −3.82230 −0.288117
\(177\) −0.0474315 −0.00356517
\(178\) −13.4095 −1.00509
\(179\) 3.78670 0.283031 0.141516 0.989936i \(-0.454802\pi\)
0.141516 + 0.989936i \(0.454802\pi\)
\(180\) 0 0
\(181\) 7.77095 0.577610 0.288805 0.957388i \(-0.406742\pi\)
0.288805 + 0.957388i \(0.406742\pi\)
\(182\) −3.96901 −0.294203
\(183\) 2.56820 0.189847
\(184\) −2.01394 −0.148470
\(185\) 0 0
\(186\) −2.60813 −0.191238
\(187\) 5.90319 0.431684
\(188\) 0.892370 0.0650828
\(189\) 4.63641 0.337249
\(190\) 0 0
\(191\) −19.2865 −1.39552 −0.697760 0.716331i \(-0.745820\pi\)
−0.697760 + 0.716331i \(0.745820\pi\)
\(192\) 1.49303 0.107750
\(193\) −7.76965 −0.559272 −0.279636 0.960106i \(-0.590214\pi\)
−0.279636 + 0.960106i \(0.590214\pi\)
\(194\) 8.96701 0.643794
\(195\) 0 0
\(196\) −0.997397 −0.0712426
\(197\) −19.8618 −1.41510 −0.707548 0.706665i \(-0.750199\pi\)
−0.707548 + 0.706665i \(0.750199\pi\)
\(198\) 4.10661 0.291844
\(199\) 3.55464 0.251982 0.125991 0.992031i \(-0.459789\pi\)
0.125991 + 0.992031i \(0.459789\pi\)
\(200\) 0 0
\(201\) 2.63194 0.185642
\(202\) 11.4216 0.803624
\(203\) 14.3977 1.01052
\(204\) −0.0903700 −0.00632716
\(205\) 0 0
\(206\) 15.5878 1.08605
\(207\) 2.07138 0.143971
\(208\) 2.53501 0.175771
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) −20.0441 −1.37990 −0.689948 0.723859i \(-0.742367\pi\)
−0.689948 + 0.723859i \(0.742367\pi\)
\(212\) −0.690767 −0.0474421
\(213\) −1.73282 −0.118731
\(214\) 16.5860 1.13380
\(215\) 0 0
\(216\) −3.09332 −0.210474
\(217\) −45.3767 −3.08037
\(218\) −0.261218 −0.0176919
\(219\) 0.144403 0.00975788
\(220\) 0 0
\(221\) −3.91509 −0.263357
\(222\) 2.68700 0.180340
\(223\) −6.29774 −0.421728 −0.210864 0.977515i \(-0.567628\pi\)
−0.210864 + 0.977515i \(0.567628\pi\)
\(224\) 2.08341 0.139204
\(225\) 0 0
\(226\) −16.9765 −1.12926
\(227\) 23.7789 1.57826 0.789131 0.614225i \(-0.210531\pi\)
0.789131 + 0.614225i \(0.210531\pi\)
\(228\) 0.0153087 0.00101384
\(229\) −4.01952 −0.265618 −0.132809 0.991142i \(-0.542400\pi\)
−0.132809 + 0.991142i \(0.542400\pi\)
\(230\) 0 0
\(231\) −0.776914 −0.0511172
\(232\) −9.60587 −0.630656
\(233\) −26.6054 −1.74298 −0.871488 0.490416i \(-0.836845\pi\)
−0.871488 + 0.490416i \(0.836845\pi\)
\(234\) −2.72357 −0.178045
\(235\) 0 0
\(236\) 0.0225006 0.00146466
\(237\) 1.48316 0.0963414
\(238\) 35.3276 2.28995
\(239\) 24.8559 1.60780 0.803899 0.594766i \(-0.202755\pi\)
0.803899 + 0.594766i \(0.202755\pi\)
\(240\) 0 0
\(241\) −1.33695 −0.0861208 −0.0430604 0.999072i \(-0.513711\pi\)
−0.0430604 + 0.999072i \(0.513711\pi\)
\(242\) −1.38376 −0.0889512
\(243\) 4.78092 0.306696
\(244\) −1.21830 −0.0779940
\(245\) 0 0
\(246\) −1.57004 −0.100102
\(247\) 0.663216 0.0421994
\(248\) 30.2744 1.92243
\(249\) −0.250057 −0.0158467
\(250\) 0 0
\(251\) 6.30211 0.397786 0.198893 0.980021i \(-0.436265\pi\)
0.198893 + 0.980021i \(0.436265\pi\)
\(252\) −1.09377 −0.0689007
\(253\) −0.697968 −0.0438809
\(254\) −10.2693 −0.644354
\(255\) 0 0
\(256\) −2.04147 −0.127592
\(257\) −10.4318 −0.650718 −0.325359 0.945591i \(-0.605485\pi\)
−0.325359 + 0.945591i \(0.605485\pi\)
\(258\) 0.686896 0.0427643
\(259\) 46.7488 2.90483
\(260\) 0 0
\(261\) 9.87984 0.611547
\(262\) 1.59878 0.0987728
\(263\) −21.5818 −1.33079 −0.665396 0.746490i \(-0.731737\pi\)
−0.665396 + 0.746490i \(0.731737\pi\)
\(264\) 0.518342 0.0319017
\(265\) 0 0
\(266\) −5.98449 −0.366933
\(267\) 1.74084 0.106538
\(268\) −1.24854 −0.0762667
\(269\) 17.5582 1.07054 0.535272 0.844680i \(-0.320209\pi\)
0.535272 + 0.844680i \(0.320209\pi\)
\(270\) 0 0
\(271\) −17.3550 −1.05424 −0.527122 0.849790i \(-0.676729\pi\)
−0.527122 + 0.849790i \(0.676729\pi\)
\(272\) −22.5638 −1.36813
\(273\) 0.515261 0.0311850
\(274\) −20.3989 −1.23235
\(275\) 0 0
\(276\) 0.0106850 0.000643159 0
\(277\) −19.6803 −1.18247 −0.591237 0.806498i \(-0.701360\pi\)
−0.591237 + 0.806498i \(0.701360\pi\)
\(278\) −8.83530 −0.529906
\(279\) −31.1379 −1.86418
\(280\) 0 0
\(281\) −3.35786 −0.200313 −0.100157 0.994972i \(-0.531934\pi\)
−0.100157 + 0.994972i \(0.531934\pi\)
\(282\) 2.60302 0.155008
\(283\) −17.1783 −1.02114 −0.510571 0.859836i \(-0.670566\pi\)
−0.510571 + 0.859836i \(0.670566\pi\)
\(284\) 0.822015 0.0487776
\(285\) 0 0
\(286\) 0.917729 0.0542665
\(287\) −27.3158 −1.61240
\(288\) 1.42966 0.0842433
\(289\) 17.8477 1.04986
\(290\) 0 0
\(291\) −1.16411 −0.0682412
\(292\) −0.0685022 −0.00400879
\(293\) 23.9141 1.39707 0.698537 0.715574i \(-0.253835\pi\)
0.698537 + 0.715574i \(0.253835\pi\)
\(294\) −2.90938 −0.169678
\(295\) 0 0
\(296\) −31.1899 −1.81288
\(297\) −1.07205 −0.0622065
\(298\) 10.6118 0.614727
\(299\) 0.462903 0.0267704
\(300\) 0 0
\(301\) 11.9507 0.688828
\(302\) −10.0191 −0.576537
\(303\) −1.48277 −0.0851829
\(304\) 3.82230 0.219224
\(305\) 0 0
\(306\) 24.2421 1.38583
\(307\) 29.5003 1.68367 0.841835 0.539735i \(-0.181476\pi\)
0.841835 + 0.539735i \(0.181476\pi\)
\(308\) 0.368553 0.0210003
\(309\) −2.02363 −0.115120
\(310\) 0 0
\(311\) −24.1117 −1.36725 −0.683625 0.729834i \(-0.739598\pi\)
−0.683625 + 0.729834i \(0.739598\pi\)
\(312\) −0.343773 −0.0194623
\(313\) 8.25794 0.466766 0.233383 0.972385i \(-0.425020\pi\)
0.233383 + 0.972385i \(0.425020\pi\)
\(314\) 9.39773 0.530344
\(315\) 0 0
\(316\) −0.703581 −0.0395795
\(317\) −7.65440 −0.429914 −0.214957 0.976624i \(-0.568961\pi\)
−0.214957 + 0.976624i \(0.568961\pi\)
\(318\) −2.01495 −0.112993
\(319\) −3.32909 −0.186393
\(320\) 0 0
\(321\) −2.15322 −0.120181
\(322\) −4.17698 −0.232774
\(323\) −5.90319 −0.328462
\(324\) −0.742302 −0.0412390
\(325\) 0 0
\(326\) 16.1735 0.895765
\(327\) 0.0339116 0.00187532
\(328\) 18.2246 1.00629
\(329\) 45.2877 2.49679
\(330\) 0 0
\(331\) 11.6128 0.638299 0.319149 0.947704i \(-0.396603\pi\)
0.319149 + 0.947704i \(0.396603\pi\)
\(332\) 0.118622 0.00651025
\(333\) 32.0795 1.75794
\(334\) 6.20182 0.339349
\(335\) 0 0
\(336\) 2.96960 0.162005
\(337\) −20.4093 −1.11177 −0.555883 0.831261i \(-0.687620\pi\)
−0.555883 + 0.831261i \(0.687620\pi\)
\(338\) 17.3802 0.945357
\(339\) 2.20390 0.119700
\(340\) 0 0
\(341\) 10.4922 0.568182
\(342\) −4.10661 −0.222060
\(343\) −20.3441 −1.09848
\(344\) −7.97328 −0.429891
\(345\) 0 0
\(346\) 25.8256 1.38839
\(347\) 26.5715 1.42643 0.713216 0.700945i \(-0.247238\pi\)
0.713216 + 0.700945i \(0.247238\pi\)
\(348\) 0.0509639 0.00273195
\(349\) 1.10501 0.0591499 0.0295749 0.999563i \(-0.490585\pi\)
0.0295749 + 0.999563i \(0.490585\pi\)
\(350\) 0 0
\(351\) 0.710999 0.0379503
\(352\) −0.481734 −0.0256765
\(353\) −29.9830 −1.59584 −0.797918 0.602766i \(-0.794065\pi\)
−0.797918 + 0.602766i \(0.794065\pi\)
\(354\) 0.0656336 0.00348839
\(355\) 0 0
\(356\) −0.825822 −0.0437685
\(357\) −4.58627 −0.242731
\(358\) −5.23987 −0.276936
\(359\) −14.7854 −0.780343 −0.390172 0.920742i \(-0.627584\pi\)
−0.390172 + 0.920742i \(0.627584\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −10.7531 −0.565170
\(363\) 0.179641 0.00942870
\(364\) −0.244430 −0.0128116
\(365\) 0 0
\(366\) −3.55376 −0.185758
\(367\) −27.7019 −1.44602 −0.723012 0.690835i \(-0.757243\pi\)
−0.723012 + 0.690835i \(0.757243\pi\)
\(368\) 2.66784 0.139071
\(369\) −18.7444 −0.975794
\(370\) 0 0
\(371\) −35.0564 −1.82004
\(372\) −0.160621 −0.00832781
\(373\) 7.75317 0.401444 0.200722 0.979648i \(-0.435671\pi\)
0.200722 + 0.979648i \(0.435671\pi\)
\(374\) −8.16858 −0.422387
\(375\) 0 0
\(376\) −30.2151 −1.55822
\(377\) 2.20791 0.113713
\(378\) −6.41566 −0.329986
\(379\) −30.5335 −1.56840 −0.784200 0.620508i \(-0.786926\pi\)
−0.784200 + 0.620508i \(0.786926\pi\)
\(380\) 0 0
\(381\) 1.33317 0.0683006
\(382\) 26.6878 1.36547
\(383\) 24.7369 1.26400 0.631999 0.774969i \(-0.282235\pi\)
0.631999 + 0.774969i \(0.282235\pi\)
\(384\) −1.89291 −0.0965974
\(385\) 0 0
\(386\) 10.7513 0.547227
\(387\) 8.20069 0.416864
\(388\) 0.552230 0.0280353
\(389\) 26.9954 1.36872 0.684360 0.729144i \(-0.260082\pi\)
0.684360 + 0.729144i \(0.260082\pi\)
\(390\) 0 0
\(391\) −4.12024 −0.208369
\(392\) 33.7712 1.70570
\(393\) −0.207555 −0.0104698
\(394\) 27.4839 1.38462
\(395\) 0 0
\(396\) 0.252904 0.0127089
\(397\) −6.77500 −0.340028 −0.170014 0.985442i \(-0.554381\pi\)
−0.170014 + 0.985442i \(0.554381\pi\)
\(398\) −4.91876 −0.246555
\(399\) 0.776914 0.0388943
\(400\) 0 0
\(401\) 20.9804 1.04771 0.523856 0.851807i \(-0.324493\pi\)
0.523856 + 0.851807i \(0.324493\pi\)
\(402\) −3.64196 −0.181644
\(403\) −6.95857 −0.346631
\(404\) 0.703398 0.0349953
\(405\) 0 0
\(406\) −19.9229 −0.988758
\(407\) −10.8094 −0.535804
\(408\) 3.05987 0.151486
\(409\) −22.0672 −1.09115 −0.545575 0.838062i \(-0.683689\pi\)
−0.545575 + 0.838062i \(0.683689\pi\)
\(410\) 0 0
\(411\) 2.64821 0.130627
\(412\) 0.959969 0.0472943
\(413\) 1.14190 0.0561894
\(414\) −2.86629 −0.140870
\(415\) 0 0
\(416\) 0.319494 0.0156645
\(417\) 1.14701 0.0561693
\(418\) 1.38376 0.0676817
\(419\) −26.1712 −1.27855 −0.639273 0.768980i \(-0.720764\pi\)
−0.639273 + 0.768980i \(0.720764\pi\)
\(420\) 0 0
\(421\) 16.2156 0.790300 0.395150 0.918617i \(-0.370693\pi\)
0.395150 + 0.918617i \(0.370693\pi\)
\(422\) 27.7362 1.35018
\(423\) 31.0769 1.51101
\(424\) 23.3889 1.13587
\(425\) 0 0
\(426\) 2.39780 0.116174
\(427\) −61.8289 −2.99211
\(428\) 1.02145 0.0493734
\(429\) −0.119141 −0.00575217
\(430\) 0 0
\(431\) −39.4232 −1.89895 −0.949474 0.313847i \(-0.898382\pi\)
−0.949474 + 0.313847i \(0.898382\pi\)
\(432\) 4.09769 0.197150
\(433\) 21.8969 1.05230 0.526148 0.850393i \(-0.323636\pi\)
0.526148 + 0.850393i \(0.323636\pi\)
\(434\) 62.7903 3.01403
\(435\) 0 0
\(436\) −0.0160870 −0.000770429 0
\(437\) 0.697968 0.0333883
\(438\) −0.199819 −0.00954773
\(439\) −23.8693 −1.13922 −0.569609 0.821916i \(-0.692905\pi\)
−0.569609 + 0.821916i \(0.692905\pi\)
\(440\) 0 0
\(441\) −34.7344 −1.65402
\(442\) 5.41753 0.257686
\(443\) 5.09632 0.242134 0.121067 0.992644i \(-0.461368\pi\)
0.121067 + 0.992644i \(0.461368\pi\)
\(444\) 0.165478 0.00785323
\(445\) 0 0
\(446\) 8.71454 0.412645
\(447\) −1.37764 −0.0651601
\(448\) −35.9444 −1.69821
\(449\) 5.00982 0.236428 0.118214 0.992988i \(-0.462283\pi\)
0.118214 + 0.992988i \(0.462283\pi\)
\(450\) 0 0
\(451\) 6.31607 0.297412
\(452\) −1.04549 −0.0491757
\(453\) 1.30070 0.0611120
\(454\) −32.9042 −1.54427
\(455\) 0 0
\(456\) −0.518342 −0.0242736
\(457\) −1.49477 −0.0699222 −0.0349611 0.999389i \(-0.511131\pi\)
−0.0349611 + 0.999389i \(0.511131\pi\)
\(458\) 5.56204 0.259897
\(459\) −6.32850 −0.295389
\(460\) 0 0
\(461\) −13.3892 −0.623598 −0.311799 0.950148i \(-0.600932\pi\)
−0.311799 + 0.950148i \(0.600932\pi\)
\(462\) 1.07506 0.0500163
\(463\) 22.0666 1.02552 0.512761 0.858531i \(-0.328623\pi\)
0.512761 + 0.858531i \(0.328623\pi\)
\(464\) 12.7248 0.590733
\(465\) 0 0
\(466\) 36.8154 1.70544
\(467\) 6.60761 0.305764 0.152882 0.988244i \(-0.451145\pi\)
0.152882 + 0.988244i \(0.451145\pi\)
\(468\) −0.167730 −0.00775333
\(469\) −63.3633 −2.92585
\(470\) 0 0
\(471\) −1.22002 −0.0562157
\(472\) −0.761856 −0.0350672
\(473\) −2.76329 −0.127056
\(474\) −2.05233 −0.0942665
\(475\) 0 0
\(476\) 2.17564 0.0997202
\(477\) −24.0560 −1.10145
\(478\) −34.3946 −1.57317
\(479\) −21.8942 −1.00037 −0.500185 0.865919i \(-0.666735\pi\)
−0.500185 + 0.865919i \(0.666735\pi\)
\(480\) 0 0
\(481\) 7.16899 0.326878
\(482\) 1.85002 0.0842661
\(483\) 0.542261 0.0246737
\(484\) −0.0852182 −0.00387355
\(485\) 0 0
\(486\) −6.61563 −0.300091
\(487\) 3.48029 0.157707 0.0788535 0.996886i \(-0.474874\pi\)
0.0788535 + 0.996886i \(0.474874\pi\)
\(488\) 41.2510 1.86735
\(489\) −2.09966 −0.0949498
\(490\) 0 0
\(491\) −25.4084 −1.14666 −0.573332 0.819323i \(-0.694349\pi\)
−0.573332 + 0.819323i \(0.694349\pi\)
\(492\) −0.0966906 −0.00435915
\(493\) −19.6523 −0.885093
\(494\) −0.917729 −0.0412906
\(495\) 0 0
\(496\) −40.1042 −1.80073
\(497\) 41.7172 1.87127
\(498\) 0.346018 0.0155054
\(499\) 8.27283 0.370343 0.185171 0.982706i \(-0.440716\pi\)
0.185171 + 0.982706i \(0.440716\pi\)
\(500\) 0 0
\(501\) −0.805128 −0.0359705
\(502\) −8.72059 −0.389219
\(503\) −31.4555 −1.40253 −0.701265 0.712901i \(-0.747381\pi\)
−0.701265 + 0.712901i \(0.747381\pi\)
\(504\) 37.0342 1.64963
\(505\) 0 0
\(506\) 0.965818 0.0429358
\(507\) −2.25631 −0.100206
\(508\) −0.632433 −0.0280597
\(509\) −28.6481 −1.26981 −0.634903 0.772592i \(-0.718960\pi\)
−0.634903 + 0.772592i \(0.718960\pi\)
\(510\) 0 0
\(511\) −3.47648 −0.153791
\(512\) 23.8993 1.05621
\(513\) 1.07205 0.0473321
\(514\) 14.4351 0.636704
\(515\) 0 0
\(516\) 0.0423022 0.00186225
\(517\) −10.4716 −0.460540
\(518\) −64.6889 −2.84227
\(519\) −3.35270 −0.147167
\(520\) 0 0
\(521\) 14.2218 0.623067 0.311534 0.950235i \(-0.399157\pi\)
0.311534 + 0.950235i \(0.399157\pi\)
\(522\) −13.6713 −0.598376
\(523\) −32.8102 −1.43469 −0.717345 0.696718i \(-0.754643\pi\)
−0.717345 + 0.696718i \(0.754643\pi\)
\(524\) 0.0984602 0.00430125
\(525\) 0 0
\(526\) 29.8640 1.30213
\(527\) 61.9372 2.69803
\(528\) −0.686641 −0.0298822
\(529\) −22.5128 −0.978819
\(530\) 0 0
\(531\) 0.783584 0.0340047
\(532\) −0.368553 −0.0159788
\(533\) −4.18892 −0.181442
\(534\) −2.40890 −0.104243
\(535\) 0 0
\(536\) 42.2748 1.82599
\(537\) 0.680246 0.0293548
\(538\) −24.2963 −1.04749
\(539\) 11.7040 0.504128
\(540\) 0 0
\(541\) −12.2505 −0.526689 −0.263345 0.964702i \(-0.584826\pi\)
−0.263345 + 0.964702i \(0.584826\pi\)
\(542\) 24.0151 1.03154
\(543\) 1.39598 0.0599072
\(544\) −2.84377 −0.121926
\(545\) 0 0
\(546\) −0.712996 −0.0305134
\(547\) 3.35144 0.143297 0.0716487 0.997430i \(-0.477174\pi\)
0.0716487 + 0.997430i \(0.477174\pi\)
\(548\) −1.25626 −0.0536649
\(549\) −42.4275 −1.81076
\(550\) 0 0
\(551\) 3.32909 0.141824
\(552\) −0.361786 −0.0153986
\(553\) −35.7067 −1.51840
\(554\) 27.2327 1.15701
\(555\) 0 0
\(556\) −0.544119 −0.0230758
\(557\) 32.8902 1.39360 0.696801 0.717264i \(-0.254606\pi\)
0.696801 + 0.717264i \(0.254606\pi\)
\(558\) 43.0873 1.82403
\(559\) 1.83266 0.0775131
\(560\) 0 0
\(561\) 1.06045 0.0447724
\(562\) 4.64647 0.195999
\(563\) −17.9736 −0.757495 −0.378747 0.925500i \(-0.623645\pi\)
−0.378747 + 0.925500i \(0.623645\pi\)
\(564\) 0.160306 0.00675011
\(565\) 0 0
\(566\) 23.7705 0.999150
\(567\) −37.6718 −1.58207
\(568\) −27.8329 −1.16784
\(569\) 0.968772 0.0406130 0.0203065 0.999794i \(-0.493536\pi\)
0.0203065 + 0.999794i \(0.493536\pi\)
\(570\) 0 0
\(571\) −18.5167 −0.774901 −0.387450 0.921891i \(-0.626644\pi\)
−0.387450 + 0.921891i \(0.626644\pi\)
\(572\) 0.0565180 0.00236314
\(573\) −3.46464 −0.144737
\(574\) 37.7985 1.57768
\(575\) 0 0
\(576\) −24.6654 −1.02773
\(577\) −13.3376 −0.555252 −0.277626 0.960689i \(-0.589548\pi\)
−0.277626 + 0.960689i \(0.589548\pi\)
\(578\) −24.6968 −1.02725
\(579\) −1.39575 −0.0580052
\(580\) 0 0
\(581\) 6.02008 0.249755
\(582\) 1.61084 0.0667715
\(583\) 8.10587 0.335711
\(584\) 2.31944 0.0959792
\(585\) 0 0
\(586\) −33.0912 −1.36699
\(587\) −30.6611 −1.26552 −0.632760 0.774348i \(-0.718078\pi\)
−0.632760 + 0.774348i \(0.718078\pi\)
\(588\) −0.179173 −0.00738898
\(589\) −10.4922 −0.432322
\(590\) 0 0
\(591\) −3.56799 −0.146768
\(592\) 41.3169 1.69811
\(593\) −14.1679 −0.581806 −0.290903 0.956753i \(-0.593956\pi\)
−0.290903 + 0.956753i \(0.593956\pi\)
\(594\) 1.48345 0.0608668
\(595\) 0 0
\(596\) 0.653526 0.0267695
\(597\) 0.638559 0.0261345
\(598\) −0.640546 −0.0261939
\(599\) 3.17933 0.129904 0.0649519 0.997888i \(-0.479311\pi\)
0.0649519 + 0.997888i \(0.479311\pi\)
\(600\) 0 0
\(601\) 26.0743 1.06359 0.531796 0.846872i \(-0.321517\pi\)
0.531796 + 0.846872i \(0.321517\pi\)
\(602\) −16.5369 −0.673993
\(603\) −43.4805 −1.77066
\(604\) −0.617025 −0.0251064
\(605\) 0 0
\(606\) 2.05179 0.0833484
\(607\) −30.1201 −1.22254 −0.611269 0.791423i \(-0.709341\pi\)
−0.611269 + 0.791423i \(0.709341\pi\)
\(608\) 0.481734 0.0195369
\(609\) 2.58642 0.104807
\(610\) 0 0
\(611\) 6.94493 0.280962
\(612\) 1.49294 0.0603487
\(613\) 26.6754 1.07741 0.538704 0.842495i \(-0.318914\pi\)
0.538704 + 0.842495i \(0.318914\pi\)
\(614\) −40.8212 −1.64741
\(615\) 0 0
\(616\) −12.4790 −0.502792
\(617\) 5.41171 0.217867 0.108934 0.994049i \(-0.465256\pi\)
0.108934 + 0.994049i \(0.465256\pi\)
\(618\) 2.80020 0.112641
\(619\) 35.8748 1.44193 0.720965 0.692971i \(-0.243699\pi\)
0.720965 + 0.692971i \(0.243699\pi\)
\(620\) 0 0
\(621\) 0.748255 0.0300264
\(622\) 33.3647 1.33780
\(623\) −41.9104 −1.67910
\(624\) 0.455392 0.0182303
\(625\) 0 0
\(626\) −11.4270 −0.456714
\(627\) −0.179641 −0.00717416
\(628\) 0.578756 0.0230949
\(629\) −63.8101 −2.54428
\(630\) 0 0
\(631\) 10.3550 0.412226 0.206113 0.978528i \(-0.433919\pi\)
0.206113 + 0.978528i \(0.433919\pi\)
\(632\) 23.8228 0.947621
\(633\) −3.60075 −0.143117
\(634\) 10.5918 0.420655
\(635\) 0 0
\(636\) −0.124090 −0.00492049
\(637\) −7.76230 −0.307554
\(638\) 4.60665 0.182379
\(639\) 28.6267 1.13246
\(640\) 0 0
\(641\) 6.49289 0.256454 0.128227 0.991745i \(-0.459071\pi\)
0.128227 + 0.991745i \(0.459071\pi\)
\(642\) 2.97953 0.117593
\(643\) 12.9459 0.510538 0.255269 0.966870i \(-0.417836\pi\)
0.255269 + 0.966870i \(0.417836\pi\)
\(644\) −0.257238 −0.0101366
\(645\) 0 0
\(646\) 8.16858 0.321388
\(647\) 8.43170 0.331484 0.165742 0.986169i \(-0.446998\pi\)
0.165742 + 0.986169i \(0.446998\pi\)
\(648\) 25.1339 0.987352
\(649\) −0.264035 −0.0103643
\(650\) 0 0
\(651\) −8.15150 −0.319483
\(652\) 0.996037 0.0390078
\(653\) 37.3640 1.46217 0.731083 0.682288i \(-0.239015\pi\)
0.731083 + 0.682288i \(0.239015\pi\)
\(654\) −0.0469255 −0.00183493
\(655\) 0 0
\(656\) −24.1419 −0.942584
\(657\) −2.38560 −0.0930709
\(658\) −62.6672 −2.44302
\(659\) 14.3895 0.560535 0.280267 0.959922i \(-0.409577\pi\)
0.280267 + 0.959922i \(0.409577\pi\)
\(660\) 0 0
\(661\) −18.1075 −0.704300 −0.352150 0.935944i \(-0.614549\pi\)
−0.352150 + 0.935944i \(0.614549\pi\)
\(662\) −16.0693 −0.624552
\(663\) −0.703310 −0.0273143
\(664\) −4.01648 −0.155870
\(665\) 0 0
\(666\) −44.3902 −1.72008
\(667\) 2.32360 0.0899701
\(668\) 0.381937 0.0147776
\(669\) −1.13133 −0.0437398
\(670\) 0 0
\(671\) 14.2963 0.551903
\(672\) 0.374266 0.0144376
\(673\) −24.7942 −0.955747 −0.477873 0.878429i \(-0.658592\pi\)
−0.477873 + 0.878429i \(0.658592\pi\)
\(674\) 28.2415 1.08782
\(675\) 0 0
\(676\) 1.07035 0.0411674
\(677\) −5.73973 −0.220596 −0.110298 0.993899i \(-0.535180\pi\)
−0.110298 + 0.993899i \(0.535180\pi\)
\(678\) −3.04966 −0.117122
\(679\) 28.0257 1.07553
\(680\) 0 0
\(681\) 4.27166 0.163690
\(682\) −14.5186 −0.555946
\(683\) 16.8301 0.643986 0.321993 0.946742i \(-0.395647\pi\)
0.321993 + 0.946742i \(0.395647\pi\)
\(684\) −0.252904 −0.00967004
\(685\) 0 0
\(686\) 28.1513 1.07482
\(687\) −0.722071 −0.0275487
\(688\) 10.5621 0.402677
\(689\) −5.37594 −0.204807
\(690\) 0 0
\(691\) −11.7122 −0.445553 −0.222776 0.974870i \(-0.571512\pi\)
−0.222776 + 0.974870i \(0.571512\pi\)
\(692\) 1.59046 0.0604602
\(693\) 12.8349 0.487557
\(694\) −36.7684 −1.39571
\(695\) 0 0
\(696\) −1.72561 −0.0654089
\(697\) 37.2850 1.41227
\(698\) −1.52907 −0.0578760
\(699\) −4.77941 −0.180774
\(700\) 0 0
\(701\) −21.2822 −0.803818 −0.401909 0.915680i \(-0.631653\pi\)
−0.401909 + 0.915680i \(0.631653\pi\)
\(702\) −0.983850 −0.0371330
\(703\) 10.8094 0.407685
\(704\) 8.31121 0.313240
\(705\) 0 0
\(706\) 41.4892 1.56147
\(707\) 35.6974 1.34254
\(708\) 0.00404202 0.000151909 0
\(709\) −33.7749 −1.26844 −0.634222 0.773151i \(-0.718679\pi\)
−0.634222 + 0.773151i \(0.718679\pi\)
\(710\) 0 0
\(711\) −24.5023 −0.918907
\(712\) 27.9618 1.04791
\(713\) −7.32319 −0.274256
\(714\) 6.34628 0.237504
\(715\) 0 0
\(716\) −0.322696 −0.0120597
\(717\) 4.46514 0.166754
\(718\) 20.4594 0.763537
\(719\) −17.4677 −0.651434 −0.325717 0.945467i \(-0.605606\pi\)
−0.325717 + 0.945467i \(0.605606\pi\)
\(720\) 0 0
\(721\) 48.7184 1.81437
\(722\) −1.38376 −0.0514981
\(723\) −0.240172 −0.00893208
\(724\) −0.662226 −0.0246114
\(725\) 0 0
\(726\) −0.248579 −0.00922564
\(727\) −43.0612 −1.59705 −0.798525 0.601961i \(-0.794386\pi\)
−0.798525 + 0.601961i \(0.794386\pi\)
\(728\) 8.27625 0.306738
\(729\) −25.2730 −0.936036
\(730\) 0 0
\(731\) −16.3122 −0.603329
\(732\) −0.218857 −0.00808920
\(733\) 32.1119 1.18608 0.593039 0.805173i \(-0.297928\pi\)
0.593039 + 0.805173i \(0.297928\pi\)
\(734\) 38.3326 1.41488
\(735\) 0 0
\(736\) 0.336235 0.0123938
\(737\) 14.6511 0.539680
\(738\) 25.9377 0.954779
\(739\) 8.07571 0.297070 0.148535 0.988907i \(-0.452544\pi\)
0.148535 + 0.988907i \(0.452544\pi\)
\(740\) 0 0
\(741\) 0.119141 0.00437674
\(742\) 48.5095 1.78084
\(743\) −14.9512 −0.548507 −0.274253 0.961657i \(-0.588431\pi\)
−0.274253 + 0.961657i \(0.588431\pi\)
\(744\) 5.43853 0.199386
\(745\) 0 0
\(746\) −10.7285 −0.392798
\(747\) 4.13103 0.151147
\(748\) −0.503059 −0.0183937
\(749\) 51.8383 1.89413
\(750\) 0 0
\(751\) 48.7426 1.77864 0.889322 0.457282i \(-0.151177\pi\)
0.889322 + 0.457282i \(0.151177\pi\)
\(752\) 40.0256 1.45958
\(753\) 1.13212 0.0412566
\(754\) −3.05520 −0.111264
\(755\) 0 0
\(756\) −0.395106 −0.0143699
\(757\) 47.6174 1.73068 0.865342 0.501182i \(-0.167101\pi\)
0.865342 + 0.501182i \(0.167101\pi\)
\(758\) 42.2509 1.53462
\(759\) −0.125384 −0.00455113
\(760\) 0 0
\(761\) 30.9317 1.12127 0.560637 0.828062i \(-0.310556\pi\)
0.560637 + 0.828062i \(0.310556\pi\)
\(762\) −1.84479 −0.0668297
\(763\) −0.816416 −0.0295562
\(764\) 1.64356 0.0594619
\(765\) 0 0
\(766\) −34.2299 −1.23678
\(767\) 0.175112 0.00632294
\(768\) −0.366731 −0.0132333
\(769\) 27.4711 0.990633 0.495316 0.868713i \(-0.335052\pi\)
0.495316 + 0.868713i \(0.335052\pi\)
\(770\) 0 0
\(771\) −1.87398 −0.0674897
\(772\) 0.662115 0.0238301
\(773\) −32.0937 −1.15433 −0.577166 0.816627i \(-0.695841\pi\)
−0.577166 + 0.816627i \(0.695841\pi\)
\(774\) −11.3478 −0.407887
\(775\) 0 0
\(776\) −18.6982 −0.671225
\(777\) 8.39799 0.301276
\(778\) −37.3550 −1.33924
\(779\) −6.31607 −0.226297
\(780\) 0 0
\(781\) −9.64601 −0.345161
\(782\) 5.70141 0.203882
\(783\) 3.56894 0.127544
\(784\) −44.7364 −1.59773
\(785\) 0 0
\(786\) 0.287206 0.0102443
\(787\) −21.1724 −0.754715 −0.377357 0.926068i \(-0.623167\pi\)
−0.377357 + 0.926068i \(0.623167\pi\)
\(788\) 1.69259 0.0602959
\(789\) −3.87698 −0.138024
\(790\) 0 0
\(791\) −53.0585 −1.88654
\(792\) −8.56319 −0.304280
\(793\) −9.48153 −0.336699
\(794\) 9.37495 0.332705
\(795\) 0 0
\(796\) −0.302920 −0.0107367
\(797\) −0.325852 −0.0115423 −0.00577114 0.999983i \(-0.501837\pi\)
−0.00577114 + 0.999983i \(0.501837\pi\)
\(798\) −1.07506 −0.0380567
\(799\) −61.8158 −2.18689
\(800\) 0 0
\(801\) −28.7593 −1.01616
\(802\) −29.0318 −1.02515
\(803\) 0.803845 0.0283671
\(804\) −0.224289 −0.00791006
\(805\) 0 0
\(806\) 9.62896 0.339166
\(807\) 3.15418 0.111032
\(808\) −23.8166 −0.837865
\(809\) −5.48167 −0.192725 −0.0963626 0.995346i \(-0.530721\pi\)
−0.0963626 + 0.995346i \(0.530721\pi\)
\(810\) 0 0
\(811\) −46.4009 −1.62935 −0.814677 0.579915i \(-0.803086\pi\)
−0.814677 + 0.579915i \(0.803086\pi\)
\(812\) −1.22695 −0.0430574
\(813\) −3.11767 −0.109342
\(814\) 14.9576 0.524264
\(815\) 0 0
\(816\) −4.05338 −0.141897
\(817\) 2.76329 0.0966752
\(818\) 30.5356 1.06765
\(819\) −8.51230 −0.297444
\(820\) 0 0
\(821\) −29.0800 −1.01490 −0.507449 0.861682i \(-0.669411\pi\)
−0.507449 + 0.861682i \(0.669411\pi\)
\(822\) −3.66448 −0.127814
\(823\) 29.3196 1.02202 0.511008 0.859576i \(-0.329272\pi\)
0.511008 + 0.859576i \(0.329272\pi\)
\(824\) −32.5040 −1.13233
\(825\) 0 0
\(826\) −1.58012 −0.0549793
\(827\) −11.4469 −0.398048 −0.199024 0.979995i \(-0.563777\pi\)
−0.199024 + 0.979995i \(0.563777\pi\)
\(828\) −0.176519 −0.00613447
\(829\) −50.5560 −1.75588 −0.877941 0.478770i \(-0.841083\pi\)
−0.877941 + 0.478770i \(0.841083\pi\)
\(830\) 0 0
\(831\) −3.53538 −0.122641
\(832\) −5.51212 −0.191098
\(833\) 69.0912 2.39387
\(834\) −1.58718 −0.0549596
\(835\) 0 0
\(836\) 0.0852182 0.00294733
\(837\) −11.2481 −0.388791
\(838\) 36.2145 1.25101
\(839\) 1.09625 0.0378466 0.0189233 0.999821i \(-0.493976\pi\)
0.0189233 + 0.999821i \(0.493976\pi\)
\(840\) 0 0
\(841\) −17.9172 −0.617833
\(842\) −22.4384 −0.773280
\(843\) −0.603210 −0.0207756
\(844\) 1.70813 0.0587961
\(845\) 0 0
\(846\) −43.0028 −1.47847
\(847\) −4.32482 −0.148602
\(848\) −30.9831 −1.06396
\(849\) −3.08592 −0.105908
\(850\) 0 0
\(851\) 7.54464 0.258627
\(852\) 0.147668 0.00505901
\(853\) 13.1321 0.449633 0.224816 0.974401i \(-0.427822\pi\)
0.224816 + 0.974401i \(0.427822\pi\)
\(854\) 85.5561 2.92767
\(855\) 0 0
\(856\) −34.5855 −1.18211
\(857\) −18.5275 −0.632888 −0.316444 0.948611i \(-0.602489\pi\)
−0.316444 + 0.948611i \(0.602489\pi\)
\(858\) 0.164862 0.00562828
\(859\) 24.1651 0.824502 0.412251 0.911070i \(-0.364743\pi\)
0.412251 + 0.911070i \(0.364743\pi\)
\(860\) 0 0
\(861\) −4.90704 −0.167232
\(862\) 54.5521 1.85805
\(863\) −8.57175 −0.291786 −0.145893 0.989300i \(-0.546605\pi\)
−0.145893 + 0.989300i \(0.546605\pi\)
\(864\) 0.516442 0.0175697
\(865\) 0 0
\(866\) −30.2999 −1.02963
\(867\) 3.20617 0.108887
\(868\) 3.86692 0.131252
\(869\) 8.25623 0.280074
\(870\) 0 0
\(871\) −9.71684 −0.329243
\(872\) 0.544697 0.0184458
\(873\) 19.2315 0.650886
\(874\) −0.965818 −0.0326693
\(875\) 0 0
\(876\) −0.0123058 −0.000415774 0
\(877\) −10.4764 −0.353763 −0.176881 0.984232i \(-0.556601\pi\)
−0.176881 + 0.984232i \(0.556601\pi\)
\(878\) 33.0293 1.11468
\(879\) 4.29594 0.144899
\(880\) 0 0
\(881\) −0.125726 −0.00423583 −0.00211792 0.999998i \(-0.500674\pi\)
−0.00211792 + 0.999998i \(0.500674\pi\)
\(882\) 48.0640 1.61840
\(883\) −53.1047 −1.78712 −0.893558 0.448947i \(-0.851799\pi\)
−0.893558 + 0.448947i \(0.851799\pi\)
\(884\) 0.333637 0.0112214
\(885\) 0 0
\(886\) −7.05207 −0.236919
\(887\) −1.80239 −0.0605183 −0.0302591 0.999542i \(-0.509633\pi\)
−0.0302591 + 0.999542i \(0.509633\pi\)
\(888\) −5.60298 −0.188024
\(889\) −32.0959 −1.07646
\(890\) 0 0
\(891\) 8.71060 0.291816
\(892\) 0.536682 0.0179694
\(893\) 10.4716 0.350419
\(894\) 1.90632 0.0637568
\(895\) 0 0
\(896\) 45.5715 1.52244
\(897\) 0.0831564 0.00277651
\(898\) −6.93237 −0.231336
\(899\) −34.9294 −1.16496
\(900\) 0 0
\(901\) 47.8505 1.59413
\(902\) −8.73990 −0.291007
\(903\) 2.14684 0.0714422
\(904\) 35.3996 1.17737
\(905\) 0 0
\(906\) −1.79985 −0.0597959
\(907\) 1.11693 0.0370869 0.0185435 0.999828i \(-0.494097\pi\)
0.0185435 + 0.999828i \(0.494097\pi\)
\(908\) −2.02639 −0.0672483
\(909\) 24.4959 0.812477
\(910\) 0 0
\(911\) 46.5149 1.54111 0.770553 0.637376i \(-0.219980\pi\)
0.770553 + 0.637376i \(0.219980\pi\)
\(912\) 0.686641 0.0227370
\(913\) −1.39198 −0.0460680
\(914\) 2.06839 0.0684163
\(915\) 0 0
\(916\) 0.342537 0.0113177
\(917\) 4.99685 0.165010
\(918\) 8.75711 0.289028
\(919\) −40.3852 −1.33218 −0.666092 0.745870i \(-0.732034\pi\)
−0.666092 + 0.745870i \(0.732034\pi\)
\(920\) 0 0
\(921\) 5.29945 0.174623
\(922\) 18.5274 0.610168
\(923\) 6.39739 0.210573
\(924\) 0.0662072 0.00217806
\(925\) 0 0
\(926\) −30.5348 −1.00344
\(927\) 33.4310 1.09802
\(928\) 1.60374 0.0526452
\(929\) −23.3618 −0.766476 −0.383238 0.923650i \(-0.625191\pi\)
−0.383238 + 0.923650i \(0.625191\pi\)
\(930\) 0 0
\(931\) −11.7040 −0.383584
\(932\) 2.26726 0.0742666
\(933\) −4.33145 −0.141805
\(934\) −9.14332 −0.299179
\(935\) 0 0
\(936\) 5.67924 0.185632
\(937\) −13.6780 −0.446842 −0.223421 0.974722i \(-0.571722\pi\)
−0.223421 + 0.974722i \(0.571722\pi\)
\(938\) 87.6794 2.86283
\(939\) 1.48346 0.0484110
\(940\) 0 0
\(941\) 41.2212 1.34377 0.671887 0.740654i \(-0.265484\pi\)
0.671887 + 0.740654i \(0.265484\pi\)
\(942\) 1.68822 0.0550050
\(943\) −4.40842 −0.143558
\(944\) 1.00922 0.0328474
\(945\) 0 0
\(946\) 3.82372 0.124320
\(947\) −7.43620 −0.241644 −0.120822 0.992674i \(-0.538553\pi\)
−0.120822 + 0.992674i \(0.538553\pi\)
\(948\) −0.126392 −0.00410502
\(949\) −0.533123 −0.0173059
\(950\) 0 0
\(951\) −1.37504 −0.0445888
\(952\) −73.6658 −2.38752
\(953\) 3.32135 0.107589 0.0537945 0.998552i \(-0.482868\pi\)
0.0537945 + 0.998552i \(0.482868\pi\)
\(954\) 33.2877 1.07773
\(955\) 0 0
\(956\) −2.11818 −0.0685068
\(957\) −0.598040 −0.0193319
\(958\) 30.2962 0.978825
\(959\) −63.7552 −2.05876
\(960\) 0 0
\(961\) 79.0855 2.55114
\(962\) −9.92013 −0.319838
\(963\) 35.5719 1.14629
\(964\) 0.113933 0.00366953
\(965\) 0 0
\(966\) −0.750357 −0.0241423
\(967\) 40.1346 1.29064 0.645321 0.763911i \(-0.276723\pi\)
0.645321 + 0.763911i \(0.276723\pi\)
\(968\) 2.88543 0.0927414
\(969\) −1.06045 −0.0340667
\(970\) 0 0
\(971\) 27.7956 0.892005 0.446002 0.895032i \(-0.352847\pi\)
0.446002 + 0.895032i \(0.352847\pi\)
\(972\) −0.407422 −0.0130681
\(973\) −27.6140 −0.885265
\(974\) −4.81588 −0.154311
\(975\) 0 0
\(976\) −54.6448 −1.74914
\(977\) 55.0675 1.76177 0.880883 0.473335i \(-0.156950\pi\)
0.880883 + 0.473335i \(0.156950\pi\)
\(978\) 2.90541 0.0929049
\(979\) 9.69068 0.309715
\(980\) 0 0
\(981\) −0.560232 −0.0178868
\(982\) 35.1590 1.12197
\(983\) 22.0581 0.703545 0.351773 0.936085i \(-0.385579\pi\)
0.351773 + 0.936085i \(0.385579\pi\)
\(984\) 3.27388 0.104368
\(985\) 0 0
\(986\) 27.1939 0.866031
\(987\) 8.13552 0.258957
\(988\) −0.0565180 −0.00179808
\(989\) 1.92869 0.0613287
\(990\) 0 0
\(991\) 13.3501 0.424080 0.212040 0.977261i \(-0.431989\pi\)
0.212040 + 0.977261i \(0.431989\pi\)
\(992\) −5.05443 −0.160478
\(993\) 2.08614 0.0662016
\(994\) −57.7265 −1.83097
\(995\) 0 0
\(996\) 0.0213094 0.000675215 0
\(997\) 40.0293 1.26774 0.633870 0.773439i \(-0.281465\pi\)
0.633870 + 0.773439i \(0.281465\pi\)
\(998\) −11.4476 −0.362367
\(999\) 11.5882 0.366635
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 5225.2.a.s.1.5 15
5.4 even 2 5225.2.a.x.1.11 yes 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5225.2.a.s.1.5 15 1.1 even 1 trivial
5225.2.a.x.1.11 yes 15 5.4 even 2