Properties

Label 4025.2.a.h.1.1
Level $4025$
Weight $2$
Character 4025.1
Self dual yes
Analytic conductor $32.140$
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] = [4025,2,Mod(1,4025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4025, 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("4025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4025 = 5^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4025.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: \(32.1397868136\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 805)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 4025.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.61803 q^{2} +2.23607 q^{3} +4.85410 q^{4} -5.85410 q^{6} +1.00000 q^{7} -7.47214 q^{8} +2.00000 q^{9} +O(q^{10})\) \(q-2.61803 q^{2} +2.23607 q^{3} +4.85410 q^{4} -5.85410 q^{6} +1.00000 q^{7} -7.47214 q^{8} +2.00000 q^{9} +6.00000 q^{11} +10.8541 q^{12} -3.00000 q^{13} -2.61803 q^{14} +9.85410 q^{16} -3.23607 q^{17} -5.23607 q^{18} -0.472136 q^{19} +2.23607 q^{21} -15.7082 q^{22} -1.00000 q^{23} -16.7082 q^{24} +7.85410 q^{26} -2.23607 q^{27} +4.85410 q^{28} -1.47214 q^{29} +3.47214 q^{31} -10.8541 q^{32} +13.4164 q^{33} +8.47214 q^{34} +9.70820 q^{36} +2.76393 q^{37} +1.23607 q^{38} -6.70820 q^{39} -6.70820 q^{41} -5.85410 q^{42} +9.70820 q^{43} +29.1246 q^{44} +2.61803 q^{46} -0.236068 q^{47} +22.0344 q^{48} +1.00000 q^{49} -7.23607 q^{51} -14.5623 q^{52} +13.2361 q^{53} +5.85410 q^{54} -7.47214 q^{56} -1.05573 q^{57} +3.85410 q^{58} +2.47214 q^{59} +11.7082 q^{61} -9.09017 q^{62} +2.00000 q^{63} +8.70820 q^{64} -35.1246 q^{66} +1.23607 q^{67} -15.7082 q^{68} -2.23607 q^{69} +0.236068 q^{71} -14.9443 q^{72} -9.00000 q^{73} -7.23607 q^{74} -2.29180 q^{76} +6.00000 q^{77} +17.5623 q^{78} +13.4164 q^{79} -11.0000 q^{81} +17.5623 q^{82} +9.70820 q^{83} +10.8541 q^{84} -25.4164 q^{86} -3.29180 q^{87} -44.8328 q^{88} -4.76393 q^{89} -3.00000 q^{91} -4.85410 q^{92} +7.76393 q^{93} +0.618034 q^{94} -24.2705 q^{96} +13.7082 q^{97} -2.61803 q^{98} +12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} + 3 q^{4} - 5 q^{6} + 2 q^{7} - 6 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} + 3 q^{4} - 5 q^{6} + 2 q^{7} - 6 q^{8} + 4 q^{9} + 12 q^{11} + 15 q^{12} - 6 q^{13} - 3 q^{14} + 13 q^{16} - 2 q^{17} - 6 q^{18} + 8 q^{19} - 18 q^{22} - 2 q^{23} - 20 q^{24} + 9 q^{26} + 3 q^{28} + 6 q^{29} - 2 q^{31} - 15 q^{32} + 8 q^{34} + 6 q^{36} + 10 q^{37} - 2 q^{38} - 5 q^{42} + 6 q^{43} + 18 q^{44} + 3 q^{46} + 4 q^{47} + 15 q^{48} + 2 q^{49} - 10 q^{51} - 9 q^{52} + 22 q^{53} + 5 q^{54} - 6 q^{56} - 20 q^{57} + q^{58} - 4 q^{59} + 10 q^{61} - 7 q^{62} + 4 q^{63} + 4 q^{64} - 30 q^{66} - 2 q^{67} - 18 q^{68} - 4 q^{71} - 12 q^{72} - 18 q^{73} - 10 q^{74} - 18 q^{76} + 12 q^{77} + 15 q^{78} - 22 q^{81} + 15 q^{82} + 6 q^{83} + 15 q^{84} - 24 q^{86} - 20 q^{87} - 36 q^{88} - 14 q^{89} - 6 q^{91} - 3 q^{92} + 20 q^{93} - q^{94} - 15 q^{96} + 14 q^{97} - 3 q^{98} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.61803 −1.85123 −0.925615 0.378467i \(-0.876451\pi\)
−0.925615 + 0.378467i \(0.876451\pi\)
\(3\) 2.23607 1.29099 0.645497 0.763763i \(-0.276650\pi\)
0.645497 + 0.763763i \(0.276650\pi\)
\(4\) 4.85410 2.42705
\(5\) 0 0
\(6\) −5.85410 −2.38993
\(7\) 1.00000 0.377964
\(8\) −7.47214 −2.64180
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) 6.00000 1.80907 0.904534 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) 10.8541 3.13331
\(13\) −3.00000 −0.832050 −0.416025 0.909353i \(-0.636577\pi\)
−0.416025 + 0.909353i \(0.636577\pi\)
\(14\) −2.61803 −0.699699
\(15\) 0 0
\(16\) 9.85410 2.46353
\(17\) −3.23607 −0.784862 −0.392431 0.919781i \(-0.628366\pi\)
−0.392431 + 0.919781i \(0.628366\pi\)
\(18\) −5.23607 −1.23415
\(19\) −0.472136 −0.108315 −0.0541577 0.998532i \(-0.517247\pi\)
−0.0541577 + 0.998532i \(0.517247\pi\)
\(20\) 0 0
\(21\) 2.23607 0.487950
\(22\) −15.7082 −3.34900
\(23\) −1.00000 −0.208514
\(24\) −16.7082 −3.41055
\(25\) 0 0
\(26\) 7.85410 1.54032
\(27\) −2.23607 −0.430331
\(28\) 4.85410 0.917339
\(29\) −1.47214 −0.273369 −0.136684 0.990615i \(-0.543645\pi\)
−0.136684 + 0.990615i \(0.543645\pi\)
\(30\) 0 0
\(31\) 3.47214 0.623614 0.311807 0.950145i \(-0.399066\pi\)
0.311807 + 0.950145i \(0.399066\pi\)
\(32\) −10.8541 −1.91875
\(33\) 13.4164 2.33550
\(34\) 8.47214 1.45296
\(35\) 0 0
\(36\) 9.70820 1.61803
\(37\) 2.76393 0.454388 0.227194 0.973850i \(-0.427045\pi\)
0.227194 + 0.973850i \(0.427045\pi\)
\(38\) 1.23607 0.200517
\(39\) −6.70820 −1.07417
\(40\) 0 0
\(41\) −6.70820 −1.04765 −0.523823 0.851827i \(-0.675495\pi\)
−0.523823 + 0.851827i \(0.675495\pi\)
\(42\) −5.85410 −0.903308
\(43\) 9.70820 1.48049 0.740244 0.672339i \(-0.234710\pi\)
0.740244 + 0.672339i \(0.234710\pi\)
\(44\) 29.1246 4.39070
\(45\) 0 0
\(46\) 2.61803 0.386008
\(47\) −0.236068 −0.0344341 −0.0172170 0.999852i \(-0.505481\pi\)
−0.0172170 + 0.999852i \(0.505481\pi\)
\(48\) 22.0344 3.18040
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −7.23607 −1.01325
\(52\) −14.5623 −2.01943
\(53\) 13.2361 1.81811 0.909057 0.416672i \(-0.136804\pi\)
0.909057 + 0.416672i \(0.136804\pi\)
\(54\) 5.85410 0.796642
\(55\) 0 0
\(56\) −7.47214 −0.998506
\(57\) −1.05573 −0.139835
\(58\) 3.85410 0.506068
\(59\) 2.47214 0.321845 0.160922 0.986967i \(-0.448553\pi\)
0.160922 + 0.986967i \(0.448553\pi\)
\(60\) 0 0
\(61\) 11.7082 1.49908 0.749541 0.661958i \(-0.230274\pi\)
0.749541 + 0.661958i \(0.230274\pi\)
\(62\) −9.09017 −1.15445
\(63\) 2.00000 0.251976
\(64\) 8.70820 1.08853
\(65\) 0 0
\(66\) −35.1246 −4.32354
\(67\) 1.23607 0.151010 0.0755049 0.997145i \(-0.475943\pi\)
0.0755049 + 0.997145i \(0.475943\pi\)
\(68\) −15.7082 −1.90490
\(69\) −2.23607 −0.269191
\(70\) 0 0
\(71\) 0.236068 0.0280161 0.0140081 0.999902i \(-0.495541\pi\)
0.0140081 + 0.999902i \(0.495541\pi\)
\(72\) −14.9443 −1.76120
\(73\) −9.00000 −1.05337 −0.526685 0.850060i \(-0.676565\pi\)
−0.526685 + 0.850060i \(0.676565\pi\)
\(74\) −7.23607 −0.841176
\(75\) 0 0
\(76\) −2.29180 −0.262887
\(77\) 6.00000 0.683763
\(78\) 17.5623 1.98854
\(79\) 13.4164 1.50946 0.754732 0.656033i \(-0.227767\pi\)
0.754732 + 0.656033i \(0.227767\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 17.5623 1.93943
\(83\) 9.70820 1.06561 0.532807 0.846237i \(-0.321137\pi\)
0.532807 + 0.846237i \(0.321137\pi\)
\(84\) 10.8541 1.18428
\(85\) 0 0
\(86\) −25.4164 −2.74072
\(87\) −3.29180 −0.352918
\(88\) −44.8328 −4.77919
\(89\) −4.76393 −0.504976 −0.252488 0.967600i \(-0.581249\pi\)
−0.252488 + 0.967600i \(0.581249\pi\)
\(90\) 0 0
\(91\) −3.00000 −0.314485
\(92\) −4.85410 −0.506075
\(93\) 7.76393 0.805082
\(94\) 0.618034 0.0637453
\(95\) 0 0
\(96\) −24.2705 −2.47710
\(97\) 13.7082 1.39186 0.695929 0.718111i \(-0.254993\pi\)
0.695929 + 0.718111i \(0.254993\pi\)
\(98\) −2.61803 −0.264461
\(99\) 12.0000 1.20605
\(100\) 0 0
\(101\) −10.9443 −1.08900 −0.544498 0.838762i \(-0.683280\pi\)
−0.544498 + 0.838762i \(0.683280\pi\)
\(102\) 18.9443 1.87576
\(103\) 16.9443 1.66957 0.834784 0.550577i \(-0.185592\pi\)
0.834784 + 0.550577i \(0.185592\pi\)
\(104\) 22.4164 2.19811
\(105\) 0 0
\(106\) −34.6525 −3.36575
\(107\) −19.8885 −1.92270 −0.961349 0.275333i \(-0.911212\pi\)
−0.961349 + 0.275333i \(0.911212\pi\)
\(108\) −10.8541 −1.04444
\(109\) −1.70820 −0.163616 −0.0818081 0.996648i \(-0.526069\pi\)
−0.0818081 + 0.996648i \(0.526069\pi\)
\(110\) 0 0
\(111\) 6.18034 0.586612
\(112\) 9.85410 0.931125
\(113\) 19.4164 1.82654 0.913271 0.407353i \(-0.133548\pi\)
0.913271 + 0.407353i \(0.133548\pi\)
\(114\) 2.76393 0.258866
\(115\) 0 0
\(116\) −7.14590 −0.663480
\(117\) −6.00000 −0.554700
\(118\) −6.47214 −0.595808
\(119\) −3.23607 −0.296650
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) −30.6525 −2.77514
\(123\) −15.0000 −1.35250
\(124\) 16.8541 1.51354
\(125\) 0 0
\(126\) −5.23607 −0.466466
\(127\) −3.00000 −0.266207 −0.133103 0.991102i \(-0.542494\pi\)
−0.133103 + 0.991102i \(0.542494\pi\)
\(128\) −1.09017 −0.0963583
\(129\) 21.7082 1.91130
\(130\) 0 0
\(131\) 19.4721 1.70129 0.850644 0.525742i \(-0.176212\pi\)
0.850644 + 0.525742i \(0.176212\pi\)
\(132\) 65.1246 5.66837
\(133\) −0.472136 −0.0409394
\(134\) −3.23607 −0.279554
\(135\) 0 0
\(136\) 24.1803 2.07345
\(137\) 3.23607 0.276476 0.138238 0.990399i \(-0.455856\pi\)
0.138238 + 0.990399i \(0.455856\pi\)
\(138\) 5.85410 0.498334
\(139\) −2.52786 −0.214411 −0.107205 0.994237i \(-0.534190\pi\)
−0.107205 + 0.994237i \(0.534190\pi\)
\(140\) 0 0
\(141\) −0.527864 −0.0444542
\(142\) −0.618034 −0.0518643
\(143\) −18.0000 −1.50524
\(144\) 19.7082 1.64235
\(145\) 0 0
\(146\) 23.5623 1.95003
\(147\) 2.23607 0.184428
\(148\) 13.4164 1.10282
\(149\) 9.23607 0.756648 0.378324 0.925673i \(-0.376500\pi\)
0.378324 + 0.925673i \(0.376500\pi\)
\(150\) 0 0
\(151\) 17.7639 1.44561 0.722804 0.691053i \(-0.242853\pi\)
0.722804 + 0.691053i \(0.242853\pi\)
\(152\) 3.52786 0.286148
\(153\) −6.47214 −0.523241
\(154\) −15.7082 −1.26580
\(155\) 0 0
\(156\) −32.5623 −2.60707
\(157\) 19.2361 1.53521 0.767603 0.640926i \(-0.221449\pi\)
0.767603 + 0.640926i \(0.221449\pi\)
\(158\) −35.1246 −2.79436
\(159\) 29.5967 2.34717
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 28.7984 2.26261
\(163\) 17.0000 1.33154 0.665771 0.746156i \(-0.268103\pi\)
0.665771 + 0.746156i \(0.268103\pi\)
\(164\) −32.5623 −2.54269
\(165\) 0 0
\(166\) −25.4164 −1.97270
\(167\) 0.944272 0.0730700 0.0365350 0.999332i \(-0.488368\pi\)
0.0365350 + 0.999332i \(0.488368\pi\)
\(168\) −16.7082 −1.28907
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) −0.944272 −0.0722103
\(172\) 47.1246 3.59322
\(173\) −14.9443 −1.13619 −0.568096 0.822962i \(-0.692320\pi\)
−0.568096 + 0.822962i \(0.692320\pi\)
\(174\) 8.61803 0.653331
\(175\) 0 0
\(176\) 59.1246 4.45669
\(177\) 5.52786 0.415500
\(178\) 12.4721 0.934826
\(179\) 5.29180 0.395527 0.197764 0.980250i \(-0.436632\pi\)
0.197764 + 0.980250i \(0.436632\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 7.85410 0.582185
\(183\) 26.1803 1.93531
\(184\) 7.47214 0.550853
\(185\) 0 0
\(186\) −20.3262 −1.49039
\(187\) −19.4164 −1.41987
\(188\) −1.14590 −0.0835732
\(189\) −2.23607 −0.162650
\(190\) 0 0
\(191\) −8.47214 −0.613022 −0.306511 0.951867i \(-0.599162\pi\)
−0.306511 + 0.951867i \(0.599162\pi\)
\(192\) 19.4721 1.40528
\(193\) 6.23607 0.448882 0.224441 0.974488i \(-0.427944\pi\)
0.224441 + 0.974488i \(0.427944\pi\)
\(194\) −35.8885 −2.57665
\(195\) 0 0
\(196\) 4.85410 0.346722
\(197\) −18.7082 −1.33290 −0.666452 0.745548i \(-0.732188\pi\)
−0.666452 + 0.745548i \(0.732188\pi\)
\(198\) −31.4164 −2.23267
\(199\) −25.4164 −1.80172 −0.900861 0.434108i \(-0.857064\pi\)
−0.900861 + 0.434108i \(0.857064\pi\)
\(200\) 0 0
\(201\) 2.76393 0.194953
\(202\) 28.6525 2.01598
\(203\) −1.47214 −0.103324
\(204\) −35.1246 −2.45921
\(205\) 0 0
\(206\) −44.3607 −3.09076
\(207\) −2.00000 −0.139010
\(208\) −29.5623 −2.04978
\(209\) −2.83282 −0.195950
\(210\) 0 0
\(211\) 0.944272 0.0650064 0.0325032 0.999472i \(-0.489652\pi\)
0.0325032 + 0.999472i \(0.489652\pi\)
\(212\) 64.2492 4.41265
\(213\) 0.527864 0.0361686
\(214\) 52.0689 3.55936
\(215\) 0 0
\(216\) 16.7082 1.13685
\(217\) 3.47214 0.235704
\(218\) 4.47214 0.302891
\(219\) −20.1246 −1.35990
\(220\) 0 0
\(221\) 9.70820 0.653044
\(222\) −16.1803 −1.08595
\(223\) −4.94427 −0.331093 −0.165546 0.986202i \(-0.552939\pi\)
−0.165546 + 0.986202i \(0.552939\pi\)
\(224\) −10.8541 −0.725220
\(225\) 0 0
\(226\) −50.8328 −3.38135
\(227\) −7.41641 −0.492244 −0.246122 0.969239i \(-0.579156\pi\)
−0.246122 + 0.969239i \(0.579156\pi\)
\(228\) −5.12461 −0.339386
\(229\) −12.0000 −0.792982 −0.396491 0.918039i \(-0.629772\pi\)
−0.396491 + 0.918039i \(0.629772\pi\)
\(230\) 0 0
\(231\) 13.4164 0.882735
\(232\) 11.0000 0.722185
\(233\) 6.70820 0.439469 0.219735 0.975560i \(-0.429481\pi\)
0.219735 + 0.975560i \(0.429481\pi\)
\(234\) 15.7082 1.02688
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) 30.0000 1.94871
\(238\) 8.47214 0.549167
\(239\) −13.6525 −0.883105 −0.441553 0.897235i \(-0.645572\pi\)
−0.441553 + 0.897235i \(0.645572\pi\)
\(240\) 0 0
\(241\) 11.1246 0.716599 0.358300 0.933607i \(-0.383357\pi\)
0.358300 + 0.933607i \(0.383357\pi\)
\(242\) −65.4508 −4.20734
\(243\) −17.8885 −1.14755
\(244\) 56.8328 3.63835
\(245\) 0 0
\(246\) 39.2705 2.50380
\(247\) 1.41641 0.0901239
\(248\) −25.9443 −1.64746
\(249\) 21.7082 1.37570
\(250\) 0 0
\(251\) 8.94427 0.564557 0.282279 0.959332i \(-0.408910\pi\)
0.282279 + 0.959332i \(0.408910\pi\)
\(252\) 9.70820 0.611559
\(253\) −6.00000 −0.377217
\(254\) 7.85410 0.492810
\(255\) 0 0
\(256\) −14.5623 −0.910144
\(257\) −27.3607 −1.70671 −0.853356 0.521328i \(-0.825437\pi\)
−0.853356 + 0.521328i \(0.825437\pi\)
\(258\) −56.8328 −3.53826
\(259\) 2.76393 0.171742
\(260\) 0 0
\(261\) −2.94427 −0.182246
\(262\) −50.9787 −3.14948
\(263\) −25.5967 −1.57836 −0.789182 0.614160i \(-0.789495\pi\)
−0.789182 + 0.614160i \(0.789495\pi\)
\(264\) −100.249 −6.16991
\(265\) 0 0
\(266\) 1.23607 0.0757882
\(267\) −10.6525 −0.651921
\(268\) 6.00000 0.366508
\(269\) 6.70820 0.409006 0.204503 0.978866i \(-0.434442\pi\)
0.204503 + 0.978866i \(0.434442\pi\)
\(270\) 0 0
\(271\) −26.4721 −1.60807 −0.804034 0.594583i \(-0.797317\pi\)
−0.804034 + 0.594583i \(0.797317\pi\)
\(272\) −31.8885 −1.93353
\(273\) −6.70820 −0.405999
\(274\) −8.47214 −0.511820
\(275\) 0 0
\(276\) −10.8541 −0.653340
\(277\) 33.0689 1.98692 0.993458 0.114195i \(-0.0364289\pi\)
0.993458 + 0.114195i \(0.0364289\pi\)
\(278\) 6.61803 0.396923
\(279\) 6.94427 0.415743
\(280\) 0 0
\(281\) −8.18034 −0.487998 −0.243999 0.969775i \(-0.578459\pi\)
−0.243999 + 0.969775i \(0.578459\pi\)
\(282\) 1.38197 0.0822949
\(283\) −11.1246 −0.661290 −0.330645 0.943755i \(-0.607266\pi\)
−0.330645 + 0.943755i \(0.607266\pi\)
\(284\) 1.14590 0.0679965
\(285\) 0 0
\(286\) 47.1246 2.78654
\(287\) −6.70820 −0.395973
\(288\) −21.7082 −1.27917
\(289\) −6.52786 −0.383992
\(290\) 0 0
\(291\) 30.6525 1.79688
\(292\) −43.6869 −2.55658
\(293\) 32.1803 1.88000 0.939998 0.341181i \(-0.110827\pi\)
0.939998 + 0.341181i \(0.110827\pi\)
\(294\) −5.85410 −0.341418
\(295\) 0 0
\(296\) −20.6525 −1.20040
\(297\) −13.4164 −0.778499
\(298\) −24.1803 −1.40073
\(299\) 3.00000 0.173494
\(300\) 0 0
\(301\) 9.70820 0.559572
\(302\) −46.5066 −2.67615
\(303\) −24.4721 −1.40589
\(304\) −4.65248 −0.266838
\(305\) 0 0
\(306\) 16.9443 0.968640
\(307\) −19.4164 −1.10815 −0.554076 0.832466i \(-0.686928\pi\)
−0.554076 + 0.832466i \(0.686928\pi\)
\(308\) 29.1246 1.65953
\(309\) 37.8885 2.15540
\(310\) 0 0
\(311\) −10.4164 −0.590660 −0.295330 0.955395i \(-0.595430\pi\)
−0.295330 + 0.955395i \(0.595430\pi\)
\(312\) 50.1246 2.83775
\(313\) 6.29180 0.355633 0.177817 0.984064i \(-0.443097\pi\)
0.177817 + 0.984064i \(0.443097\pi\)
\(314\) −50.3607 −2.84202
\(315\) 0 0
\(316\) 65.1246 3.66355
\(317\) 3.52786 0.198145 0.0990723 0.995080i \(-0.468412\pi\)
0.0990723 + 0.995080i \(0.468412\pi\)
\(318\) −77.4853 −4.34516
\(319\) −8.83282 −0.494543
\(320\) 0 0
\(321\) −44.4721 −2.48219
\(322\) 2.61803 0.145897
\(323\) 1.52786 0.0850126
\(324\) −53.3951 −2.96640
\(325\) 0 0
\(326\) −44.5066 −2.46499
\(327\) −3.81966 −0.211228
\(328\) 50.1246 2.76767
\(329\) −0.236068 −0.0130148
\(330\) 0 0
\(331\) 20.1246 1.10615 0.553074 0.833132i \(-0.313455\pi\)
0.553074 + 0.833132i \(0.313455\pi\)
\(332\) 47.1246 2.58630
\(333\) 5.52786 0.302925
\(334\) −2.47214 −0.135269
\(335\) 0 0
\(336\) 22.0344 1.20208
\(337\) 7.41641 0.403997 0.201999 0.979386i \(-0.435256\pi\)
0.201999 + 0.979386i \(0.435256\pi\)
\(338\) 10.4721 0.569609
\(339\) 43.4164 2.35806
\(340\) 0 0
\(341\) 20.8328 1.12816
\(342\) 2.47214 0.133678
\(343\) 1.00000 0.0539949
\(344\) −72.5410 −3.91115
\(345\) 0 0
\(346\) 39.1246 2.10335
\(347\) −30.4721 −1.63583 −0.817915 0.575339i \(-0.804870\pi\)
−0.817915 + 0.575339i \(0.804870\pi\)
\(348\) −15.9787 −0.856549
\(349\) −11.7639 −0.629709 −0.314854 0.949140i \(-0.601956\pi\)
−0.314854 + 0.949140i \(0.601956\pi\)
\(350\) 0 0
\(351\) 6.70820 0.358057
\(352\) −65.1246 −3.47115
\(353\) −12.0557 −0.641662 −0.320831 0.947137i \(-0.603962\pi\)
−0.320831 + 0.947137i \(0.603962\pi\)
\(354\) −14.4721 −0.769185
\(355\) 0 0
\(356\) −23.1246 −1.22560
\(357\) −7.23607 −0.382973
\(358\) −13.8541 −0.732212
\(359\) 2.94427 0.155393 0.0776964 0.996977i \(-0.475244\pi\)
0.0776964 + 0.996977i \(0.475244\pi\)
\(360\) 0 0
\(361\) −18.7771 −0.988268
\(362\) 26.1803 1.37601
\(363\) 55.9017 2.93408
\(364\) −14.5623 −0.763272
\(365\) 0 0
\(366\) −68.5410 −3.58270
\(367\) 4.76393 0.248675 0.124338 0.992240i \(-0.460319\pi\)
0.124338 + 0.992240i \(0.460319\pi\)
\(368\) −9.85410 −0.513681
\(369\) −13.4164 −0.698430
\(370\) 0 0
\(371\) 13.2361 0.687182
\(372\) 37.6869 1.95398
\(373\) 13.2361 0.685338 0.342669 0.939456i \(-0.388669\pi\)
0.342669 + 0.939456i \(0.388669\pi\)
\(374\) 50.8328 2.62850
\(375\) 0 0
\(376\) 1.76393 0.0909678
\(377\) 4.41641 0.227457
\(378\) 5.85410 0.301103
\(379\) 31.5967 1.62302 0.811508 0.584341i \(-0.198647\pi\)
0.811508 + 0.584341i \(0.198647\pi\)
\(380\) 0 0
\(381\) −6.70820 −0.343672
\(382\) 22.1803 1.13484
\(383\) −2.47214 −0.126320 −0.0631601 0.998003i \(-0.520118\pi\)
−0.0631601 + 0.998003i \(0.520118\pi\)
\(384\) −2.43769 −0.124398
\(385\) 0 0
\(386\) −16.3262 −0.830984
\(387\) 19.4164 0.986991
\(388\) 66.5410 3.37811
\(389\) 25.8885 1.31260 0.656301 0.754499i \(-0.272120\pi\)
0.656301 + 0.754499i \(0.272120\pi\)
\(390\) 0 0
\(391\) 3.23607 0.163655
\(392\) −7.47214 −0.377400
\(393\) 43.5410 2.19635
\(394\) 48.9787 2.46751
\(395\) 0 0
\(396\) 58.2492 2.92713
\(397\) 0.416408 0.0208989 0.0104495 0.999945i \(-0.496674\pi\)
0.0104495 + 0.999945i \(0.496674\pi\)
\(398\) 66.5410 3.33540
\(399\) −1.05573 −0.0528525
\(400\) 0 0
\(401\) −13.5967 −0.678989 −0.339495 0.940608i \(-0.610256\pi\)
−0.339495 + 0.940608i \(0.610256\pi\)
\(402\) −7.23607 −0.360902
\(403\) −10.4164 −0.518878
\(404\) −53.1246 −2.64305
\(405\) 0 0
\(406\) 3.85410 0.191276
\(407\) 16.5836 0.822018
\(408\) 54.0689 2.67681
\(409\) 37.1803 1.83845 0.919225 0.393733i \(-0.128817\pi\)
0.919225 + 0.393733i \(0.128817\pi\)
\(410\) 0 0
\(411\) 7.23607 0.356929
\(412\) 82.2492 4.05213
\(413\) 2.47214 0.121646
\(414\) 5.23607 0.257339
\(415\) 0 0
\(416\) 32.5623 1.59650
\(417\) −5.65248 −0.276803
\(418\) 7.41641 0.362748
\(419\) −14.6525 −0.715820 −0.357910 0.933756i \(-0.616511\pi\)
−0.357910 + 0.933756i \(0.616511\pi\)
\(420\) 0 0
\(421\) 32.0000 1.55958 0.779792 0.626038i \(-0.215325\pi\)
0.779792 + 0.626038i \(0.215325\pi\)
\(422\) −2.47214 −0.120342
\(423\) −0.472136 −0.0229560
\(424\) −98.9017 −4.80309
\(425\) 0 0
\(426\) −1.38197 −0.0669565
\(427\) 11.7082 0.566600
\(428\) −96.5410 −4.66649
\(429\) −40.2492 −1.94325
\(430\) 0 0
\(431\) 3.05573 0.147189 0.0735946 0.997288i \(-0.476553\pi\)
0.0735946 + 0.997288i \(0.476553\pi\)
\(432\) −22.0344 −1.06013
\(433\) −4.29180 −0.206251 −0.103125 0.994668i \(-0.532884\pi\)
−0.103125 + 0.994668i \(0.532884\pi\)
\(434\) −9.09017 −0.436342
\(435\) 0 0
\(436\) −8.29180 −0.397105
\(437\) 0.472136 0.0225853
\(438\) 52.6869 2.51748
\(439\) −9.00000 −0.429547 −0.214773 0.976664i \(-0.568901\pi\)
−0.214773 + 0.976664i \(0.568901\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) −25.4164 −1.20894
\(443\) 7.36068 0.349716 0.174858 0.984594i \(-0.444053\pi\)
0.174858 + 0.984594i \(0.444053\pi\)
\(444\) 30.0000 1.42374
\(445\) 0 0
\(446\) 12.9443 0.612929
\(447\) 20.6525 0.976829
\(448\) 8.70820 0.411424
\(449\) −25.4164 −1.19947 −0.599737 0.800197i \(-0.704728\pi\)
−0.599737 + 0.800197i \(0.704728\pi\)
\(450\) 0 0
\(451\) −40.2492 −1.89526
\(452\) 94.2492 4.43311
\(453\) 39.7214 1.86627
\(454\) 19.4164 0.911257
\(455\) 0 0
\(456\) 7.88854 0.369415
\(457\) 28.7639 1.34552 0.672760 0.739861i \(-0.265109\pi\)
0.672760 + 0.739861i \(0.265109\pi\)
\(458\) 31.4164 1.46799
\(459\) 7.23607 0.337751
\(460\) 0 0
\(461\) 3.65248 0.170113 0.0850564 0.996376i \(-0.472893\pi\)
0.0850564 + 0.996376i \(0.472893\pi\)
\(462\) −35.1246 −1.63414
\(463\) −34.8328 −1.61882 −0.809409 0.587245i \(-0.800212\pi\)
−0.809409 + 0.587245i \(0.800212\pi\)
\(464\) −14.5066 −0.673451
\(465\) 0 0
\(466\) −17.5623 −0.813558
\(467\) −9.05573 −0.419049 −0.209525 0.977803i \(-0.567192\pi\)
−0.209525 + 0.977803i \(0.567192\pi\)
\(468\) −29.1246 −1.34629
\(469\) 1.23607 0.0570763
\(470\) 0 0
\(471\) 43.0132 1.98194
\(472\) −18.4721 −0.850249
\(473\) 58.2492 2.67830
\(474\) −78.5410 −3.60751
\(475\) 0 0
\(476\) −15.7082 −0.719984
\(477\) 26.4721 1.21208
\(478\) 35.7426 1.63483
\(479\) −26.8328 −1.22602 −0.613011 0.790074i \(-0.710042\pi\)
−0.613011 + 0.790074i \(0.710042\pi\)
\(480\) 0 0
\(481\) −8.29180 −0.378073
\(482\) −29.1246 −1.32659
\(483\) −2.23607 −0.101745
\(484\) 121.353 5.51602
\(485\) 0 0
\(486\) 46.8328 2.12438
\(487\) 40.4164 1.83144 0.915721 0.401814i \(-0.131620\pi\)
0.915721 + 0.401814i \(0.131620\pi\)
\(488\) −87.4853 −3.96027
\(489\) 38.0132 1.71901
\(490\) 0 0
\(491\) −9.18034 −0.414303 −0.207151 0.978309i \(-0.566419\pi\)
−0.207151 + 0.978309i \(0.566419\pi\)
\(492\) −72.8115 −3.28260
\(493\) 4.76393 0.214557
\(494\) −3.70820 −0.166840
\(495\) 0 0
\(496\) 34.2148 1.53629
\(497\) 0.236068 0.0105891
\(498\) −56.8328 −2.54674
\(499\) −8.70820 −0.389833 −0.194916 0.980820i \(-0.562444\pi\)
−0.194916 + 0.980820i \(0.562444\pi\)
\(500\) 0 0
\(501\) 2.11146 0.0943329
\(502\) −23.4164 −1.04513
\(503\) −26.7639 −1.19334 −0.596672 0.802485i \(-0.703511\pi\)
−0.596672 + 0.802485i \(0.703511\pi\)
\(504\) −14.9443 −0.665671
\(505\) 0 0
\(506\) 15.7082 0.698315
\(507\) −8.94427 −0.397229
\(508\) −14.5623 −0.646098
\(509\) 40.0132 1.77355 0.886776 0.462200i \(-0.152940\pi\)
0.886776 + 0.462200i \(0.152940\pi\)
\(510\) 0 0
\(511\) −9.00000 −0.398137
\(512\) 40.3050 1.78124
\(513\) 1.05573 0.0466115
\(514\) 71.6312 3.15952
\(515\) 0 0
\(516\) 105.374 4.63882
\(517\) −1.41641 −0.0622935
\(518\) −7.23607 −0.317935
\(519\) −33.4164 −1.46682
\(520\) 0 0
\(521\) 5.88854 0.257982 0.128991 0.991646i \(-0.458826\pi\)
0.128991 + 0.991646i \(0.458826\pi\)
\(522\) 7.70820 0.337379
\(523\) −32.5410 −1.42292 −0.711460 0.702727i \(-0.751966\pi\)
−0.711460 + 0.702727i \(0.751966\pi\)
\(524\) 94.5197 4.12911
\(525\) 0 0
\(526\) 67.0132 2.92191
\(527\) −11.2361 −0.489451
\(528\) 132.207 5.75356
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 4.94427 0.214563
\(532\) −2.29180 −0.0993620
\(533\) 20.1246 0.871694
\(534\) 27.8885 1.20686
\(535\) 0 0
\(536\) −9.23607 −0.398937
\(537\) 11.8328 0.510624
\(538\) −17.5623 −0.757165
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) 13.0000 0.558914 0.279457 0.960158i \(-0.409846\pi\)
0.279457 + 0.960158i \(0.409846\pi\)
\(542\) 69.3050 2.97690
\(543\) −22.3607 −0.959589
\(544\) 35.1246 1.50596
\(545\) 0 0
\(546\) 17.5623 0.751597
\(547\) −33.4721 −1.43117 −0.715583 0.698528i \(-0.753839\pi\)
−0.715583 + 0.698528i \(0.753839\pi\)
\(548\) 15.7082 0.671021
\(549\) 23.4164 0.999388
\(550\) 0 0
\(551\) 0.695048 0.0296101
\(552\) 16.7082 0.711148
\(553\) 13.4164 0.570524
\(554\) −86.5755 −3.67824
\(555\) 0 0
\(556\) −12.2705 −0.520386
\(557\) 12.0000 0.508456 0.254228 0.967144i \(-0.418179\pi\)
0.254228 + 0.967144i \(0.418179\pi\)
\(558\) −18.1803 −0.769635
\(559\) −29.1246 −1.23184
\(560\) 0 0
\(561\) −43.4164 −1.83304
\(562\) 21.4164 0.903397
\(563\) −32.1803 −1.35624 −0.678120 0.734951i \(-0.737205\pi\)
−0.678120 + 0.734951i \(0.737205\pi\)
\(564\) −2.56231 −0.107893
\(565\) 0 0
\(566\) 29.1246 1.22420
\(567\) −11.0000 −0.461957
\(568\) −1.76393 −0.0740129
\(569\) −27.5279 −1.15403 −0.577014 0.816734i \(-0.695782\pi\)
−0.577014 + 0.816734i \(0.695782\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) −87.3738 −3.65328
\(573\) −18.9443 −0.791408
\(574\) 17.5623 0.733036
\(575\) 0 0
\(576\) 17.4164 0.725684
\(577\) 5.11146 0.212793 0.106396 0.994324i \(-0.466069\pi\)
0.106396 + 0.994324i \(0.466069\pi\)
\(578\) 17.0902 0.710857
\(579\) 13.9443 0.579504
\(580\) 0 0
\(581\) 9.70820 0.402764
\(582\) −80.2492 −3.32644
\(583\) 79.4164 3.28909
\(584\) 67.2492 2.78279
\(585\) 0 0
\(586\) −84.2492 −3.48030
\(587\) 37.0689 1.53000 0.764998 0.644032i \(-0.222740\pi\)
0.764998 + 0.644032i \(0.222740\pi\)
\(588\) 10.8541 0.447616
\(589\) −1.63932 −0.0675470
\(590\) 0 0
\(591\) −41.8328 −1.72077
\(592\) 27.2361 1.11940
\(593\) −41.7771 −1.71558 −0.857790 0.514001i \(-0.828163\pi\)
−0.857790 + 0.514001i \(0.828163\pi\)
\(594\) 35.1246 1.44118
\(595\) 0 0
\(596\) 44.8328 1.83642
\(597\) −56.8328 −2.32601
\(598\) −7.85410 −0.321178
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −34.5967 −1.41123 −0.705615 0.708595i \(-0.749329\pi\)
−0.705615 + 0.708595i \(0.749329\pi\)
\(602\) −25.4164 −1.03590
\(603\) 2.47214 0.100673
\(604\) 86.2279 3.50857
\(605\) 0 0
\(606\) 64.0689 2.60262
\(607\) 23.4164 0.950443 0.475221 0.879866i \(-0.342368\pi\)
0.475221 + 0.879866i \(0.342368\pi\)
\(608\) 5.12461 0.207830
\(609\) −3.29180 −0.133390
\(610\) 0 0
\(611\) 0.708204 0.0286509
\(612\) −31.4164 −1.26993
\(613\) 33.5279 1.35418 0.677089 0.735901i \(-0.263241\pi\)
0.677089 + 0.735901i \(0.263241\pi\)
\(614\) 50.8328 2.05145
\(615\) 0 0
\(616\) −44.8328 −1.80637
\(617\) −16.4721 −0.663143 −0.331572 0.943430i \(-0.607579\pi\)
−0.331572 + 0.943430i \(0.607579\pi\)
\(618\) −99.1935 −3.99015
\(619\) −22.6525 −0.910480 −0.455240 0.890369i \(-0.650447\pi\)
−0.455240 + 0.890369i \(0.650447\pi\)
\(620\) 0 0
\(621\) 2.23607 0.0897303
\(622\) 27.2705 1.09345
\(623\) −4.76393 −0.190863
\(624\) −66.1033 −2.64625
\(625\) 0 0
\(626\) −16.4721 −0.658359
\(627\) −6.33437 −0.252970
\(628\) 93.3738 3.72602
\(629\) −8.94427 −0.356631
\(630\) 0 0
\(631\) −20.7639 −0.826599 −0.413300 0.910595i \(-0.635624\pi\)
−0.413300 + 0.910595i \(0.635624\pi\)
\(632\) −100.249 −3.98770
\(633\) 2.11146 0.0839228
\(634\) −9.23607 −0.366811
\(635\) 0 0
\(636\) 143.666 5.69671
\(637\) −3.00000 −0.118864
\(638\) 23.1246 0.915512
\(639\) 0.472136 0.0186774
\(640\) 0 0
\(641\) 41.1246 1.62432 0.812162 0.583432i \(-0.198290\pi\)
0.812162 + 0.583432i \(0.198290\pi\)
\(642\) 116.430 4.59511
\(643\) 43.8885 1.73080 0.865398 0.501086i \(-0.167066\pi\)
0.865398 + 0.501086i \(0.167066\pi\)
\(644\) −4.85410 −0.191278
\(645\) 0 0
\(646\) −4.00000 −0.157378
\(647\) 27.1803 1.06857 0.534285 0.845305i \(-0.320581\pi\)
0.534285 + 0.845305i \(0.320581\pi\)
\(648\) 82.1935 3.22887
\(649\) 14.8328 0.582239
\(650\) 0 0
\(651\) 7.76393 0.304292
\(652\) 82.5197 3.23172
\(653\) −6.59675 −0.258151 −0.129075 0.991635i \(-0.541201\pi\)
−0.129075 + 0.991635i \(0.541201\pi\)
\(654\) 10.0000 0.391031
\(655\) 0 0
\(656\) −66.1033 −2.58090
\(657\) −18.0000 −0.702247
\(658\) 0.618034 0.0240935
\(659\) 15.0557 0.586488 0.293244 0.956038i \(-0.405265\pi\)
0.293244 + 0.956038i \(0.405265\pi\)
\(660\) 0 0
\(661\) −12.2918 −0.478095 −0.239048 0.971008i \(-0.576835\pi\)
−0.239048 + 0.971008i \(0.576835\pi\)
\(662\) −52.6869 −2.04774
\(663\) 21.7082 0.843077
\(664\) −72.5410 −2.81514
\(665\) 0 0
\(666\) −14.4721 −0.560784
\(667\) 1.47214 0.0570013
\(668\) 4.58359 0.177345
\(669\) −11.0557 −0.427439
\(670\) 0 0
\(671\) 70.2492 2.71194
\(672\) −24.2705 −0.936255
\(673\) −40.7082 −1.56919 −0.784593 0.620011i \(-0.787128\pi\)
−0.784593 + 0.620011i \(0.787128\pi\)
\(674\) −19.4164 −0.747892
\(675\) 0 0
\(676\) −19.4164 −0.746785
\(677\) 13.8197 0.531133 0.265566 0.964093i \(-0.414441\pi\)
0.265566 + 0.964093i \(0.414441\pi\)
\(678\) −113.666 −4.36530
\(679\) 13.7082 0.526073
\(680\) 0 0
\(681\) −16.5836 −0.635485
\(682\) −54.5410 −2.08848
\(683\) 1.36068 0.0520650 0.0260325 0.999661i \(-0.491713\pi\)
0.0260325 + 0.999661i \(0.491713\pi\)
\(684\) −4.58359 −0.175258
\(685\) 0 0
\(686\) −2.61803 −0.0999570
\(687\) −26.8328 −1.02374
\(688\) 95.6656 3.64722
\(689\) −39.7082 −1.51276
\(690\) 0 0
\(691\) −6.83282 −0.259933 −0.129966 0.991518i \(-0.541487\pi\)
−0.129966 + 0.991518i \(0.541487\pi\)
\(692\) −72.5410 −2.75760
\(693\) 12.0000 0.455842
\(694\) 79.7771 3.02830
\(695\) 0 0
\(696\) 24.5967 0.932337
\(697\) 21.7082 0.822257
\(698\) 30.7984 1.16574
\(699\) 15.0000 0.567352
\(700\) 0 0
\(701\) 9.70820 0.366674 0.183337 0.983050i \(-0.441310\pi\)
0.183337 + 0.983050i \(0.441310\pi\)
\(702\) −17.5623 −0.662847
\(703\) −1.30495 −0.0492172
\(704\) 52.2492 1.96922
\(705\) 0 0
\(706\) 31.5623 1.18786
\(707\) −10.9443 −0.411602
\(708\) 26.8328 1.00844
\(709\) −16.9443 −0.636355 −0.318178 0.948031i \(-0.603071\pi\)
−0.318178 + 0.948031i \(0.603071\pi\)
\(710\) 0 0
\(711\) 26.8328 1.00631
\(712\) 35.5967 1.33404
\(713\) −3.47214 −0.130033
\(714\) 18.9443 0.708972
\(715\) 0 0
\(716\) 25.6869 0.959965
\(717\) −30.5279 −1.14008
\(718\) −7.70820 −0.287668
\(719\) 28.9443 1.07944 0.539720 0.841845i \(-0.318530\pi\)
0.539720 + 0.841845i \(0.318530\pi\)
\(720\) 0 0
\(721\) 16.9443 0.631038
\(722\) 49.1591 1.82951
\(723\) 24.8754 0.925126
\(724\) −48.5410 −1.80401
\(725\) 0 0
\(726\) −146.353 −5.43165
\(727\) 18.0000 0.667583 0.333792 0.942647i \(-0.391672\pi\)
0.333792 + 0.942647i \(0.391672\pi\)
\(728\) 22.4164 0.830807
\(729\) −7.00000 −0.259259
\(730\) 0 0
\(731\) −31.4164 −1.16198
\(732\) 127.082 4.69709
\(733\) 9.70820 0.358581 0.179290 0.983796i \(-0.442620\pi\)
0.179290 + 0.983796i \(0.442620\pi\)
\(734\) −12.4721 −0.460355
\(735\) 0 0
\(736\) 10.8541 0.400088
\(737\) 7.41641 0.273187
\(738\) 35.1246 1.29295
\(739\) 20.1246 0.740296 0.370148 0.928973i \(-0.379307\pi\)
0.370148 + 0.928973i \(0.379307\pi\)
\(740\) 0 0
\(741\) 3.16718 0.116349
\(742\) −34.6525 −1.27213
\(743\) 38.0689 1.39661 0.698306 0.715799i \(-0.253938\pi\)
0.698306 + 0.715799i \(0.253938\pi\)
\(744\) −58.0132 −2.12687
\(745\) 0 0
\(746\) −34.6525 −1.26872
\(747\) 19.4164 0.710409
\(748\) −94.2492 −3.44609
\(749\) −19.8885 −0.726712
\(750\) 0 0
\(751\) 6.87539 0.250886 0.125443 0.992101i \(-0.459965\pi\)
0.125443 + 0.992101i \(0.459965\pi\)
\(752\) −2.32624 −0.0848292
\(753\) 20.0000 0.728841
\(754\) −11.5623 −0.421074
\(755\) 0 0
\(756\) −10.8541 −0.394760
\(757\) 27.3050 0.992415 0.496208 0.868204i \(-0.334725\pi\)
0.496208 + 0.868204i \(0.334725\pi\)
\(758\) −82.7214 −3.00458
\(759\) −13.4164 −0.486985
\(760\) 0 0
\(761\) −2.23607 −0.0810574 −0.0405287 0.999178i \(-0.512904\pi\)
−0.0405287 + 0.999178i \(0.512904\pi\)
\(762\) 17.5623 0.636215
\(763\) −1.70820 −0.0618411
\(764\) −41.1246 −1.48784
\(765\) 0 0
\(766\) 6.47214 0.233848
\(767\) −7.41641 −0.267791
\(768\) −32.5623 −1.17499
\(769\) 16.8328 0.607007 0.303503 0.952830i \(-0.401844\pi\)
0.303503 + 0.952830i \(0.401844\pi\)
\(770\) 0 0
\(771\) −61.1803 −2.20336
\(772\) 30.2705 1.08946
\(773\) 2.83282 0.101889 0.0509446 0.998701i \(-0.483777\pi\)
0.0509446 + 0.998701i \(0.483777\pi\)
\(774\) −50.8328 −1.82715
\(775\) 0 0
\(776\) −102.430 −3.67701
\(777\) 6.18034 0.221718
\(778\) −67.7771 −2.42993
\(779\) 3.16718 0.113476
\(780\) 0 0
\(781\) 1.41641 0.0506831
\(782\) −8.47214 −0.302963
\(783\) 3.29180 0.117639
\(784\) 9.85410 0.351932
\(785\) 0 0
\(786\) −113.992 −4.06596
\(787\) −3.63932 −0.129728 −0.0648639 0.997894i \(-0.520661\pi\)
−0.0648639 + 0.997894i \(0.520661\pi\)
\(788\) −90.8115 −3.23503
\(789\) −57.2361 −2.03766
\(790\) 0 0
\(791\) 19.4164 0.690368
\(792\) −89.6656 −3.18613
\(793\) −35.1246 −1.24731
\(794\) −1.09017 −0.0386887
\(795\) 0 0
\(796\) −123.374 −4.37287
\(797\) −10.9443 −0.387666 −0.193833 0.981035i \(-0.562092\pi\)
−0.193833 + 0.981035i \(0.562092\pi\)
\(798\) 2.76393 0.0978421
\(799\) 0.763932 0.0270260
\(800\) 0 0
\(801\) −9.52786 −0.336651
\(802\) 35.5967 1.25696
\(803\) −54.0000 −1.90562
\(804\) 13.4164 0.473160
\(805\) 0 0
\(806\) 27.2705 0.960563
\(807\) 15.0000 0.528025
\(808\) 81.7771 2.87691
\(809\) −0.111456 −0.00391859 −0.00195930 0.999998i \(-0.500624\pi\)
−0.00195930 + 0.999998i \(0.500624\pi\)
\(810\) 0 0
\(811\) 9.00000 0.316033 0.158016 0.987436i \(-0.449490\pi\)
0.158016 + 0.987436i \(0.449490\pi\)
\(812\) −7.14590 −0.250772
\(813\) −59.1935 −2.07601
\(814\) −43.4164 −1.52174
\(815\) 0 0
\(816\) −71.3050 −2.49617
\(817\) −4.58359 −0.160360
\(818\) −97.3394 −3.40339
\(819\) −6.00000 −0.209657
\(820\) 0 0
\(821\) −10.3607 −0.361590 −0.180795 0.983521i \(-0.557867\pi\)
−0.180795 + 0.983521i \(0.557867\pi\)
\(822\) −18.9443 −0.660757
\(823\) −19.8328 −0.691328 −0.345664 0.938358i \(-0.612346\pi\)
−0.345664 + 0.938358i \(0.612346\pi\)
\(824\) −126.610 −4.41066
\(825\) 0 0
\(826\) −6.47214 −0.225194
\(827\) −29.3050 −1.01903 −0.509517 0.860461i \(-0.670176\pi\)
−0.509517 + 0.860461i \(0.670176\pi\)
\(828\) −9.70820 −0.337383
\(829\) −30.0000 −1.04194 −0.520972 0.853574i \(-0.674430\pi\)
−0.520972 + 0.853574i \(0.674430\pi\)
\(830\) 0 0
\(831\) 73.9443 2.56510
\(832\) −26.1246 −0.905708
\(833\) −3.23607 −0.112123
\(834\) 14.7984 0.512426
\(835\) 0 0
\(836\) −13.7508 −0.475581
\(837\) −7.76393 −0.268361
\(838\) 38.3607 1.32515
\(839\) 6.87539 0.237365 0.118682 0.992932i \(-0.462133\pi\)
0.118682 + 0.992932i \(0.462133\pi\)
\(840\) 0 0
\(841\) −26.8328 −0.925270
\(842\) −83.7771 −2.88715
\(843\) −18.2918 −0.630003
\(844\) 4.58359 0.157774
\(845\) 0 0
\(846\) 1.23607 0.0424969
\(847\) 25.0000 0.859010
\(848\) 130.430 4.47897
\(849\) −24.8754 −0.853721
\(850\) 0 0
\(851\) −2.76393 −0.0947464
\(852\) 2.56231 0.0877832
\(853\) −23.3050 −0.797946 −0.398973 0.916963i \(-0.630633\pi\)
−0.398973 + 0.916963i \(0.630633\pi\)
\(854\) −30.6525 −1.04891
\(855\) 0 0
\(856\) 148.610 5.07938
\(857\) 17.9443 0.612965 0.306482 0.951876i \(-0.400848\pi\)
0.306482 + 0.951876i \(0.400848\pi\)
\(858\) 105.374 3.59740
\(859\) −29.4721 −1.00558 −0.502788 0.864410i \(-0.667692\pi\)
−0.502788 + 0.864410i \(0.667692\pi\)
\(860\) 0 0
\(861\) −15.0000 −0.511199
\(862\) −8.00000 −0.272481
\(863\) 17.3607 0.590964 0.295482 0.955348i \(-0.404520\pi\)
0.295482 + 0.955348i \(0.404520\pi\)
\(864\) 24.2705 0.825700
\(865\) 0 0
\(866\) 11.2361 0.381817
\(867\) −14.5967 −0.495732
\(868\) 16.8541 0.572065
\(869\) 80.4984 2.73072
\(870\) 0 0
\(871\) −3.70820 −0.125648
\(872\) 12.7639 0.432241
\(873\) 27.4164 0.927905
\(874\) −1.23607 −0.0418106
\(875\) 0 0
\(876\) −97.6869 −3.30054
\(877\) −33.4164 −1.12839 −0.564196 0.825641i \(-0.690814\pi\)
−0.564196 + 0.825641i \(0.690814\pi\)
\(878\) 23.5623 0.795189
\(879\) 71.9574 2.42706
\(880\) 0 0
\(881\) −38.8328 −1.30831 −0.654155 0.756360i \(-0.726976\pi\)
−0.654155 + 0.756360i \(0.726976\pi\)
\(882\) −5.23607 −0.176308
\(883\) −2.11146 −0.0710562 −0.0355281 0.999369i \(-0.511311\pi\)
−0.0355281 + 0.999369i \(0.511311\pi\)
\(884\) 47.1246 1.58497
\(885\) 0 0
\(886\) −19.2705 −0.647405
\(887\) 34.0132 1.14205 0.571025 0.820933i \(-0.306546\pi\)
0.571025 + 0.820933i \(0.306546\pi\)
\(888\) −46.1803 −1.54971
\(889\) −3.00000 −0.100617
\(890\) 0 0
\(891\) −66.0000 −2.21108
\(892\) −24.0000 −0.803579
\(893\) 0.111456 0.00372974
\(894\) −54.0689 −1.80833
\(895\) 0 0
\(896\) −1.09017 −0.0364200
\(897\) 6.70820 0.223980
\(898\) 66.5410 2.22050
\(899\) −5.11146 −0.170477
\(900\) 0 0
\(901\) −42.8328 −1.42697
\(902\) 105.374 3.50856
\(903\) 21.7082 0.722404
\(904\) −145.082 −4.82536
\(905\) 0 0
\(906\) −103.992 −3.45490
\(907\) −36.2492 −1.20364 −0.601818 0.798633i \(-0.705557\pi\)
−0.601818 + 0.798633i \(0.705557\pi\)
\(908\) −36.0000 −1.19470
\(909\) −21.8885 −0.725997
\(910\) 0 0
\(911\) 15.7082 0.520436 0.260218 0.965550i \(-0.416206\pi\)
0.260218 + 0.965550i \(0.416206\pi\)
\(912\) −10.4033 −0.344486
\(913\) 58.2492 1.92777
\(914\) −75.3050 −2.49087
\(915\) 0 0
\(916\) −58.2492 −1.92461
\(917\) 19.4721 0.643027
\(918\) −18.9443 −0.625254
\(919\) −5.12461 −0.169045 −0.0845227 0.996422i \(-0.526937\pi\)
−0.0845227 + 0.996422i \(0.526937\pi\)
\(920\) 0 0
\(921\) −43.4164 −1.43062
\(922\) −9.56231 −0.314918
\(923\) −0.708204 −0.0233108
\(924\) 65.1246 2.14244
\(925\) 0 0
\(926\) 91.1935 2.99680
\(927\) 33.8885 1.11305
\(928\) 15.9787 0.524527
\(929\) 46.9574 1.54062 0.770312 0.637668i \(-0.220101\pi\)
0.770312 + 0.637668i \(0.220101\pi\)
\(930\) 0 0
\(931\) −0.472136 −0.0154736
\(932\) 32.5623 1.06661
\(933\) −23.2918 −0.762539
\(934\) 23.7082 0.775756
\(935\) 0 0
\(936\) 44.8328 1.46541
\(937\) −28.6525 −0.936036 −0.468018 0.883719i \(-0.655032\pi\)
−0.468018 + 0.883719i \(0.655032\pi\)
\(938\) −3.23607 −0.105661
\(939\) 14.0689 0.459121
\(940\) 0 0
\(941\) 41.3050 1.34650 0.673251 0.739414i \(-0.264897\pi\)
0.673251 + 0.739414i \(0.264897\pi\)
\(942\) −112.610 −3.66903
\(943\) 6.70820 0.218449
\(944\) 24.3607 0.792873
\(945\) 0 0
\(946\) −152.498 −4.95815
\(947\) −4.41641 −0.143514 −0.0717570 0.997422i \(-0.522861\pi\)
−0.0717570 + 0.997422i \(0.522861\pi\)
\(948\) 145.623 4.72962
\(949\) 27.0000 0.876457
\(950\) 0 0
\(951\) 7.88854 0.255804
\(952\) 24.1803 0.783689
\(953\) 14.9443 0.484092 0.242046 0.970265i \(-0.422181\pi\)
0.242046 + 0.970265i \(0.422181\pi\)
\(954\) −69.3050 −2.24383
\(955\) 0 0
\(956\) −66.2705 −2.14334
\(957\) −19.7508 −0.638452
\(958\) 70.2492 2.26965
\(959\) 3.23607 0.104498
\(960\) 0 0
\(961\) −18.9443 −0.611106
\(962\) 21.7082 0.699901
\(963\) −39.7771 −1.28180
\(964\) 54.0000 1.73922
\(965\) 0 0
\(966\) 5.85410 0.188353
\(967\) −17.1115 −0.550267 −0.275134 0.961406i \(-0.588722\pi\)
−0.275134 + 0.961406i \(0.588722\pi\)
\(968\) −186.803 −6.00409
\(969\) 3.41641 0.109751
\(970\) 0 0
\(971\) 2.18034 0.0699704 0.0349852 0.999388i \(-0.488862\pi\)
0.0349852 + 0.999388i \(0.488862\pi\)
\(972\) −86.8328 −2.78516
\(973\) −2.52786 −0.0810396
\(974\) −105.812 −3.39042
\(975\) 0 0
\(976\) 115.374 3.69303
\(977\) −45.2361 −1.44723 −0.723615 0.690204i \(-0.757521\pi\)
−0.723615 + 0.690204i \(0.757521\pi\)
\(978\) −99.5197 −3.18229
\(979\) −28.5836 −0.913536
\(980\) 0 0
\(981\) −3.41641 −0.109078
\(982\) 24.0344 0.766970
\(983\) −25.0132 −0.797796 −0.398898 0.916995i \(-0.630607\pi\)
−0.398898 + 0.916995i \(0.630607\pi\)
\(984\) 112.082 3.57304
\(985\) 0 0
\(986\) −12.4721 −0.397194
\(987\) −0.527864 −0.0168021
\(988\) 6.87539 0.218735
\(989\) −9.70820 −0.308703
\(990\) 0 0
\(991\) 4.00000 0.127064 0.0635321 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) −37.6869 −1.19656
\(993\) 45.0000 1.42803
\(994\) −0.618034 −0.0196028
\(995\) 0 0
\(996\) 105.374 3.33890
\(997\) −48.8328 −1.54655 −0.773275 0.634070i \(-0.781383\pi\)
−0.773275 + 0.634070i \(0.781383\pi\)
\(998\) 22.7984 0.721670
\(999\) −6.18034 −0.195537
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 4025.2.a.h.1.1 2
5.4 even 2 805.2.a.e.1.2 2
15.14 odd 2 7245.2.a.v.1.1 2
35.34 odd 2 5635.2.a.q.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
805.2.a.e.1.2 2 5.4 even 2
4025.2.a.h.1.1 2 1.1 even 1 trivial
5635.2.a.q.1.2 2 35.34 odd 2
7245.2.a.v.1.1 2 15.14 odd 2