Properties

Label 6825.2.a.bg.1.1
Level $6825$
Weight $2$
Character 6825.1
Self dual yes
Analytic conductor $54.498$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6825,2,Mod(1,6825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6825, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6825.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6825 = 3 \cdot 5^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6825.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: \(54.4978993795\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.17428.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + 4x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 273)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.36865\) of defining polynomial
Character \(\chi\) \(=\) 6825.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.61050 q^{2} -1.00000 q^{3} +4.81471 q^{4} +2.61050 q^{6} -1.00000 q^{7} -7.34780 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.61050 q^{2} -1.00000 q^{3} +4.81471 q^{4} +2.61050 q^{6} -1.00000 q^{7} -7.34780 q^{8} +1.00000 q^{9} -4.73730 q^{11} -4.81471 q^{12} -1.00000 q^{13} +2.61050 q^{14} +9.55201 q^{16} +5.22100 q^{17} -2.61050 q^{18} +2.92259 q^{19} +1.00000 q^{21} +12.3667 q^{22} -3.33101 q^{23} +7.34780 q^{24} +2.61050 q^{26} -1.00000 q^{27} -4.81471 q^{28} -0.922589 q^{29} -7.51941 q^{31} -10.2399 q^{32} +4.73730 q^{33} -13.6294 q^{34} +4.81471 q^{36} -0.154821 q^{37} -7.62942 q^{38} +1.00000 q^{39} +6.36672 q^{41} -2.61050 q^{42} +6.55201 q^{43} -22.8087 q^{44} +8.69560 q^{46} -9.03571 q^{47} -9.55201 q^{48} +1.00000 q^{49} -5.22100 q^{51} -4.81471 q^{52} -8.55201 q^{53} +2.61050 q^{54} +7.34780 q^{56} -2.92259 q^{57} +2.40842 q^{58} +3.95830 q^{59} +12.4420 q^{61} +19.6294 q^{62} -1.00000 q^{63} +7.62729 q^{64} -12.3667 q^{66} +10.6620 q^{67} +25.1376 q^{68} +3.33101 q^{69} -6.58248 q^{71} -7.34780 q^{72} +7.73517 q^{73} +0.404161 q^{74} +14.0714 q^{76} +4.73730 q^{77} -2.61050 q^{78} +13.3646 q^{79} +1.00000 q^{81} -16.6203 q^{82} +1.40629 q^{83} +4.81471 q^{84} -17.1040 q^{86} +0.922589 q^{87} +34.8087 q^{88} -1.96953 q^{89} +1.00000 q^{91} -16.0378 q^{92} +7.51941 q^{93} +23.5877 q^{94} +10.2399 q^{96} +2.11001 q^{97} -2.61050 q^{98} -4.73730 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} - 4 q^{3} + 7 q^{4} + q^{6} - 4 q^{7} - 3 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} - 4 q^{3} + 7 q^{4} + q^{6} - 4 q^{7} - 3 q^{8} + 4 q^{9} - 2 q^{11} - 7 q^{12} - 4 q^{13} + q^{14} + 9 q^{16} + 2 q^{17} - q^{18} + 7 q^{19} + 4 q^{21} + 8 q^{22} - 3 q^{23} + 3 q^{24} + q^{26} - 4 q^{27} - 7 q^{28} + q^{29} + 3 q^{31} - 7 q^{32} + 2 q^{33} - 30 q^{34} + 7 q^{36} - 10 q^{37} - 6 q^{38} + 4 q^{39} - 16 q^{41} - q^{42} - 3 q^{43} - 12 q^{44} - 18 q^{46} - 5 q^{47} - 9 q^{48} + 4 q^{49} - 2 q^{51} - 7 q^{52} - 5 q^{53} + q^{54} + 3 q^{56} - 7 q^{57} + 4 q^{58} - 20 q^{59} + 12 q^{61} + 54 q^{62} - 4 q^{63} + 5 q^{64} - 8 q^{66} + 22 q^{67} + 10 q^{68} + 3 q^{69} - 3 q^{72} + 13 q^{73} - 6 q^{74} - 6 q^{76} + 2 q^{77} - q^{78} + 11 q^{79} + 4 q^{81} - 10 q^{82} - q^{83} + 7 q^{84} - 10 q^{86} - q^{87} + 60 q^{88} - 5 q^{89} + 4 q^{91} - 34 q^{92} - 3 q^{93} + 34 q^{94} + 7 q^{96} + 17 q^{97} - q^{98} - 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.61050 −1.84590 −0.922951 0.384917i \(-0.874230\pi\)
−0.922951 + 0.384917i \(0.874230\pi\)
\(3\) −1.00000 −0.577350
\(4\) 4.81471 2.40735
\(5\) 0 0
\(6\) 2.61050 1.06573
\(7\) −1.00000 −0.377964
\(8\) −7.34780 −2.59784
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.73730 −1.42835 −0.714175 0.699968i \(-0.753198\pi\)
−0.714175 + 0.699968i \(0.753198\pi\)
\(12\) −4.81471 −1.38989
\(13\) −1.00000 −0.277350
\(14\) 2.61050 0.697685
\(15\) 0 0
\(16\) 9.55201 2.38800
\(17\) 5.22100 1.26628 0.633139 0.774038i \(-0.281766\pi\)
0.633139 + 0.774038i \(0.281766\pi\)
\(18\) −2.61050 −0.615301
\(19\) 2.92259 0.670488 0.335244 0.942131i \(-0.391181\pi\)
0.335244 + 0.942131i \(0.391181\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 12.3667 2.63659
\(23\) −3.33101 −0.694563 −0.347282 0.937761i \(-0.612895\pi\)
−0.347282 + 0.937761i \(0.612895\pi\)
\(24\) 7.34780 1.49986
\(25\) 0 0
\(26\) 2.61050 0.511961
\(27\) −1.00000 −0.192450
\(28\) −4.81471 −0.909895
\(29\) −0.922589 −0.171321 −0.0856603 0.996324i \(-0.527300\pi\)
−0.0856603 + 0.996324i \(0.527300\pi\)
\(30\) 0 0
\(31\) −7.51941 −1.35053 −0.675263 0.737577i \(-0.735970\pi\)
−0.675263 + 0.737577i \(0.735970\pi\)
\(32\) −10.2399 −1.81018
\(33\) 4.73730 0.824658
\(34\) −13.6294 −2.33743
\(35\) 0 0
\(36\) 4.81471 0.802452
\(37\) −0.154821 −0.0254525 −0.0127262 0.999919i \(-0.504051\pi\)
−0.0127262 + 0.999919i \(0.504051\pi\)
\(38\) −7.62942 −1.23766
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 6.36672 0.994314 0.497157 0.867661i \(-0.334377\pi\)
0.497157 + 0.867661i \(0.334377\pi\)
\(42\) −2.61050 −0.402809
\(43\) 6.55201 0.999172 0.499586 0.866264i \(-0.333486\pi\)
0.499586 + 0.866264i \(0.333486\pi\)
\(44\) −22.8087 −3.43854
\(45\) 0 0
\(46\) 8.69560 1.28210
\(47\) −9.03571 −1.31799 −0.658997 0.752146i \(-0.729019\pi\)
−0.658997 + 0.752146i \(0.729019\pi\)
\(48\) −9.55201 −1.37871
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −5.22100 −0.731086
\(52\) −4.81471 −0.667680
\(53\) −8.55201 −1.17471 −0.587354 0.809330i \(-0.699830\pi\)
−0.587354 + 0.809330i \(0.699830\pi\)
\(54\) 2.61050 0.355244
\(55\) 0 0
\(56\) 7.34780 0.981891
\(57\) −2.92259 −0.387106
\(58\) 2.40842 0.316241
\(59\) 3.95830 0.515327 0.257663 0.966235i \(-0.417047\pi\)
0.257663 + 0.966235i \(0.417047\pi\)
\(60\) 0 0
\(61\) 12.4420 1.59303 0.796517 0.604616i \(-0.206673\pi\)
0.796517 + 0.604616i \(0.206673\pi\)
\(62\) 19.6294 2.49294
\(63\) −1.00000 −0.125988
\(64\) 7.62729 0.953411
\(65\) 0 0
\(66\) −12.3667 −1.52224
\(67\) 10.6620 1.30257 0.651286 0.758832i \(-0.274230\pi\)
0.651286 + 0.758832i \(0.274230\pi\)
\(68\) 25.1376 3.04838
\(69\) 3.33101 0.401006
\(70\) 0 0
\(71\) −6.58248 −0.781196 −0.390598 0.920561i \(-0.627732\pi\)
−0.390598 + 0.920561i \(0.627732\pi\)
\(72\) −7.34780 −0.865946
\(73\) 7.73517 0.905333 0.452667 0.891680i \(-0.350473\pi\)
0.452667 + 0.891680i \(0.350473\pi\)
\(74\) 0.404161 0.0469828
\(75\) 0 0
\(76\) 14.0714 1.61410
\(77\) 4.73730 0.539865
\(78\) −2.61050 −0.295581
\(79\) 13.3646 1.50363 0.751817 0.659372i \(-0.229178\pi\)
0.751817 + 0.659372i \(0.229178\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −16.6203 −1.83541
\(83\) 1.40629 0.154360 0.0771802 0.997017i \(-0.475408\pi\)
0.0771802 + 0.997017i \(0.475408\pi\)
\(84\) 4.81471 0.525328
\(85\) 0 0
\(86\) −17.1040 −1.84437
\(87\) 0.922589 0.0989120
\(88\) 34.8087 3.71062
\(89\) −1.96953 −0.208770 −0.104385 0.994537i \(-0.533287\pi\)
−0.104385 + 0.994537i \(0.533287\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) −16.0378 −1.67206
\(93\) 7.51941 0.779727
\(94\) 23.5877 2.43289
\(95\) 0 0
\(96\) 10.2399 1.04511
\(97\) 2.11001 0.214239 0.107119 0.994246i \(-0.465837\pi\)
0.107119 + 0.994246i \(0.465837\pi\)
\(98\) −2.61050 −0.263700
\(99\) −4.73730 −0.476116
\(100\) 0 0
\(101\) 0.850419 0.0846198 0.0423099 0.999105i \(-0.486528\pi\)
0.0423099 + 0.999105i \(0.486528\pi\)
\(102\) 13.6294 1.34951
\(103\) 1.47460 0.145296 0.0726482 0.997358i \(-0.476855\pi\)
0.0726482 + 0.997358i \(0.476855\pi\)
\(104\) 7.34780 0.720511
\(105\) 0 0
\(106\) 22.3250 2.16840
\(107\) −2.62418 −0.253689 −0.126844 0.991923i \(-0.540485\pi\)
−0.126844 + 0.991923i \(0.540485\pi\)
\(108\) −4.81471 −0.463296
\(109\) −17.9166 −1.71610 −0.858049 0.513567i \(-0.828324\pi\)
−0.858049 + 0.513567i \(0.828324\pi\)
\(110\) 0 0
\(111\) 0.154821 0.0146950
\(112\) −9.55201 −0.902580
\(113\) −0.922589 −0.0867899 −0.0433950 0.999058i \(-0.513817\pi\)
−0.0433950 + 0.999058i \(0.513817\pi\)
\(114\) 7.62942 0.714561
\(115\) 0 0
\(116\) −4.44200 −0.412429
\(117\) −1.00000 −0.0924500
\(118\) −10.3331 −0.951242
\(119\) −5.22100 −0.478608
\(120\) 0 0
\(121\) 11.4420 1.04018
\(122\) −32.4798 −2.94059
\(123\) −6.36672 −0.574068
\(124\) −36.2038 −3.25119
\(125\) 0 0
\(126\) 2.61050 0.232562
\(127\) −17.4746 −1.55062 −0.775310 0.631581i \(-0.782406\pi\)
−0.775310 + 0.631581i \(0.782406\pi\)
\(128\) 0.568798 0.0502751
\(129\) −6.55201 −0.576872
\(130\) 0 0
\(131\) 0.967402 0.0845223 0.0422611 0.999107i \(-0.486544\pi\)
0.0422611 + 0.999107i \(0.486544\pi\)
\(132\) 22.8087 1.98524
\(133\) −2.92259 −0.253421
\(134\) −27.8332 −2.40442
\(135\) 0 0
\(136\) −38.3629 −3.28959
\(137\) −3.29628 −0.281620 −0.140810 0.990037i \(-0.544971\pi\)
−0.140810 + 0.990037i \(0.544971\pi\)
\(138\) −8.69560 −0.740218
\(139\) 0.370581 0.0314323 0.0157161 0.999876i \(-0.494997\pi\)
0.0157161 + 0.999876i \(0.494997\pi\)
\(140\) 0 0
\(141\) 9.03571 0.760944
\(142\) 17.1836 1.44201
\(143\) 4.73730 0.396153
\(144\) 9.55201 0.796001
\(145\) 0 0
\(146\) −20.1927 −1.67116
\(147\) −1.00000 −0.0824786
\(148\) −0.745420 −0.0612732
\(149\) −15.7425 −1.28968 −0.644840 0.764318i \(-0.723076\pi\)
−0.644840 + 0.764318i \(0.723076\pi\)
\(150\) 0 0
\(151\) 10.2914 0.837505 0.418753 0.908100i \(-0.362467\pi\)
0.418753 + 0.908100i \(0.362467\pi\)
\(152\) −21.4746 −1.74182
\(153\) 5.22100 0.422093
\(154\) −12.3667 −0.996539
\(155\) 0 0
\(156\) 4.81471 0.385485
\(157\) 11.4137 0.910909 0.455455 0.890259i \(-0.349477\pi\)
0.455455 + 0.890259i \(0.349477\pi\)
\(158\) −34.8883 −2.77556
\(159\) 8.55201 0.678218
\(160\) 0 0
\(161\) 3.33101 0.262520
\(162\) −2.61050 −0.205100
\(163\) 13.4746 1.05541 0.527706 0.849427i \(-0.323052\pi\)
0.527706 + 0.849427i \(0.323052\pi\)
\(164\) 30.6539 2.39367
\(165\) 0 0
\(166\) −3.67112 −0.284934
\(167\) −19.1905 −1.48501 −0.742504 0.669842i \(-0.766362\pi\)
−0.742504 + 0.669842i \(0.766362\pi\)
\(168\) −7.34780 −0.566895
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 2.92259 0.223496
\(172\) 31.5460 2.40536
\(173\) 19.5124 1.48350 0.741752 0.670675i \(-0.233995\pi\)
0.741752 + 0.670675i \(0.233995\pi\)
\(174\) −2.40842 −0.182582
\(175\) 0 0
\(176\) −45.2507 −3.41090
\(177\) −3.95830 −0.297524
\(178\) 5.14146 0.385369
\(179\) −16.5856 −1.23967 −0.619833 0.784734i \(-0.712799\pi\)
−0.619833 + 0.784734i \(0.712799\pi\)
\(180\) 0 0
\(181\) −2.81684 −0.209374 −0.104687 0.994505i \(-0.533384\pi\)
−0.104687 + 0.994505i \(0.533384\pi\)
\(182\) −2.61050 −0.193503
\(183\) −12.4420 −0.919739
\(184\) 24.4756 1.80436
\(185\) 0 0
\(186\) −19.6294 −1.43930
\(187\) −24.7334 −1.80869
\(188\) −43.5043 −3.17288
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 15.1601 1.09694 0.548472 0.836169i \(-0.315210\pi\)
0.548472 + 0.836169i \(0.315210\pi\)
\(192\) −7.62729 −0.550452
\(193\) −0.0651962 −0.00469293 −0.00234646 0.999997i \(-0.500747\pi\)
−0.00234646 + 0.999997i \(0.500747\pi\)
\(194\) −5.50818 −0.395464
\(195\) 0 0
\(196\) 4.81471 0.343908
\(197\) 17.1415 1.22128 0.610639 0.791909i \(-0.290913\pi\)
0.610639 + 0.791909i \(0.290913\pi\)
\(198\) 12.3667 0.878864
\(199\) 6.44200 0.456661 0.228331 0.973584i \(-0.426673\pi\)
0.228331 + 0.973584i \(0.426673\pi\)
\(200\) 0 0
\(201\) −10.6620 −0.752041
\(202\) −2.22002 −0.156200
\(203\) 0.922589 0.0647531
\(204\) −25.1376 −1.75998
\(205\) 0 0
\(206\) −3.84944 −0.268203
\(207\) −3.33101 −0.231521
\(208\) −9.55201 −0.662313
\(209\) −13.8452 −0.957691
\(210\) 0 0
\(211\) −16.0266 −1.10332 −0.551659 0.834070i \(-0.686005\pi\)
−0.551659 + 0.834070i \(0.686005\pi\)
\(212\) −41.1754 −2.82794
\(213\) 6.58248 0.451024
\(214\) 6.85042 0.468285
\(215\) 0 0
\(216\) 7.34780 0.499954
\(217\) 7.51941 0.510451
\(218\) 46.7713 3.16775
\(219\) −7.73517 −0.522694
\(220\) 0 0
\(221\) −5.22100 −0.351202
\(222\) −0.404161 −0.0271255
\(223\) −20.0266 −1.34108 −0.670540 0.741873i \(-0.733938\pi\)
−0.670540 + 0.741873i \(0.733938\pi\)
\(224\) 10.2399 0.684183
\(225\) 0 0
\(226\) 2.40842 0.160206
\(227\) −19.2171 −1.27549 −0.637743 0.770249i \(-0.720132\pi\)
−0.637743 + 0.770249i \(0.720132\pi\)
\(228\) −14.0714 −0.931902
\(229\) 5.25884 0.347514 0.173757 0.984789i \(-0.444409\pi\)
0.173757 + 0.984789i \(0.444409\pi\)
\(230\) 0 0
\(231\) −4.73730 −0.311691
\(232\) 6.77900 0.445063
\(233\) 26.6234 1.74416 0.872079 0.489365i \(-0.162771\pi\)
0.872079 + 0.489365i \(0.162771\pi\)
\(234\) 2.61050 0.170654
\(235\) 0 0
\(236\) 19.0581 1.24057
\(237\) −13.3646 −0.868123
\(238\) 13.6294 0.883464
\(239\) 4.29104 0.277564 0.138782 0.990323i \(-0.455681\pi\)
0.138782 + 0.990323i \(0.455681\pi\)
\(240\) 0 0
\(241\) 7.52367 0.484642 0.242321 0.970196i \(-0.422091\pi\)
0.242321 + 0.970196i \(0.422091\pi\)
\(242\) −29.8693 −1.92007
\(243\) −1.00000 −0.0641500
\(244\) 59.9046 3.83500
\(245\) 0 0
\(246\) 16.6203 1.05967
\(247\) −2.92259 −0.185960
\(248\) 55.2511 3.50845
\(249\) −1.40629 −0.0891200
\(250\) 0 0
\(251\) 11.3198 0.714498 0.357249 0.934009i \(-0.383715\pi\)
0.357249 + 0.934009i \(0.383715\pi\)
\(252\) −4.81471 −0.303298
\(253\) 15.7800 0.992079
\(254\) 45.6174 2.86229
\(255\) 0 0
\(256\) −16.7394 −1.04621
\(257\) 24.8504 1.55013 0.775063 0.631884i \(-0.217718\pi\)
0.775063 + 0.631884i \(0.217718\pi\)
\(258\) 17.1040 1.06485
\(259\) 0.154821 0.00962013
\(260\) 0 0
\(261\) −0.922589 −0.0571068
\(262\) −2.52540 −0.156020
\(263\) −17.1762 −1.05913 −0.529565 0.848270i \(-0.677645\pi\)
−0.529565 + 0.848270i \(0.677645\pi\)
\(264\) −34.8087 −2.14233
\(265\) 0 0
\(266\) 7.62942 0.467790
\(267\) 1.96953 0.120533
\(268\) 51.3345 3.13575
\(269\) −7.74640 −0.472306 −0.236153 0.971716i \(-0.575887\pi\)
−0.236153 + 0.971716i \(0.575887\pi\)
\(270\) 0 0
\(271\) −29.7008 −1.80420 −0.902099 0.431530i \(-0.857974\pi\)
−0.902099 + 0.431530i \(0.857974\pi\)
\(272\) 49.8710 3.02388
\(273\) −1.00000 −0.0605228
\(274\) 8.60494 0.519844
\(275\) 0 0
\(276\) 16.0378 0.965364
\(277\) −25.1488 −1.51105 −0.755523 0.655122i \(-0.772617\pi\)
−0.755523 + 0.655122i \(0.772617\pi\)
\(278\) −0.967402 −0.0580209
\(279\) −7.51941 −0.450175
\(280\) 0 0
\(281\) 8.40030 0.501120 0.250560 0.968101i \(-0.419385\pi\)
0.250560 + 0.968101i \(0.419385\pi\)
\(282\) −23.5877 −1.40463
\(283\) 21.3912 1.27157 0.635787 0.771864i \(-0.280676\pi\)
0.635787 + 0.771864i \(0.280676\pi\)
\(284\) −31.6927 −1.88062
\(285\) 0 0
\(286\) −12.3667 −0.731259
\(287\) −6.36672 −0.375815
\(288\) −10.2399 −0.603393
\(289\) 10.2588 0.603461
\(290\) 0 0
\(291\) −2.11001 −0.123691
\(292\) 37.2426 2.17946
\(293\) 9.59895 0.560777 0.280388 0.959887i \(-0.409537\pi\)
0.280388 + 0.959887i \(0.409537\pi\)
\(294\) 2.61050 0.152247
\(295\) 0 0
\(296\) 1.13760 0.0661215
\(297\) 4.73730 0.274886
\(298\) 41.0959 2.38062
\(299\) 3.33101 0.192637
\(300\) 0 0
\(301\) −6.55201 −0.377651
\(302\) −26.8658 −1.54595
\(303\) −0.850419 −0.0488553
\(304\) 27.9166 1.60113
\(305\) 0 0
\(306\) −13.6294 −0.779142
\(307\) 15.1488 0.864589 0.432295 0.901732i \(-0.357704\pi\)
0.432295 + 0.901732i \(0.357704\pi\)
\(308\) 22.8087 1.29965
\(309\) −1.47460 −0.0838869
\(310\) 0 0
\(311\) −4.37058 −0.247833 −0.123916 0.992293i \(-0.539545\pi\)
−0.123916 + 0.992293i \(0.539545\pi\)
\(312\) −7.34780 −0.415987
\(313\) 1.49280 0.0843783 0.0421891 0.999110i \(-0.486567\pi\)
0.0421891 + 0.999110i \(0.486567\pi\)
\(314\) −29.7954 −1.68145
\(315\) 0 0
\(316\) 64.3466 3.61978
\(317\) −23.2129 −1.30377 −0.651883 0.758320i \(-0.726021\pi\)
−0.651883 + 0.758320i \(0.726021\pi\)
\(318\) −22.3250 −1.25192
\(319\) 4.37058 0.244706
\(320\) 0 0
\(321\) 2.62418 0.146467
\(322\) −8.69560 −0.484587
\(323\) 15.2588 0.849024
\(324\) 4.81471 0.267484
\(325\) 0 0
\(326\) −35.1754 −1.94819
\(327\) 17.9166 0.990790
\(328\) −46.7814 −2.58307
\(329\) 9.03571 0.498155
\(330\) 0 0
\(331\) 1.10402 0.0606822 0.0303411 0.999540i \(-0.490341\pi\)
0.0303411 + 0.999540i \(0.490341\pi\)
\(332\) 6.77088 0.371600
\(333\) −0.154821 −0.00848416
\(334\) 50.0969 2.74118
\(335\) 0 0
\(336\) 9.55201 0.521105
\(337\) 4.24237 0.231096 0.115548 0.993302i \(-0.463138\pi\)
0.115548 + 0.993302i \(0.463138\pi\)
\(338\) −2.61050 −0.141992
\(339\) 0.922589 0.0501082
\(340\) 0 0
\(341\) 35.6217 1.92902
\(342\) −7.62942 −0.412552
\(343\) −1.00000 −0.0539949
\(344\) −48.1428 −2.59569
\(345\) 0 0
\(346\) −50.9372 −2.73840
\(347\) −14.0336 −0.753362 −0.376681 0.926343i \(-0.622935\pi\)
−0.376681 + 0.926343i \(0.622935\pi\)
\(348\) 4.44200 0.238116
\(349\) 4.10575 0.219776 0.109888 0.993944i \(-0.464951\pi\)
0.109888 + 0.993944i \(0.464951\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 48.5096 2.58557
\(353\) 16.0753 0.855601 0.427800 0.903873i \(-0.359289\pi\)
0.427800 + 0.903873i \(0.359289\pi\)
\(354\) 10.3331 0.549200
\(355\) 0 0
\(356\) −9.48272 −0.502583
\(357\) 5.22100 0.276325
\(358\) 43.2967 2.28830
\(359\) −18.1510 −0.957971 −0.478985 0.877823i \(-0.658995\pi\)
−0.478985 + 0.877823i \(0.658995\pi\)
\(360\) 0 0
\(361\) −10.4585 −0.550446
\(362\) 7.35336 0.386484
\(363\) −11.4420 −0.600549
\(364\) 4.81471 0.252359
\(365\) 0 0
\(366\) 32.4798 1.69775
\(367\) 23.3955 1.22123 0.610616 0.791927i \(-0.290922\pi\)
0.610616 + 0.791927i \(0.290922\pi\)
\(368\) −31.8178 −1.65862
\(369\) 6.36672 0.331438
\(370\) 0 0
\(371\) 8.55201 0.443998
\(372\) 36.2038 1.87708
\(373\) 4.75164 0.246031 0.123015 0.992405i \(-0.460744\pi\)
0.123015 + 0.992405i \(0.460744\pi\)
\(374\) 64.5666 3.33866
\(375\) 0 0
\(376\) 66.3926 3.42394
\(377\) 0.922589 0.0475158
\(378\) −2.61050 −0.134270
\(379\) −6.81258 −0.349939 −0.174969 0.984574i \(-0.555983\pi\)
−0.174969 + 0.984574i \(0.555983\pi\)
\(380\) 0 0
\(381\) 17.4746 0.895251
\(382\) −39.5753 −2.02485
\(383\) −2.25746 −0.115351 −0.0576754 0.998335i \(-0.518369\pi\)
−0.0576754 + 0.998335i \(0.518369\pi\)
\(384\) −0.568798 −0.0290264
\(385\) 0 0
\(386\) 0.170195 0.00866268
\(387\) 6.55201 0.333057
\(388\) 10.1591 0.515749
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) −17.3912 −0.879511
\(392\) −7.34780 −0.371120
\(393\) −0.967402 −0.0487990
\(394\) −44.7478 −2.25436
\(395\) 0 0
\(396\) −22.8087 −1.14618
\(397\) −24.9897 −1.25420 −0.627100 0.778939i \(-0.715758\pi\)
−0.627100 + 0.778939i \(0.715758\pi\)
\(398\) −16.8168 −0.842952
\(399\) 2.92259 0.146312
\(400\) 0 0
\(401\) 27.6529 1.38092 0.690460 0.723370i \(-0.257408\pi\)
0.690460 + 0.723370i \(0.257408\pi\)
\(402\) 27.8332 1.38819
\(403\) 7.51941 0.374568
\(404\) 4.09452 0.203710
\(405\) 0 0
\(406\) −2.40842 −0.119528
\(407\) 0.733435 0.0363550
\(408\) 38.3629 1.89924
\(409\) 13.1488 0.650168 0.325084 0.945685i \(-0.394607\pi\)
0.325084 + 0.945685i \(0.394607\pi\)
\(410\) 0 0
\(411\) 3.29628 0.162594
\(412\) 7.09976 0.349780
\(413\) −3.95830 −0.194775
\(414\) 8.69560 0.427365
\(415\) 0 0
\(416\) 10.2399 0.502053
\(417\) −0.370581 −0.0181474
\(418\) 36.1428 1.76780
\(419\) −5.69462 −0.278200 −0.139100 0.990278i \(-0.544421\pi\)
−0.139100 + 0.990278i \(0.544421\pi\)
\(420\) 0 0
\(421\) −23.0206 −1.12196 −0.560978 0.827831i \(-0.689575\pi\)
−0.560978 + 0.827831i \(0.689575\pi\)
\(422\) 41.8375 2.03662
\(423\) −9.03571 −0.439331
\(424\) 62.8384 3.05170
\(425\) 0 0
\(426\) −17.1836 −0.832546
\(427\) −12.4420 −0.602111
\(428\) −12.6347 −0.610719
\(429\) −4.73730 −0.228719
\(430\) 0 0
\(431\) 20.8045 1.00212 0.501058 0.865414i \(-0.332944\pi\)
0.501058 + 0.865414i \(0.332944\pi\)
\(432\) −9.55201 −0.459571
\(433\) −31.4808 −1.51287 −0.756436 0.654068i \(-0.773061\pi\)
−0.756436 + 0.654068i \(0.773061\pi\)
\(434\) −19.6294 −0.942242
\(435\) 0 0
\(436\) −86.2632 −4.13126
\(437\) −9.73517 −0.465696
\(438\) 20.1927 0.964843
\(439\) 22.3811 1.06819 0.534095 0.845425i \(-0.320653\pi\)
0.534095 + 0.845425i \(0.320653\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 13.6294 0.648285
\(443\) −8.52465 −0.405018 −0.202509 0.979280i \(-0.564910\pi\)
−0.202509 + 0.979280i \(0.564910\pi\)
\(444\) 0.745420 0.0353761
\(445\) 0 0
\(446\) 52.2795 2.47550
\(447\) 15.7425 0.744597
\(448\) −7.62729 −0.360356
\(449\) 18.7142 0.883178 0.441589 0.897218i \(-0.354415\pi\)
0.441589 + 0.897218i \(0.354415\pi\)
\(450\) 0 0
\(451\) −30.1610 −1.42023
\(452\) −4.44200 −0.208934
\(453\) −10.2914 −0.483534
\(454\) 50.1663 2.35442
\(455\) 0 0
\(456\) 21.4746 1.00564
\(457\) 14.1366 0.661283 0.330641 0.943756i \(-0.392735\pi\)
0.330641 + 0.943756i \(0.392735\pi\)
\(458\) −13.7282 −0.641476
\(459\) −5.22100 −0.243695
\(460\) 0 0
\(461\) −2.67636 −0.124651 −0.0623253 0.998056i \(-0.519852\pi\)
−0.0623253 + 0.998056i \(0.519852\pi\)
\(462\) 12.3667 0.575352
\(463\) 2.53162 0.117655 0.0588273 0.998268i \(-0.481264\pi\)
0.0588273 + 0.998268i \(0.481264\pi\)
\(464\) −8.81258 −0.409114
\(465\) 0 0
\(466\) −69.5005 −3.21955
\(467\) −2.00426 −0.0927460 −0.0463730 0.998924i \(-0.514766\pi\)
−0.0463730 + 0.998924i \(0.514766\pi\)
\(468\) −4.81471 −0.222560
\(469\) −10.6620 −0.492326
\(470\) 0 0
\(471\) −11.4137 −0.525914
\(472\) −29.0848 −1.33874
\(473\) −31.0388 −1.42717
\(474\) 34.8883 1.60247
\(475\) 0 0
\(476\) −25.1376 −1.15218
\(477\) −8.55201 −0.391570
\(478\) −11.2018 −0.512357
\(479\) −14.6609 −0.669872 −0.334936 0.942241i \(-0.608715\pi\)
−0.334936 + 0.942241i \(0.608715\pi\)
\(480\) 0 0
\(481\) 0.154821 0.00705925
\(482\) −19.6405 −0.894602
\(483\) −3.33101 −0.151566
\(484\) 55.0899 2.50409
\(485\) 0 0
\(486\) 2.61050 0.118415
\(487\) −38.2143 −1.73165 −0.865827 0.500344i \(-0.833207\pi\)
−0.865827 + 0.500344i \(0.833207\pi\)
\(488\) −91.4213 −4.13845
\(489\) −13.4746 −0.609342
\(490\) 0 0
\(491\) 21.3758 0.964677 0.482339 0.875985i \(-0.339787\pi\)
0.482339 + 0.875985i \(0.339787\pi\)
\(492\) −30.6539 −1.38198
\(493\) −4.81684 −0.216939
\(494\) 7.62942 0.343264
\(495\) 0 0
\(496\) −71.8255 −3.22506
\(497\) 6.58248 0.295264
\(498\) 3.67112 0.164507
\(499\) 27.0920 1.21281 0.606403 0.795158i \(-0.292612\pi\)
0.606403 + 0.795158i \(0.292612\pi\)
\(500\) 0 0
\(501\) 19.1905 0.857370
\(502\) −29.5503 −1.31889
\(503\) −24.8778 −1.10925 −0.554623 0.832102i \(-0.687137\pi\)
−0.554623 + 0.832102i \(0.687137\pi\)
\(504\) 7.34780 0.327297
\(505\) 0 0
\(506\) −41.1936 −1.83128
\(507\) −1.00000 −0.0444116
\(508\) −84.1351 −3.73289
\(509\) 2.70297 0.119807 0.0599034 0.998204i \(-0.480921\pi\)
0.0599034 + 0.998204i \(0.480921\pi\)
\(510\) 0 0
\(511\) −7.73517 −0.342184
\(512\) 42.5607 1.88093
\(513\) −2.92259 −0.129035
\(514\) −64.8720 −2.86138
\(515\) 0 0
\(516\) −31.5460 −1.38874
\(517\) 42.8049 1.88256
\(518\) −0.404161 −0.0177578
\(519\) −19.5124 −0.856501
\(520\) 0 0
\(521\) −33.4472 −1.46535 −0.732675 0.680579i \(-0.761728\pi\)
−0.732675 + 0.680579i \(0.761728\pi\)
\(522\) 2.40842 0.105414
\(523\) −19.3198 −0.844795 −0.422397 0.906411i \(-0.638811\pi\)
−0.422397 + 0.906411i \(0.638811\pi\)
\(524\) 4.65776 0.203475
\(525\) 0 0
\(526\) 44.8384 1.95505
\(527\) −39.2588 −1.71014
\(528\) 45.2507 1.96928
\(529\) −11.9044 −0.517582
\(530\) 0 0
\(531\) 3.95830 0.171776
\(532\) −14.0714 −0.610073
\(533\) −6.36672 −0.275773
\(534\) −5.14146 −0.222493
\(535\) 0 0
\(536\) −78.3424 −3.38387
\(537\) 16.5856 0.715721
\(538\) 20.2220 0.871832
\(539\) −4.73730 −0.204050
\(540\) 0 0
\(541\) −24.9554 −1.07292 −0.536459 0.843927i \(-0.680238\pi\)
−0.536459 + 0.843927i \(0.680238\pi\)
\(542\) 77.5340 3.33037
\(543\) 2.81684 0.120882
\(544\) −53.4626 −2.29219
\(545\) 0 0
\(546\) 2.61050 0.111719
\(547\) −3.80037 −0.162492 −0.0812460 0.996694i \(-0.525890\pi\)
−0.0812460 + 0.996694i \(0.525890\pi\)
\(548\) −15.8706 −0.677960
\(549\) 12.4420 0.531012
\(550\) 0 0
\(551\) −2.69635 −0.114868
\(552\) −24.4756 −1.04175
\(553\) −13.3646 −0.568320
\(554\) 65.6510 2.78924
\(555\) 0 0
\(556\) 1.78424 0.0756686
\(557\) −43.8792 −1.85922 −0.929610 0.368546i \(-0.879856\pi\)
−0.929610 + 0.368546i \(0.879856\pi\)
\(558\) 19.6294 0.830980
\(559\) −6.55201 −0.277120
\(560\) 0 0
\(561\) 24.7334 1.04425
\(562\) −21.9290 −0.925018
\(563\) −12.3768 −0.521620 −0.260810 0.965390i \(-0.583990\pi\)
−0.260810 + 0.965390i \(0.583990\pi\)
\(564\) 43.5043 1.83186
\(565\) 0 0
\(566\) −55.8417 −2.34720
\(567\) −1.00000 −0.0419961
\(568\) 48.3667 2.02942
\(569\) −26.6234 −1.11611 −0.558056 0.829803i \(-0.688453\pi\)
−0.558056 + 0.829803i \(0.688453\pi\)
\(570\) 0 0
\(571\) 10.9983 0.460263 0.230132 0.973160i \(-0.426084\pi\)
0.230132 + 0.973160i \(0.426084\pi\)
\(572\) 22.8087 0.953680
\(573\) −15.1601 −0.633321
\(574\) 16.6203 0.693719
\(575\) 0 0
\(576\) 7.62729 0.317804
\(577\) −30.4238 −1.26656 −0.633280 0.773923i \(-0.718292\pi\)
−0.633280 + 0.773923i \(0.718292\pi\)
\(578\) −26.7807 −1.11393
\(579\) 0.0651962 0.00270946
\(580\) 0 0
\(581\) −1.40629 −0.0583428
\(582\) 5.50818 0.228321
\(583\) 40.5134 1.67789
\(584\) −56.8365 −2.35191
\(585\) 0 0
\(586\) −25.0581 −1.03514
\(587\) −3.84207 −0.158579 −0.0792895 0.996852i \(-0.525265\pi\)
−0.0792895 + 0.996852i \(0.525265\pi\)
\(588\) −4.81471 −0.198555
\(589\) −21.9761 −0.905511
\(590\) 0 0
\(591\) −17.1415 −0.705105
\(592\) −1.47886 −0.0607806
\(593\) 23.8904 0.981061 0.490530 0.871424i \(-0.336803\pi\)
0.490530 + 0.871424i \(0.336803\pi\)
\(594\) −12.3667 −0.507413
\(595\) 0 0
\(596\) −75.7958 −3.10471
\(597\) −6.44200 −0.263653
\(598\) −8.69560 −0.355589
\(599\) −15.4206 −0.630070 −0.315035 0.949080i \(-0.602016\pi\)
−0.315035 + 0.949080i \(0.602016\pi\)
\(600\) 0 0
\(601\) −1.49280 −0.0608928 −0.0304464 0.999536i \(-0.509693\pi\)
−0.0304464 + 0.999536i \(0.509693\pi\)
\(602\) 17.1040 0.697108
\(603\) 10.6620 0.434191
\(604\) 49.5503 2.01617
\(605\) 0 0
\(606\) 2.22002 0.0901821
\(607\) 4.44626 0.180468 0.0902340 0.995921i \(-0.471239\pi\)
0.0902340 + 0.995921i \(0.471239\pi\)
\(608\) −29.9271 −1.21370
\(609\) −0.922589 −0.0373852
\(610\) 0 0
\(611\) 9.03571 0.365546
\(612\) 25.1376 1.01613
\(613\) −26.6640 −1.07695 −0.538474 0.842642i \(-0.680999\pi\)
−0.538474 + 0.842642i \(0.680999\pi\)
\(614\) −39.5460 −1.59595
\(615\) 0 0
\(616\) −34.8087 −1.40248
\(617\) −27.3495 −1.10105 −0.550525 0.834819i \(-0.685572\pi\)
−0.550525 + 0.834819i \(0.685572\pi\)
\(618\) 3.84944 0.154847
\(619\) −9.24836 −0.371723 −0.185861 0.982576i \(-0.559508\pi\)
−0.185861 + 0.982576i \(0.559508\pi\)
\(620\) 0 0
\(621\) 3.33101 0.133669
\(622\) 11.4094 0.457475
\(623\) 1.96953 0.0789076
\(624\) 9.55201 0.382386
\(625\) 0 0
\(626\) −3.89697 −0.155754
\(627\) 13.8452 0.552923
\(628\) 54.9535 2.19288
\(629\) −0.808323 −0.0322299
\(630\) 0 0
\(631\) 24.5134 0.975864 0.487932 0.872882i \(-0.337751\pi\)
0.487932 + 0.872882i \(0.337751\pi\)
\(632\) −98.2003 −3.90620
\(633\) 16.0266 0.637000
\(634\) 60.5972 2.40662
\(635\) 0 0
\(636\) 41.1754 1.63271
\(637\) −1.00000 −0.0396214
\(638\) −11.4094 −0.451703
\(639\) −6.58248 −0.260399
\(640\) 0 0
\(641\) 34.3972 1.35861 0.679304 0.733857i \(-0.262282\pi\)
0.679304 + 0.733857i \(0.262282\pi\)
\(642\) −6.85042 −0.270364
\(643\) 7.69036 0.303278 0.151639 0.988436i \(-0.451545\pi\)
0.151639 + 0.988436i \(0.451545\pi\)
\(644\) 16.0378 0.631979
\(645\) 0 0
\(646\) −39.8332 −1.56722
\(647\) −41.9123 −1.64774 −0.823872 0.566776i \(-0.808191\pi\)
−0.823872 + 0.566776i \(0.808191\pi\)
\(648\) −7.34780 −0.288649
\(649\) −18.7516 −0.736066
\(650\) 0 0
\(651\) −7.51941 −0.294709
\(652\) 64.8763 2.54075
\(653\) −34.2262 −1.33938 −0.669688 0.742642i \(-0.733572\pi\)
−0.669688 + 0.742642i \(0.733572\pi\)
\(654\) −46.7713 −1.82890
\(655\) 0 0
\(656\) 60.8149 2.37442
\(657\) 7.73517 0.301778
\(658\) −23.5877 −0.919545
\(659\) 45.1685 1.75951 0.879757 0.475424i \(-0.157705\pi\)
0.879757 + 0.475424i \(0.157705\pi\)
\(660\) 0 0
\(661\) 11.9839 0.466119 0.233059 0.972463i \(-0.425126\pi\)
0.233059 + 0.972463i \(0.425126\pi\)
\(662\) −2.88203 −0.112013
\(663\) 5.22100 0.202767
\(664\) −10.3331 −0.401004
\(665\) 0 0
\(666\) 0.404161 0.0156609
\(667\) 3.07315 0.118993
\(668\) −92.3968 −3.57494
\(669\) 20.0266 0.774273
\(670\) 0 0
\(671\) −58.9415 −2.27541
\(672\) −10.2399 −0.395013
\(673\) 29.5117 1.13759 0.568796 0.822479i \(-0.307409\pi\)
0.568796 + 0.822479i \(0.307409\pi\)
\(674\) −11.0747 −0.426581
\(675\) 0 0
\(676\) 4.81471 0.185181
\(677\) 31.0662 1.19397 0.596985 0.802252i \(-0.296365\pi\)
0.596985 + 0.802252i \(0.296365\pi\)
\(678\) −2.40842 −0.0924948
\(679\) −2.11001 −0.0809747
\(680\) 0 0
\(681\) 19.2171 0.736402
\(682\) −92.9904 −3.56079
\(683\) −5.79433 −0.221714 −0.110857 0.993836i \(-0.535360\pi\)
−0.110857 + 0.993836i \(0.535360\pi\)
\(684\) 14.0714 0.538034
\(685\) 0 0
\(686\) 2.61050 0.0996693
\(687\) −5.25884 −0.200637
\(688\) 62.5848 2.38602
\(689\) 8.55201 0.325806
\(690\) 0 0
\(691\) 18.8617 0.717531 0.358766 0.933428i \(-0.383198\pi\)
0.358766 + 0.933428i \(0.383198\pi\)
\(692\) 93.9467 3.57132
\(693\) 4.73730 0.179955
\(694\) 36.6347 1.39063
\(695\) 0 0
\(696\) −6.77900 −0.256957
\(697\) 33.2406 1.25908
\(698\) −10.7181 −0.405685
\(699\) −26.6234 −1.00699
\(700\) 0 0
\(701\) −32.3180 −1.22064 −0.610318 0.792157i \(-0.708958\pi\)
−0.610318 + 0.792157i \(0.708958\pi\)
\(702\) −2.61050 −0.0985270
\(703\) −0.452479 −0.0170656
\(704\) −36.1328 −1.36180
\(705\) 0 0
\(706\) −41.9645 −1.57936
\(707\) −0.850419 −0.0319833
\(708\) −19.0581 −0.716246
\(709\) 4.72296 0.177374 0.0886872 0.996060i \(-0.471733\pi\)
0.0886872 + 0.996060i \(0.471733\pi\)
\(710\) 0 0
\(711\) 13.3646 0.501211
\(712\) 14.4717 0.542350
\(713\) 25.0472 0.938026
\(714\) −13.6294 −0.510068
\(715\) 0 0
\(716\) −79.8548 −2.98431
\(717\) −4.29104 −0.160252
\(718\) 47.3831 1.76832
\(719\) −40.8902 −1.52495 −0.762474 0.647019i \(-0.776015\pi\)
−0.762474 + 0.647019i \(0.776015\pi\)
\(720\) 0 0
\(721\) −1.47460 −0.0549169
\(722\) 27.3018 1.01607
\(723\) −7.52367 −0.279808
\(724\) −13.5623 −0.504037
\(725\) 0 0
\(726\) 29.8693 1.10856
\(727\) −18.7292 −0.694627 −0.347313 0.937749i \(-0.612906\pi\)
−0.347313 + 0.937749i \(0.612906\pi\)
\(728\) −7.34780 −0.272328
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 34.2080 1.26523
\(732\) −59.9046 −2.21414
\(733\) 14.1100 0.521165 0.260583 0.965452i \(-0.416085\pi\)
0.260583 + 0.965452i \(0.416085\pi\)
\(734\) −61.0738 −2.25428
\(735\) 0 0
\(736\) 34.1093 1.25728
\(737\) −50.5092 −1.86053
\(738\) −16.6203 −0.611802
\(739\) −49.9209 −1.83637 −0.918184 0.396154i \(-0.870345\pi\)
−0.918184 + 0.396154i \(0.870345\pi\)
\(740\) 0 0
\(741\) 2.92259 0.107364
\(742\) −22.3250 −0.819577
\(743\) −2.15868 −0.0791945 −0.0395972 0.999216i \(-0.512607\pi\)
−0.0395972 + 0.999216i \(0.512607\pi\)
\(744\) −55.2511 −2.02560
\(745\) 0 0
\(746\) −12.4042 −0.454149
\(747\) 1.40629 0.0514535
\(748\) −119.084 −4.35415
\(749\) 2.62418 0.0958854
\(750\) 0 0
\(751\) 16.8372 0.614399 0.307199 0.951645i \(-0.400608\pi\)
0.307199 + 0.951645i \(0.400608\pi\)
\(752\) −86.3092 −3.14737
\(753\) −11.3198 −0.412516
\(754\) −2.40842 −0.0877095
\(755\) 0 0
\(756\) 4.81471 0.175109
\(757\) 17.6028 0.639785 0.319893 0.947454i \(-0.396353\pi\)
0.319893 + 0.947454i \(0.396353\pi\)
\(758\) 17.7842 0.645953
\(759\) −15.7800 −0.572777
\(760\) 0 0
\(761\) 19.2179 0.696648 0.348324 0.937374i \(-0.386751\pi\)
0.348324 + 0.937374i \(0.386751\pi\)
\(762\) −45.6174 −1.65255
\(763\) 17.9166 0.648624
\(764\) 72.9913 2.64073
\(765\) 0 0
\(766\) 5.89310 0.212926
\(767\) −3.95830 −0.142926
\(768\) 16.7394 0.604032
\(769\) −14.0406 −0.506315 −0.253158 0.967425i \(-0.581469\pi\)
−0.253158 + 0.967425i \(0.581469\pi\)
\(770\) 0 0
\(771\) −24.8504 −0.894966
\(772\) −0.313901 −0.0112975
\(773\) −22.4339 −0.806891 −0.403445 0.915004i \(-0.632187\pi\)
−0.403445 + 0.915004i \(0.632187\pi\)
\(774\) −17.1040 −0.614791
\(775\) 0 0
\(776\) −15.5039 −0.556558
\(777\) −0.154821 −0.00555419
\(778\) −15.6630 −0.561546
\(779\) 18.6073 0.666676
\(780\) 0 0
\(781\) 31.1832 1.11582
\(782\) 45.3997 1.62349
\(783\) 0.922589 0.0329707
\(784\) 9.55201 0.341143
\(785\) 0 0
\(786\) 2.52540 0.0900781
\(787\) 0.926847 0.0330385 0.0165193 0.999864i \(-0.494742\pi\)
0.0165193 + 0.999864i \(0.494742\pi\)
\(788\) 82.5311 2.94005
\(789\) 17.1762 0.611488
\(790\) 0 0
\(791\) 0.922589 0.0328035
\(792\) 34.8087 1.23687
\(793\) −12.4420 −0.441828
\(794\) 65.2357 2.31513
\(795\) 0 0
\(796\) 31.0164 1.09935
\(797\) 26.0154 0.921512 0.460756 0.887527i \(-0.347578\pi\)
0.460756 + 0.887527i \(0.347578\pi\)
\(798\) −7.62942 −0.270079
\(799\) −47.1754 −1.66895
\(800\) 0 0
\(801\) −1.96953 −0.0695900
\(802\) −72.1879 −2.54904
\(803\) −36.6438 −1.29313
\(804\) −51.3345 −1.81043
\(805\) 0 0
\(806\) −19.6294 −0.691417
\(807\) 7.74640 0.272686
\(808\) −6.24870 −0.219829
\(809\) −44.8539 −1.57698 −0.788490 0.615048i \(-0.789137\pi\)
−0.788490 + 0.615048i \(0.789137\pi\)
\(810\) 0 0
\(811\) 27.2511 0.956916 0.478458 0.878110i \(-0.341196\pi\)
0.478458 + 0.878110i \(0.341196\pi\)
\(812\) 4.44200 0.155884
\(813\) 29.7008 1.04165
\(814\) −1.91463 −0.0671078
\(815\) 0 0
\(816\) −49.8710 −1.74584
\(817\) 19.1488 0.669933
\(818\) −34.3250 −1.20015
\(819\) 1.00000 0.0349428
\(820\) 0 0
\(821\) 20.4003 0.711975 0.355988 0.934491i \(-0.384145\pi\)
0.355988 + 0.934491i \(0.384145\pi\)
\(822\) −8.60494 −0.300132
\(823\) −54.3509 −1.89455 −0.947276 0.320418i \(-0.896176\pi\)
−0.947276 + 0.320418i \(0.896176\pi\)
\(824\) −10.8350 −0.377457
\(825\) 0 0
\(826\) 10.3331 0.359536
\(827\) −3.48272 −0.121106 −0.0605530 0.998165i \(-0.519286\pi\)
−0.0605530 + 0.998165i \(0.519286\pi\)
\(828\) −16.0378 −0.557353
\(829\) −29.8417 −1.03645 −0.518223 0.855246i \(-0.673406\pi\)
−0.518223 + 0.855246i \(0.673406\pi\)
\(830\) 0 0
\(831\) 25.1488 0.872403
\(832\) −7.62729 −0.264429
\(833\) 5.22100 0.180897
\(834\) 0.967402 0.0334984
\(835\) 0 0
\(836\) −66.6605 −2.30550
\(837\) 7.51941 0.259909
\(838\) 14.8658 0.513530
\(839\) −43.0033 −1.48464 −0.742320 0.670045i \(-0.766275\pi\)
−0.742320 + 0.670045i \(0.766275\pi\)
\(840\) 0 0
\(841\) −28.1488 −0.970649
\(842\) 60.0953 2.07102
\(843\) −8.40030 −0.289322
\(844\) −77.1634 −2.65608
\(845\) 0 0
\(846\) 23.5877 0.810962
\(847\) −11.4420 −0.393152
\(848\) −81.6889 −2.80521
\(849\) −21.3912 −0.734144
\(850\) 0 0
\(851\) 0.515711 0.0176784
\(852\) 31.6927 1.08577
\(853\) −34.1023 −1.16764 −0.583820 0.811883i \(-0.698443\pi\)
−0.583820 + 0.811883i \(0.698443\pi\)
\(854\) 32.4798 1.11144
\(855\) 0 0
\(856\) 19.2819 0.659043
\(857\) 45.7344 1.56226 0.781129 0.624370i \(-0.214644\pi\)
0.781129 + 0.624370i \(0.214644\pi\)
\(858\) 12.3667 0.422193
\(859\) 15.9861 0.545437 0.272719 0.962094i \(-0.412077\pi\)
0.272719 + 0.962094i \(0.412077\pi\)
\(860\) 0 0
\(861\) 6.36672 0.216977
\(862\) −54.3100 −1.84981
\(863\) 24.5096 0.834315 0.417157 0.908834i \(-0.363026\pi\)
0.417157 + 0.908834i \(0.363026\pi\)
\(864\) 10.2399 0.348369
\(865\) 0 0
\(866\) 82.1807 2.79261
\(867\) −10.2588 −0.348408
\(868\) 36.2038 1.22884
\(869\) −63.3120 −2.14771
\(870\) 0 0
\(871\) −10.6620 −0.361269
\(872\) 131.648 4.45815
\(873\) 2.11001 0.0714130
\(874\) 25.4137 0.859630
\(875\) 0 0
\(876\) −37.2426 −1.25831
\(877\) −53.4913 −1.80627 −0.903136 0.429354i \(-0.858741\pi\)
−0.903136 + 0.429354i \(0.858741\pi\)
\(878\) −58.4258 −1.97177
\(879\) −9.59895 −0.323765
\(880\) 0 0
\(881\) 26.1745 0.881840 0.440920 0.897546i \(-0.354652\pi\)
0.440920 + 0.897546i \(0.354652\pi\)
\(882\) −2.61050 −0.0879001
\(883\) −23.1840 −0.780202 −0.390101 0.920772i \(-0.627560\pi\)
−0.390101 + 0.920772i \(0.627560\pi\)
\(884\) −25.1376 −0.845469
\(885\) 0 0
\(886\) 22.2536 0.747624
\(887\) 13.7008 0.460029 0.230015 0.973187i \(-0.426123\pi\)
0.230015 + 0.973187i \(0.426123\pi\)
\(888\) −1.13760 −0.0381752
\(889\) 17.4746 0.586079
\(890\) 0 0
\(891\) −4.73730 −0.158705
\(892\) −96.4223 −3.22846
\(893\) −26.4077 −0.883699
\(894\) −41.0959 −1.37445
\(895\) 0 0
\(896\) −0.568798 −0.0190022
\(897\) −3.33101 −0.111219
\(898\) −48.8534 −1.63026
\(899\) 6.93733 0.231373
\(900\) 0 0
\(901\) −44.6500 −1.48751
\(902\) 78.7354 2.62160
\(903\) 6.55201 0.218037
\(904\) 6.77900 0.225466
\(905\) 0 0
\(906\) 26.8658 0.892556
\(907\) −25.5621 −0.848777 −0.424388 0.905480i \(-0.639511\pi\)
−0.424388 + 0.905480i \(0.639511\pi\)
\(908\) −92.5249 −3.07055
\(909\) 0.850419 0.0282066
\(910\) 0 0
\(911\) 2.07643 0.0687951 0.0343976 0.999408i \(-0.489049\pi\)
0.0343976 + 0.999408i \(0.489049\pi\)
\(912\) −27.9166 −0.924411
\(913\) −6.66202 −0.220481
\(914\) −36.9036 −1.22066
\(915\) 0 0
\(916\) 25.3198 0.836589
\(917\) −0.967402 −0.0319464
\(918\) 13.6294 0.449838
\(919\) 17.8514 0.588863 0.294432 0.955673i \(-0.404870\pi\)
0.294432 + 0.955673i \(0.404870\pi\)
\(920\) 0 0
\(921\) −15.1488 −0.499171
\(922\) 6.98664 0.230093
\(923\) 6.58248 0.216665
\(924\) −22.8087 −0.750352
\(925\) 0 0
\(926\) −6.60881 −0.217179
\(927\) 1.47460 0.0484321
\(928\) 9.44724 0.310121
\(929\) −37.0553 −1.21575 −0.607873 0.794034i \(-0.707977\pi\)
−0.607873 + 0.794034i \(0.707977\pi\)
\(930\) 0 0
\(931\) 2.92259 0.0957840
\(932\) 128.184 4.19881
\(933\) 4.37058 0.143086
\(934\) 5.23212 0.171200
\(935\) 0 0
\(936\) 7.34780 0.240170
\(937\) −39.7540 −1.29871 −0.649354 0.760486i \(-0.724961\pi\)
−0.649354 + 0.760486i \(0.724961\pi\)
\(938\) 27.8332 0.908786
\(939\) −1.49280 −0.0487158
\(940\) 0 0
\(941\) −47.3530 −1.54366 −0.771832 0.635827i \(-0.780659\pi\)
−0.771832 + 0.635827i \(0.780659\pi\)
\(942\) 29.7954 0.970785
\(943\) −21.2076 −0.690614
\(944\) 37.8097 1.23060
\(945\) 0 0
\(946\) 81.0268 2.63441
\(947\) −8.80446 −0.286106 −0.143053 0.989715i \(-0.545692\pi\)
−0.143053 + 0.989715i \(0.545692\pi\)
\(948\) −64.3466 −2.08988
\(949\) −7.73517 −0.251094
\(950\) 0 0
\(951\) 23.2129 0.752729
\(952\) 38.3629 1.24335
\(953\) −7.72895 −0.250365 −0.125183 0.992134i \(-0.539952\pi\)
−0.125183 + 0.992134i \(0.539952\pi\)
\(954\) 22.3250 0.722799
\(955\) 0 0
\(956\) 20.6601 0.668196
\(957\) −4.37058 −0.141281
\(958\) 38.2722 1.23652
\(959\) 3.29628 0.106442
\(960\) 0 0
\(961\) 25.5415 0.823920
\(962\) −0.404161 −0.0130307
\(963\) −2.62418 −0.0845630
\(964\) 36.2243 1.16671
\(965\) 0 0
\(966\) 8.69560 0.279776
\(967\) −14.3054 −0.460030 −0.230015 0.973187i \(-0.573878\pi\)
−0.230015 + 0.973187i \(0.573878\pi\)
\(968\) −84.0735 −2.70222
\(969\) −15.2588 −0.490184
\(970\) 0 0
\(971\) −15.1692 −0.486803 −0.243402 0.969926i \(-0.578263\pi\)
−0.243402 + 0.969926i \(0.578263\pi\)
\(972\) −4.81471 −0.154432
\(973\) −0.370581 −0.0118803
\(974\) 99.7583 3.19646
\(975\) 0 0
\(976\) 118.846 3.80417
\(977\) 8.25746 0.264180 0.132090 0.991238i \(-0.457831\pi\)
0.132090 + 0.991238i \(0.457831\pi\)
\(978\) 35.1754 1.12479
\(979\) 9.33026 0.298196
\(980\) 0 0
\(981\) −17.9166 −0.572033
\(982\) −55.8016 −1.78070
\(983\) 25.8525 0.824568 0.412284 0.911055i \(-0.364731\pi\)
0.412284 + 0.911055i \(0.364731\pi\)
\(984\) 46.7814 1.49134
\(985\) 0 0
\(986\) 12.5744 0.400449
\(987\) −9.03571 −0.287610
\(988\) −14.0714 −0.447671
\(989\) −21.8248 −0.693988
\(990\) 0 0
\(991\) −56.2100 −1.78557 −0.892785 0.450484i \(-0.851252\pi\)
−0.892785 + 0.450484i \(0.851252\pi\)
\(992\) 76.9981 2.44469
\(993\) −1.10402 −0.0350349
\(994\) −17.1836 −0.545029
\(995\) 0 0
\(996\) −6.77088 −0.214544
\(997\) −37.4789 −1.18697 −0.593484 0.804846i \(-0.702248\pi\)
−0.593484 + 0.804846i \(0.702248\pi\)
\(998\) −70.7237 −2.23872
\(999\) 0.154821 0.00489833
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 6825.2.a.bg.1.1 4
5.4 even 2 273.2.a.e.1.4 4
15.14 odd 2 819.2.a.k.1.1 4
20.19 odd 2 4368.2.a.br.1.1 4
35.34 odd 2 1911.2.a.s.1.4 4
65.64 even 2 3549.2.a.w.1.1 4
105.104 even 2 5733.2.a.bf.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.2.a.e.1.4 4 5.4 even 2
819.2.a.k.1.1 4 15.14 odd 2
1911.2.a.s.1.4 4 35.34 odd 2
3549.2.a.w.1.1 4 65.64 even 2
4368.2.a.br.1.1 4 20.19 odd 2
5733.2.a.bf.1.1 4 105.104 even 2
6825.2.a.bg.1.1 4 1.1 even 1 trivial