Properties

Label 1875.2.a.e.1.2
Level $1875$
Weight $2$
Character 1875.1
Self dual yes
Analytic conductor $14.972$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1875,2,Mod(1,1875)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1875, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1875.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1875 = 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1875.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: \(14.9719503790\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.5125.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 6x^{2} + 7x + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 75)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.12233\) of defining polynomial
Character \(\chi\) \(=\) 1875.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.12233 q^{2} -1.00000 q^{3} +2.50430 q^{4} +2.12233 q^{6} -4.35840 q^{7} -1.07029 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.12233 q^{2} -1.00000 q^{3} +2.50430 q^{4} +2.12233 q^{6} -4.35840 q^{7} -1.07029 q^{8} +1.00000 q^{9} -1.57991 q^{11} -2.50430 q^{12} -1.19794 q^{13} +9.24998 q^{14} -2.73708 q^{16} +1.12233 q^{17} -2.12233 q^{18} +7.67867 q^{19} +4.35840 q^{21} +3.35309 q^{22} +2.32027 q^{23} +1.07029 q^{24} +2.54243 q^{26} -1.00000 q^{27} -10.9147 q^{28} -5.50430 q^{29} +4.80206 q^{31} +7.94959 q^{32} +1.57991 q^{33} -2.38197 q^{34} +2.50430 q^{36} +6.37232 q^{37} -16.2967 q^{38} +1.19794 q^{39} -7.47214 q^{41} -9.24998 q^{42} +1.24998 q^{43} -3.95656 q^{44} -4.92439 q^{46} -4.12765 q^{47} +2.73708 q^{48} +11.9957 q^{49} -1.12233 q^{51} -3.00000 q^{52} -3.74568 q^{53} +2.12233 q^{54} +4.66476 q^{56} -7.67867 q^{57} +11.6820 q^{58} +9.15613 q^{59} +1.39588 q^{61} -10.1916 q^{62} -4.35840 q^{63} -11.3975 q^{64} -3.35309 q^{66} +3.86270 q^{67} +2.81066 q^{68} -2.32027 q^{69} -10.6051 q^{71} -1.07029 q^{72} -4.99672 q^{73} -13.5242 q^{74} +19.2297 q^{76} +6.88586 q^{77} -2.54243 q^{78} +14.5531 q^{79} +1.00000 q^{81} +15.8584 q^{82} +8.73603 q^{83} +10.9147 q^{84} -2.65288 q^{86} +5.50430 q^{87} +1.69096 q^{88} -10.0381 q^{89} +5.22110 q^{91} +5.81066 q^{92} -4.80206 q^{93} +8.76025 q^{94} -7.94959 q^{96} +7.49996 q^{97} -25.4588 q^{98} -1.57991 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 4 q^{3} + 8 q^{4} + 2 q^{6} - 2 q^{7} - 15 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 4 q^{3} + 8 q^{4} + 2 q^{6} - 2 q^{7} - 15 q^{8} + 4 q^{9} - 7 q^{11} - 8 q^{12} - q^{13} + 16 q^{14} + 4 q^{16} - 2 q^{17} - 2 q^{18} + 5 q^{19} + 2 q^{21} + 6 q^{22} - q^{23} + 15 q^{24} + 3 q^{26} - 4 q^{27} - 9 q^{28} - 20 q^{29} + 23 q^{31} - 12 q^{32} + 7 q^{33} - 14 q^{34} + 8 q^{36} - 2 q^{37} - 35 q^{38} + q^{39} - 12 q^{41} - 16 q^{42} - 16 q^{43} - 29 q^{44} - 17 q^{46} - 2 q^{47} - 4 q^{48} + 8 q^{49} + 2 q^{51} - 12 q^{52} + 4 q^{53} + 2 q^{54} + 5 q^{56} - 5 q^{57} + 25 q^{58} - 15 q^{59} - 2 q^{61} - 9 q^{62} - 2 q^{63} + 23 q^{64} - 6 q^{66} - 2 q^{67} + 11 q^{68} + q^{69} - 2 q^{71} - 15 q^{72} - 16 q^{73} - 19 q^{74} + 40 q^{76} - 19 q^{77} - 3 q^{78} + 35 q^{79} + 4 q^{81} + 6 q^{82} - 16 q^{83} + 9 q^{84} + 3 q^{86} + 20 q^{87} + 30 q^{88} - 35 q^{89} - 12 q^{91} + 23 q^{92} - 23 q^{93} - 9 q^{94} + 12 q^{96} - 12 q^{97} + q^{98} - 7 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.12233 −1.50072 −0.750358 0.661031i \(-0.770119\pi\)
−0.750358 + 0.661031i \(0.770119\pi\)
\(3\) −1.00000 −0.577350
\(4\) 2.50430 1.25215
\(5\) 0 0
\(6\) 2.12233 0.866439
\(7\) −4.35840 −1.64732 −0.823660 0.567083i \(-0.808072\pi\)
−0.823660 + 0.567083i \(0.808072\pi\)
\(8\) −1.07029 −0.378405
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.57991 −0.476360 −0.238180 0.971221i \(-0.576551\pi\)
−0.238180 + 0.971221i \(0.576551\pi\)
\(12\) −2.50430 −0.722929
\(13\) −1.19794 −0.332249 −0.166124 0.986105i \(-0.553125\pi\)
−0.166124 + 0.986105i \(0.553125\pi\)
\(14\) 9.24998 2.47216
\(15\) 0 0
\(16\) −2.73708 −0.684271
\(17\) 1.12233 0.272206 0.136103 0.990695i \(-0.456542\pi\)
0.136103 + 0.990695i \(0.456542\pi\)
\(18\) −2.12233 −0.500239
\(19\) 7.67867 1.76161 0.880804 0.473480i \(-0.157002\pi\)
0.880804 + 0.473480i \(0.157002\pi\)
\(20\) 0 0
\(21\) 4.35840 0.951081
\(22\) 3.35309 0.714881
\(23\) 2.32027 0.483810 0.241905 0.970300i \(-0.422228\pi\)
0.241905 + 0.970300i \(0.422228\pi\)
\(24\) 1.07029 0.218472
\(25\) 0 0
\(26\) 2.54243 0.498611
\(27\) −1.00000 −0.192450
\(28\) −10.9147 −2.06269
\(29\) −5.50430 −1.02212 −0.511061 0.859544i \(-0.670748\pi\)
−0.511061 + 0.859544i \(0.670748\pi\)
\(30\) 0 0
\(31\) 4.80206 0.862475 0.431238 0.902238i \(-0.358077\pi\)
0.431238 + 0.902238i \(0.358077\pi\)
\(32\) 7.94959 1.40530
\(33\) 1.57991 0.275026
\(34\) −2.38197 −0.408504
\(35\) 0 0
\(36\) 2.50430 0.417383
\(37\) 6.37232 1.04760 0.523801 0.851841i \(-0.324514\pi\)
0.523801 + 0.851841i \(0.324514\pi\)
\(38\) −16.2967 −2.64368
\(39\) 1.19794 0.191824
\(40\) 0 0
\(41\) −7.47214 −1.16695 −0.583476 0.812131i \(-0.698308\pi\)
−0.583476 + 0.812131i \(0.698308\pi\)
\(42\) −9.24998 −1.42730
\(43\) 1.24998 0.190620 0.0953102 0.995448i \(-0.469616\pi\)
0.0953102 + 0.995448i \(0.469616\pi\)
\(44\) −3.95656 −0.596473
\(45\) 0 0
\(46\) −4.92439 −0.726062
\(47\) −4.12765 −0.602079 −0.301040 0.953612i \(-0.597334\pi\)
−0.301040 + 0.953612i \(0.597334\pi\)
\(48\) 2.73708 0.395064
\(49\) 11.9957 1.71367
\(50\) 0 0
\(51\) −1.12233 −0.157158
\(52\) −3.00000 −0.416025
\(53\) −3.74568 −0.514509 −0.257254 0.966344i \(-0.582818\pi\)
−0.257254 + 0.966344i \(0.582818\pi\)
\(54\) 2.12233 0.288813
\(55\) 0 0
\(56\) 4.66476 0.623355
\(57\) −7.67867 −1.01707
\(58\) 11.6820 1.53392
\(59\) 9.15613 1.19203 0.596013 0.802975i \(-0.296751\pi\)
0.596013 + 0.802975i \(0.296751\pi\)
\(60\) 0 0
\(61\) 1.39588 0.178724 0.0893620 0.995999i \(-0.471517\pi\)
0.0893620 + 0.995999i \(0.471517\pi\)
\(62\) −10.1916 −1.29433
\(63\) −4.35840 −0.549107
\(64\) −11.3975 −1.42469
\(65\) 0 0
\(66\) −3.35309 −0.412736
\(67\) 3.86270 0.471904 0.235952 0.971765i \(-0.424179\pi\)
0.235952 + 0.971765i \(0.424179\pi\)
\(68\) 2.81066 0.340843
\(69\) −2.32027 −0.279328
\(70\) 0 0
\(71\) −10.6051 −1.25859 −0.629297 0.777165i \(-0.716657\pi\)
−0.629297 + 0.777165i \(0.716657\pi\)
\(72\) −1.07029 −0.126135
\(73\) −4.99672 −0.584821 −0.292411 0.956293i \(-0.594457\pi\)
−0.292411 + 0.956293i \(0.594457\pi\)
\(74\) −13.5242 −1.57215
\(75\) 0 0
\(76\) 19.2297 2.20580
\(77\) 6.88586 0.784717
\(78\) −2.54243 −0.287873
\(79\) 14.5531 1.63735 0.818673 0.574259i \(-0.194710\pi\)
0.818673 + 0.574259i \(0.194710\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 15.8584 1.75126
\(83\) 8.73603 0.958904 0.479452 0.877568i \(-0.340835\pi\)
0.479452 + 0.877568i \(0.340835\pi\)
\(84\) 10.9147 1.19090
\(85\) 0 0
\(86\) −2.65288 −0.286067
\(87\) 5.50430 0.590123
\(88\) 1.69096 0.180257
\(89\) −10.0381 −1.06404 −0.532020 0.846732i \(-0.678567\pi\)
−0.532020 + 0.846732i \(0.678567\pi\)
\(90\) 0 0
\(91\) 5.22110 0.547320
\(92\) 5.81066 0.605803
\(93\) −4.80206 −0.497950
\(94\) 8.76025 0.903550
\(95\) 0 0
\(96\) −7.94959 −0.811351
\(97\) 7.49996 0.761506 0.380753 0.924677i \(-0.375665\pi\)
0.380753 + 0.924677i \(0.375665\pi\)
\(98\) −25.4588 −2.57173
\(99\) −1.57991 −0.158787
\(100\) 0 0
\(101\) −6.51821 −0.648586 −0.324293 0.945957i \(-0.605126\pi\)
−0.324293 + 0.945957i \(0.605126\pi\)
\(102\) 2.38197 0.235850
\(103\) −8.04876 −0.793068 −0.396534 0.918020i \(-0.629787\pi\)
−0.396534 + 0.918020i \(0.629787\pi\)
\(104\) 1.28215 0.125725
\(105\) 0 0
\(106\) 7.94959 0.772132
\(107\) −9.47745 −0.916220 −0.458110 0.888896i \(-0.651473\pi\)
−0.458110 + 0.888896i \(0.651473\pi\)
\(108\) −2.50430 −0.240976
\(109\) −11.6657 −1.11738 −0.558688 0.829378i \(-0.688695\pi\)
−0.558688 + 0.829378i \(0.688695\pi\)
\(110\) 0 0
\(111\) −6.37232 −0.604833
\(112\) 11.9293 1.12721
\(113\) 14.9254 1.40406 0.702030 0.712147i \(-0.252277\pi\)
0.702030 + 0.712147i \(0.252277\pi\)
\(114\) 16.2967 1.52633
\(115\) 0 0
\(116\) −13.7844 −1.27985
\(117\) −1.19794 −0.110750
\(118\) −19.4324 −1.78889
\(119\) −4.89158 −0.448410
\(120\) 0 0
\(121\) −8.50390 −0.773082
\(122\) −2.96252 −0.268214
\(123\) 7.47214 0.673740
\(124\) 12.0258 1.07995
\(125\) 0 0
\(126\) 9.24998 0.824054
\(127\) 16.9957 1.50812 0.754061 0.656805i \(-0.228092\pi\)
0.754061 + 0.656805i \(0.228092\pi\)
\(128\) 8.29014 0.732752
\(129\) −1.24998 −0.110055
\(130\) 0 0
\(131\) −0.328872 −0.0287337 −0.0143669 0.999897i \(-0.504573\pi\)
−0.0143669 + 0.999897i \(0.504573\pi\)
\(132\) 3.95656 0.344374
\(133\) −33.4667 −2.90194
\(134\) −8.19794 −0.708194
\(135\) 0 0
\(136\) −1.20122 −0.103004
\(137\) −4.56271 −0.389818 −0.194909 0.980821i \(-0.562441\pi\)
−0.194909 + 0.980821i \(0.562441\pi\)
\(138\) 4.92439 0.419192
\(139\) −4.73400 −0.401533 −0.200766 0.979639i \(-0.564343\pi\)
−0.200766 + 0.979639i \(0.564343\pi\)
\(140\) 0 0
\(141\) 4.12765 0.347611
\(142\) 22.5076 1.88879
\(143\) 1.89263 0.158270
\(144\) −2.73708 −0.228090
\(145\) 0 0
\(146\) 10.6047 0.877651
\(147\) −11.9957 −0.989386
\(148\) 15.9582 1.31175
\(149\) 4.67644 0.383109 0.191555 0.981482i \(-0.438647\pi\)
0.191555 + 0.981482i \(0.438647\pi\)
\(150\) 0 0
\(151\) −6.54178 −0.532362 −0.266181 0.963923i \(-0.585762\pi\)
−0.266181 + 0.963923i \(0.585762\pi\)
\(152\) −8.21842 −0.666602
\(153\) 1.12233 0.0907353
\(154\) −14.6141 −1.17764
\(155\) 0 0
\(156\) 3.00000 0.240192
\(157\) −3.99404 −0.318759 −0.159379 0.987217i \(-0.550949\pi\)
−0.159379 + 0.987217i \(0.550949\pi\)
\(158\) −30.8864 −2.45719
\(159\) 3.74568 0.297052
\(160\) 0 0
\(161\) −10.1127 −0.796991
\(162\) −2.12233 −0.166746
\(163\) −4.80141 −0.376075 −0.188038 0.982162i \(-0.560213\pi\)
−0.188038 + 0.982162i \(0.560213\pi\)
\(164\) −18.7125 −1.46120
\(165\) 0 0
\(166\) −18.5408 −1.43904
\(167\) −23.6600 −1.83087 −0.915434 0.402469i \(-0.868152\pi\)
−0.915434 + 0.402469i \(0.868152\pi\)
\(168\) −4.66476 −0.359894
\(169\) −11.5649 −0.889611
\(170\) 0 0
\(171\) 7.67867 0.587203
\(172\) 3.13033 0.238685
\(173\) −14.5043 −1.10274 −0.551371 0.834260i \(-0.685895\pi\)
−0.551371 + 0.834260i \(0.685895\pi\)
\(174\) −11.6820 −0.885607
\(175\) 0 0
\(176\) 4.32433 0.325959
\(177\) −9.15613 −0.688217
\(178\) 21.3043 1.59682
\(179\) 0.796746 0.0595516 0.0297758 0.999557i \(-0.490521\pi\)
0.0297758 + 0.999557i \(0.490521\pi\)
\(180\) 0 0
\(181\) −14.2185 −1.05685 −0.528425 0.848980i \(-0.677217\pi\)
−0.528425 + 0.848980i \(0.677217\pi\)
\(182\) −11.0809 −0.821372
\(183\) −1.39588 −0.103186
\(184\) −2.48337 −0.183076
\(185\) 0 0
\(186\) 10.1916 0.747282
\(187\) −1.77318 −0.129668
\(188\) −10.3369 −0.753894
\(189\) 4.35840 0.317027
\(190\) 0 0
\(191\) −10.3697 −0.750324 −0.375162 0.926959i \(-0.622413\pi\)
−0.375162 + 0.926959i \(0.622413\pi\)
\(192\) 11.3975 0.822544
\(193\) −4.34712 −0.312913 −0.156456 0.987685i \(-0.550007\pi\)
−0.156456 + 0.987685i \(0.550007\pi\)
\(194\) −15.9174 −1.14280
\(195\) 0 0
\(196\) 30.0407 2.14577
\(197\) −1.87990 −0.133937 −0.0669686 0.997755i \(-0.521333\pi\)
−0.0669686 + 0.997755i \(0.521333\pi\)
\(198\) 3.35309 0.238294
\(199\) 4.26028 0.302003 0.151002 0.988534i \(-0.451750\pi\)
0.151002 + 0.988534i \(0.451750\pi\)
\(200\) 0 0
\(201\) −3.86270 −0.272454
\(202\) 13.8338 0.973344
\(203\) 23.9899 1.68376
\(204\) −2.81066 −0.196786
\(205\) 0 0
\(206\) 17.0821 1.19017
\(207\) 2.32027 0.161270
\(208\) 3.27886 0.227348
\(209\) −12.1316 −0.839159
\(210\) 0 0
\(211\) −9.29671 −0.640012 −0.320006 0.947416i \(-0.603685\pi\)
−0.320006 + 0.947416i \(0.603685\pi\)
\(212\) −9.38031 −0.644242
\(213\) 10.6051 0.726649
\(214\) 20.1143 1.37499
\(215\) 0 0
\(216\) 1.07029 0.0728241
\(217\) −20.9293 −1.42077
\(218\) 24.7586 1.67686
\(219\) 4.99672 0.337647
\(220\) 0 0
\(221\) −1.34449 −0.0904400
\(222\) 13.5242 0.907683
\(223\) 3.39758 0.227519 0.113759 0.993508i \(-0.463711\pi\)
0.113759 + 0.993508i \(0.463711\pi\)
\(224\) −34.6475 −2.31498
\(225\) 0 0
\(226\) −31.6766 −2.10710
\(227\) −26.7973 −1.77860 −0.889300 0.457324i \(-0.848808\pi\)
−0.889300 + 0.457324i \(0.848808\pi\)
\(228\) −19.2297 −1.27352
\(229\) −1.18200 −0.0781085 −0.0390543 0.999237i \(-0.512435\pi\)
−0.0390543 + 0.999237i \(0.512435\pi\)
\(230\) 0 0
\(231\) −6.88586 −0.453057
\(232\) 5.89121 0.386777
\(233\) −7.09483 −0.464798 −0.232399 0.972621i \(-0.574658\pi\)
−0.232399 + 0.972621i \(0.574658\pi\)
\(234\) 2.54243 0.166204
\(235\) 0 0
\(236\) 22.9297 1.49260
\(237\) −14.5531 −0.945323
\(238\) 10.3816 0.672937
\(239\) 0.0427926 0.00276802 0.00138401 0.999999i \(-0.499559\pi\)
0.00138401 + 0.999999i \(0.499559\pi\)
\(240\) 0 0
\(241\) −11.7711 −0.758242 −0.379121 0.925347i \(-0.623774\pi\)
−0.379121 + 0.925347i \(0.623774\pi\)
\(242\) 18.0481 1.16018
\(243\) −1.00000 −0.0641500
\(244\) 3.49570 0.223789
\(245\) 0 0
\(246\) −15.8584 −1.01109
\(247\) −9.19859 −0.585292
\(248\) −5.13961 −0.326365
\(249\) −8.73603 −0.553623
\(250\) 0 0
\(251\) 17.4764 1.10310 0.551550 0.834142i \(-0.314036\pi\)
0.551550 + 0.834142i \(0.314036\pi\)
\(252\) −10.9147 −0.687564
\(253\) −3.66581 −0.230468
\(254\) −36.0705 −2.26326
\(255\) 0 0
\(256\) 5.20057 0.325036
\(257\) −15.4671 −0.964814 −0.482407 0.875947i \(-0.660237\pi\)
−0.482407 + 0.875947i \(0.660237\pi\)
\(258\) 2.65288 0.165161
\(259\) −27.7731 −1.72574
\(260\) 0 0
\(261\) −5.50430 −0.340708
\(262\) 0.697977 0.0431211
\(263\) −1.44759 −0.0892624 −0.0446312 0.999004i \(-0.514211\pi\)
−0.0446312 + 0.999004i \(0.514211\pi\)
\(264\) −1.69096 −0.104071
\(265\) 0 0
\(266\) 71.0276 4.35498
\(267\) 10.0381 0.614323
\(268\) 9.67336 0.590895
\(269\) −8.70617 −0.530825 −0.265412 0.964135i \(-0.585508\pi\)
−0.265412 + 0.964135i \(0.585508\pi\)
\(270\) 0 0
\(271\) −28.6338 −1.73938 −0.869689 0.493600i \(-0.835681\pi\)
−0.869689 + 0.493600i \(0.835681\pi\)
\(272\) −3.07192 −0.186263
\(273\) −5.22110 −0.315995
\(274\) 9.68359 0.585007
\(275\) 0 0
\(276\) −5.81066 −0.349761
\(277\) −5.30210 −0.318572 −0.159286 0.987232i \(-0.550919\pi\)
−0.159286 + 0.987232i \(0.550919\pi\)
\(278\) 10.0471 0.602587
\(279\) 4.80206 0.287492
\(280\) 0 0
\(281\) −24.0832 −1.43668 −0.718342 0.695691i \(-0.755098\pi\)
−0.718342 + 0.695691i \(0.755098\pi\)
\(282\) −8.76025 −0.521665
\(283\) 27.1143 1.61178 0.805889 0.592066i \(-0.201688\pi\)
0.805889 + 0.592066i \(0.201688\pi\)
\(284\) −26.5583 −1.57595
\(285\) 0 0
\(286\) −4.01680 −0.237518
\(287\) 32.5666 1.92234
\(288\) 7.94959 0.468434
\(289\) −15.7404 −0.925904
\(290\) 0 0
\(291\) −7.49996 −0.439656
\(292\) −12.5133 −0.732284
\(293\) 20.3016 1.18603 0.593017 0.805190i \(-0.297937\pi\)
0.593017 + 0.805190i \(0.297937\pi\)
\(294\) 25.4588 1.48479
\(295\) 0 0
\(296\) −6.82024 −0.396418
\(297\) 1.57991 0.0916754
\(298\) −9.92497 −0.574938
\(299\) −2.77955 −0.160745
\(300\) 0 0
\(301\) −5.44792 −0.314013
\(302\) 13.8838 0.798925
\(303\) 6.51821 0.374462
\(304\) −21.0172 −1.20542
\(305\) 0 0
\(306\) −2.38197 −0.136168
\(307\) 2.51330 0.143442 0.0717208 0.997425i \(-0.477151\pi\)
0.0717208 + 0.997425i \(0.477151\pi\)
\(308\) 17.2443 0.982583
\(309\) 8.04876 0.457878
\(310\) 0 0
\(311\) 4.82417 0.273554 0.136777 0.990602i \(-0.456326\pi\)
0.136777 + 0.990602i \(0.456326\pi\)
\(312\) −1.28215 −0.0725872
\(313\) 5.98812 0.338468 0.169234 0.985576i \(-0.445871\pi\)
0.169234 + 0.985576i \(0.445871\pi\)
\(314\) 8.47668 0.478366
\(315\) 0 0
\(316\) 36.4452 2.05020
\(317\) −17.3531 −0.974646 −0.487323 0.873222i \(-0.662027\pi\)
−0.487323 + 0.873222i \(0.662027\pi\)
\(318\) −7.94959 −0.445791
\(319\) 8.69627 0.486898
\(320\) 0 0
\(321\) 9.47745 0.528980
\(322\) 21.4625 1.19606
\(323\) 8.61803 0.479520
\(324\) 2.50430 0.139128
\(325\) 0 0
\(326\) 10.1902 0.564383
\(327\) 11.6657 0.645117
\(328\) 7.99737 0.441581
\(329\) 17.9899 0.991818
\(330\) 0 0
\(331\) 20.3355 1.11774 0.558870 0.829255i \(-0.311235\pi\)
0.558870 + 0.829255i \(0.311235\pi\)
\(332\) 21.8776 1.20069
\(333\) 6.37232 0.349201
\(334\) 50.2145 2.74761
\(335\) 0 0
\(336\) −11.9293 −0.650797
\(337\) −34.2571 −1.86610 −0.933052 0.359741i \(-0.882865\pi\)
−0.933052 + 0.359741i \(0.882865\pi\)
\(338\) 24.5447 1.33505
\(339\) −14.9254 −0.810635
\(340\) 0 0
\(341\) −7.58680 −0.410848
\(342\) −16.2967 −0.881225
\(343\) −21.7731 −1.17564
\(344\) −1.33784 −0.0721318
\(345\) 0 0
\(346\) 30.7830 1.65490
\(347\) 0.0493616 0.00264987 0.00132494 0.999999i \(-0.499578\pi\)
0.00132494 + 0.999999i \(0.499578\pi\)
\(348\) 13.7844 0.738922
\(349\) −7.47437 −0.400094 −0.200047 0.979786i \(-0.564109\pi\)
−0.200047 + 0.979786i \(0.564109\pi\)
\(350\) 0 0
\(351\) 1.19794 0.0639413
\(352\) −12.5596 −0.669429
\(353\) 18.5650 0.988115 0.494057 0.869429i \(-0.335513\pi\)
0.494057 + 0.869429i \(0.335513\pi\)
\(354\) 19.4324 1.03282
\(355\) 0 0
\(356\) −25.1385 −1.33234
\(357\) 4.89158 0.258890
\(358\) −1.69096 −0.0893700
\(359\) 10.6044 0.559681 0.279841 0.960046i \(-0.409718\pi\)
0.279841 + 0.960046i \(0.409718\pi\)
\(360\) 0 0
\(361\) 39.9620 2.10327
\(362\) 30.1763 1.58603
\(363\) 8.50390 0.446339
\(364\) 13.0752 0.685327
\(365\) 0 0
\(366\) 2.96252 0.154854
\(367\) 11.6176 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(368\) −6.35078 −0.331057
\(369\) −7.47214 −0.388984
\(370\) 0 0
\(371\) 16.3252 0.847561
\(372\) −12.0258 −0.623509
\(373\) 23.1272 1.19748 0.598742 0.800942i \(-0.295668\pi\)
0.598742 + 0.800942i \(0.295668\pi\)
\(374\) 3.76328 0.194595
\(375\) 0 0
\(376\) 4.41779 0.227830
\(377\) 6.59382 0.339599
\(378\) −9.24998 −0.475768
\(379\) 24.8912 1.27857 0.639287 0.768968i \(-0.279230\pi\)
0.639287 + 0.768968i \(0.279230\pi\)
\(380\) 0 0
\(381\) −16.9957 −0.870714
\(382\) 22.0079 1.12602
\(383\) 15.7221 0.803363 0.401681 0.915780i \(-0.368426\pi\)
0.401681 + 0.915780i \(0.368426\pi\)
\(384\) −8.29014 −0.423054
\(385\) 0 0
\(386\) 9.22604 0.469593
\(387\) 1.24998 0.0635401
\(388\) 18.7822 0.953519
\(389\) −8.50633 −0.431288 −0.215644 0.976472i \(-0.569185\pi\)
−0.215644 + 0.976472i \(0.569185\pi\)
\(390\) 0 0
\(391\) 2.60412 0.131696
\(392\) −12.8389 −0.648460
\(393\) 0.328872 0.0165894
\(394\) 3.98977 0.201002
\(395\) 0 0
\(396\) −3.95656 −0.198824
\(397\) −26.2549 −1.31770 −0.658848 0.752276i \(-0.728956\pi\)
−0.658848 + 0.752276i \(0.728956\pi\)
\(398\) −9.04174 −0.453222
\(399\) 33.4667 1.67543
\(400\) 0 0
\(401\) −25.2815 −1.26250 −0.631250 0.775579i \(-0.717458\pi\)
−0.631250 + 0.775579i \(0.717458\pi\)
\(402\) 8.19794 0.408876
\(403\) −5.75258 −0.286556
\(404\) −16.3236 −0.812127
\(405\) 0 0
\(406\) −50.9147 −2.52685
\(407\) −10.0677 −0.499035
\(408\) 1.20122 0.0594695
\(409\) 33.7932 1.67097 0.835483 0.549517i \(-0.185188\pi\)
0.835483 + 0.549517i \(0.185188\pi\)
\(410\) 0 0
\(411\) 4.56271 0.225062
\(412\) −20.1565 −0.993039
\(413\) −39.9061 −1.96365
\(414\) −4.92439 −0.242021
\(415\) 0 0
\(416\) −9.52313 −0.466910
\(417\) 4.73400 0.231825
\(418\) 25.7473 1.25934
\(419\) 7.93332 0.387568 0.193784 0.981044i \(-0.437924\pi\)
0.193784 + 0.981044i \(0.437924\pi\)
\(420\) 0 0
\(421\) 7.48795 0.364941 0.182470 0.983211i \(-0.441591\pi\)
0.182470 + 0.983211i \(0.441591\pi\)
\(422\) 19.7307 0.960476
\(423\) −4.12765 −0.200693
\(424\) 4.00897 0.194693
\(425\) 0 0
\(426\) −22.5076 −1.09049
\(427\) −6.08380 −0.294416
\(428\) −23.7344 −1.14724
\(429\) −1.89263 −0.0913771
\(430\) 0 0
\(431\) 23.3470 1.12459 0.562294 0.826937i \(-0.309919\pi\)
0.562294 + 0.826937i \(0.309919\pi\)
\(432\) 2.73708 0.131688
\(433\) 3.35677 0.161316 0.0806581 0.996742i \(-0.474298\pi\)
0.0806581 + 0.996742i \(0.474298\pi\)
\(434\) 44.4190 2.13218
\(435\) 0 0
\(436\) −29.2145 −1.39912
\(437\) 17.8166 0.852285
\(438\) −10.6047 −0.506712
\(439\) −9.48402 −0.452648 −0.226324 0.974052i \(-0.572671\pi\)
−0.226324 + 0.974052i \(0.572671\pi\)
\(440\) 0 0
\(441\) 11.9957 0.571222
\(442\) 2.85345 0.135725
\(443\) −9.65446 −0.458697 −0.229349 0.973344i \(-0.573660\pi\)
−0.229349 + 0.973344i \(0.573660\pi\)
\(444\) −15.9582 −0.757342
\(445\) 0 0
\(446\) −7.21080 −0.341441
\(447\) −4.67644 −0.221188
\(448\) 49.6749 2.34692
\(449\) 31.5260 1.48780 0.743902 0.668289i \(-0.232973\pi\)
0.743902 + 0.668289i \(0.232973\pi\)
\(450\) 0 0
\(451\) 11.8053 0.555888
\(452\) 37.3776 1.75809
\(453\) 6.54178 0.307360
\(454\) 56.8729 2.66918
\(455\) 0 0
\(456\) 8.21842 0.384863
\(457\) 9.94467 0.465192 0.232596 0.972573i \(-0.425278\pi\)
0.232596 + 0.972573i \(0.425278\pi\)
\(458\) 2.50859 0.117219
\(459\) −1.12233 −0.0523860
\(460\) 0 0
\(461\) −23.6836 −1.10305 −0.551527 0.834157i \(-0.685955\pi\)
−0.551527 + 0.834157i \(0.685955\pi\)
\(462\) 14.6141 0.679909
\(463\) −27.6504 −1.28502 −0.642511 0.766277i \(-0.722107\pi\)
−0.642511 + 0.766277i \(0.722107\pi\)
\(464\) 15.0657 0.699409
\(465\) 0 0
\(466\) 15.0576 0.697530
\(467\) −4.35840 −0.201683 −0.100841 0.994903i \(-0.532153\pi\)
−0.100841 + 0.994903i \(0.532153\pi\)
\(468\) −3.00000 −0.138675
\(469\) −16.8352 −0.777377
\(470\) 0 0
\(471\) 3.99404 0.184035
\(472\) −9.79973 −0.451069
\(473\) −1.97485 −0.0908038
\(474\) 30.8864 1.41866
\(475\) 0 0
\(476\) −12.2500 −0.561477
\(477\) −3.74568 −0.171503
\(478\) −0.0908201 −0.00415401
\(479\) 12.8143 0.585499 0.292750 0.956189i \(-0.405430\pi\)
0.292750 + 0.956189i \(0.405430\pi\)
\(480\) 0 0
\(481\) −7.63365 −0.348064
\(482\) 24.9822 1.13791
\(483\) 10.1127 0.460143
\(484\) −21.2963 −0.968014
\(485\) 0 0
\(486\) 2.12233 0.0962710
\(487\) 6.92209 0.313670 0.156835 0.987625i \(-0.449871\pi\)
0.156835 + 0.987625i \(0.449871\pi\)
\(488\) −1.49400 −0.0676301
\(489\) 4.80141 0.217127
\(490\) 0 0
\(491\) 26.8161 1.21020 0.605098 0.796151i \(-0.293134\pi\)
0.605098 + 0.796151i \(0.293134\pi\)
\(492\) 18.7125 0.843623
\(493\) −6.17766 −0.278228
\(494\) 19.5225 0.878358
\(495\) 0 0
\(496\) −13.1436 −0.590167
\(497\) 46.2213 2.07331
\(498\) 18.5408 0.830832
\(499\) 2.75460 0.123313 0.0616565 0.998097i \(-0.480362\pi\)
0.0616565 + 0.998097i \(0.480362\pi\)
\(500\) 0 0
\(501\) 23.6600 1.05705
\(502\) −37.0907 −1.65544
\(503\) −15.7102 −0.700483 −0.350241 0.936659i \(-0.613900\pi\)
−0.350241 + 0.936659i \(0.613900\pi\)
\(504\) 4.66476 0.207785
\(505\) 0 0
\(506\) 7.78008 0.345867
\(507\) 11.5649 0.513617
\(508\) 42.5622 1.88839
\(509\) −41.0884 −1.82121 −0.910606 0.413277i \(-0.864384\pi\)
−0.910606 + 0.413277i \(0.864384\pi\)
\(510\) 0 0
\(511\) 21.7777 0.963388
\(512\) −27.6176 −1.22054
\(513\) −7.67867 −0.339022
\(514\) 32.8264 1.44791
\(515\) 0 0
\(516\) −3.13033 −0.137805
\(517\) 6.52129 0.286806
\(518\) 58.9438 2.58984
\(519\) 14.5043 0.636668
\(520\) 0 0
\(521\) 36.1710 1.58468 0.792341 0.610079i \(-0.208862\pi\)
0.792341 + 0.610079i \(0.208862\pi\)
\(522\) 11.6820 0.511305
\(523\) −28.0490 −1.22650 −0.613248 0.789891i \(-0.710137\pi\)
−0.613248 + 0.789891i \(0.710137\pi\)
\(524\) −0.823595 −0.0359789
\(525\) 0 0
\(526\) 3.07228 0.133958
\(527\) 5.38951 0.234771
\(528\) −4.32433 −0.188192
\(529\) −17.6163 −0.765927
\(530\) 0 0
\(531\) 9.15613 0.397342
\(532\) −83.8108 −3.63366
\(533\) 8.95117 0.387718
\(534\) −21.3043 −0.921925
\(535\) 0 0
\(536\) −4.13422 −0.178571
\(537\) −0.796746 −0.0343821
\(538\) 18.4774 0.796617
\(539\) −18.9520 −0.816321
\(540\) 0 0
\(541\) 3.45822 0.148681 0.0743403 0.997233i \(-0.476315\pi\)
0.0743403 + 0.997233i \(0.476315\pi\)
\(542\) 60.7704 2.61031
\(543\) 14.2185 0.610173
\(544\) 8.92209 0.382531
\(545\) 0 0
\(546\) 11.0809 0.474220
\(547\) −13.9635 −0.597034 −0.298517 0.954404i \(-0.596492\pi\)
−0.298517 + 0.954404i \(0.596492\pi\)
\(548\) −11.4264 −0.488111
\(549\) 1.39588 0.0595747
\(550\) 0 0
\(551\) −42.2657 −1.80058
\(552\) 2.48337 0.105699
\(553\) −63.4281 −2.69724
\(554\) 11.2528 0.478086
\(555\) 0 0
\(556\) −11.8554 −0.502779
\(557\) 6.59585 0.279475 0.139738 0.990189i \(-0.455374\pi\)
0.139738 + 0.990189i \(0.455374\pi\)
\(558\) −10.1916 −0.431444
\(559\) −1.49740 −0.0633334
\(560\) 0 0
\(561\) 1.77318 0.0748638
\(562\) 51.1126 2.15605
\(563\) 16.0218 0.675238 0.337619 0.941283i \(-0.390379\pi\)
0.337619 + 0.941283i \(0.390379\pi\)
\(564\) 10.3369 0.435261
\(565\) 0 0
\(566\) −57.5456 −2.41882
\(567\) −4.35840 −0.183036
\(568\) 11.3505 0.476258
\(569\) 23.0914 0.968042 0.484021 0.875056i \(-0.339176\pi\)
0.484021 + 0.875056i \(0.339176\pi\)
\(570\) 0 0
\(571\) −34.0102 −1.42328 −0.711640 0.702544i \(-0.752047\pi\)
−0.711640 + 0.702544i \(0.752047\pi\)
\(572\) 4.73972 0.198178
\(573\) 10.3697 0.433200
\(574\) −69.1171 −2.88489
\(575\) 0 0
\(576\) −11.3975 −0.474896
\(577\) 6.46727 0.269236 0.134618 0.990898i \(-0.457019\pi\)
0.134618 + 0.990898i \(0.457019\pi\)
\(578\) 33.4063 1.38952
\(579\) 4.34712 0.180660
\(580\) 0 0
\(581\) −38.0751 −1.57962
\(582\) 15.9174 0.659798
\(583\) 5.91782 0.245091
\(584\) 5.34794 0.221299
\(585\) 0 0
\(586\) −43.0868 −1.77990
\(587\) −38.5012 −1.58911 −0.794557 0.607189i \(-0.792297\pi\)
−0.794557 + 0.607189i \(0.792297\pi\)
\(588\) −30.0407 −1.23886
\(589\) 36.8735 1.51934
\(590\) 0 0
\(591\) 1.87990 0.0773287
\(592\) −17.4416 −0.716843
\(593\) −4.93069 −0.202479 −0.101240 0.994862i \(-0.532281\pi\)
−0.101240 + 0.994862i \(0.532281\pi\)
\(594\) −3.35309 −0.137579
\(595\) 0 0
\(596\) 11.7112 0.479710
\(597\) −4.26028 −0.174362
\(598\) 5.89913 0.241233
\(599\) −35.0268 −1.43116 −0.715578 0.698533i \(-0.753836\pi\)
−0.715578 + 0.698533i \(0.753836\pi\)
\(600\) 0 0
\(601\) −4.90570 −0.200108 −0.100054 0.994982i \(-0.531901\pi\)
−0.100054 + 0.994982i \(0.531901\pi\)
\(602\) 11.5623 0.471244
\(603\) 3.86270 0.157301
\(604\) −16.3826 −0.666597
\(605\) 0 0
\(606\) −13.8338 −0.561961
\(607\) −48.6955 −1.97649 −0.988244 0.152884i \(-0.951144\pi\)
−0.988244 + 0.152884i \(0.951144\pi\)
\(608\) 61.0423 2.47559
\(609\) −23.9899 −0.972122
\(610\) 0 0
\(611\) 4.94467 0.200040
\(612\) 2.81066 0.113614
\(613\) −17.2139 −0.695262 −0.347631 0.937631i \(-0.613014\pi\)
−0.347631 + 0.937631i \(0.613014\pi\)
\(614\) −5.33406 −0.215265
\(615\) 0 0
\(616\) −7.36988 −0.296941
\(617\) 31.6092 1.27254 0.636268 0.771468i \(-0.280477\pi\)
0.636268 + 0.771468i \(0.280477\pi\)
\(618\) −17.0821 −0.687145
\(619\) −22.8196 −0.917198 −0.458599 0.888643i \(-0.651649\pi\)
−0.458599 + 0.888643i \(0.651649\pi\)
\(620\) 0 0
\(621\) −2.32027 −0.0931094
\(622\) −10.2385 −0.410526
\(623\) 43.7502 1.75281
\(624\) −3.27886 −0.131259
\(625\) 0 0
\(626\) −12.7088 −0.507945
\(627\) 12.1316 0.484489
\(628\) −10.0023 −0.399134
\(629\) 7.15186 0.285163
\(630\) 0 0
\(631\) 38.0499 1.51474 0.757372 0.652984i \(-0.226483\pi\)
0.757372 + 0.652984i \(0.226483\pi\)
\(632\) −15.5760 −0.619581
\(633\) 9.29671 0.369511
\(634\) 36.8290 1.46267
\(635\) 0 0
\(636\) 9.38031 0.371953
\(637\) −14.3701 −0.569363
\(638\) −18.4564 −0.730696
\(639\) −10.6051 −0.419531
\(640\) 0 0
\(641\) 26.5108 1.04711 0.523557 0.851991i \(-0.324605\pi\)
0.523557 + 0.851991i \(0.324605\pi\)
\(642\) −20.1143 −0.793849
\(643\) 36.0014 1.41976 0.709879 0.704324i \(-0.248750\pi\)
0.709879 + 0.704324i \(0.248750\pi\)
\(644\) −25.3252 −0.997952
\(645\) 0 0
\(646\) −18.2903 −0.719624
\(647\) −41.7840 −1.64270 −0.821349 0.570426i \(-0.806778\pi\)
−0.821349 + 0.570426i \(0.806778\pi\)
\(648\) −1.07029 −0.0420450
\(649\) −14.4658 −0.567833
\(650\) 0 0
\(651\) 20.9293 0.820284
\(652\) −12.0242 −0.470903
\(653\) −21.4404 −0.839027 −0.419513 0.907749i \(-0.637799\pi\)
−0.419513 + 0.907749i \(0.637799\pi\)
\(654\) −24.7586 −0.968137
\(655\) 0 0
\(656\) 20.4519 0.798511
\(657\) −4.99672 −0.194940
\(658\) −38.1807 −1.48844
\(659\) −16.8319 −0.655678 −0.327839 0.944734i \(-0.606320\pi\)
−0.327839 + 0.944734i \(0.606320\pi\)
\(660\) 0 0
\(661\) 2.03846 0.0792868 0.0396434 0.999214i \(-0.487378\pi\)
0.0396434 + 0.999214i \(0.487378\pi\)
\(662\) −43.1587 −1.67741
\(663\) 1.34449 0.0522156
\(664\) −9.35010 −0.362854
\(665\) 0 0
\(666\) −13.5242 −0.524051
\(667\) −12.7715 −0.494514
\(668\) −59.2518 −2.29252
\(669\) −3.39758 −0.131358
\(670\) 0 0
\(671\) −2.20536 −0.0851369
\(672\) 34.6475 1.33656
\(673\) −35.3776 −1.36371 −0.681854 0.731489i \(-0.738826\pi\)
−0.681854 + 0.731489i \(0.738826\pi\)
\(674\) 72.7050 2.80049
\(675\) 0 0
\(676\) −28.9621 −1.11393
\(677\) −44.4173 −1.70710 −0.853548 0.521014i \(-0.825554\pi\)
−0.853548 + 0.521014i \(0.825554\pi\)
\(678\) 31.6766 1.21653
\(679\) −32.6879 −1.25444
\(680\) 0 0
\(681\) 26.7973 1.02688
\(682\) 16.1017 0.616567
\(683\) −42.4952 −1.62603 −0.813016 0.582241i \(-0.802176\pi\)
−0.813016 + 0.582241i \(0.802176\pi\)
\(684\) 19.2297 0.735266
\(685\) 0 0
\(686\) 46.2098 1.76430
\(687\) 1.18200 0.0450960
\(688\) −3.42130 −0.130436
\(689\) 4.48710 0.170945
\(690\) 0 0
\(691\) 28.0808 1.06825 0.534123 0.845407i \(-0.320642\pi\)
0.534123 + 0.845407i \(0.320642\pi\)
\(692\) −36.3231 −1.38080
\(693\) 6.88586 0.261572
\(694\) −0.104762 −0.00397671
\(695\) 0 0
\(696\) −5.89121 −0.223306
\(697\) −8.38623 −0.317651
\(698\) 15.8631 0.600427
\(699\) 7.09483 0.268351
\(700\) 0 0
\(701\) 46.4314 1.75369 0.876845 0.480772i \(-0.159644\pi\)
0.876845 + 0.480772i \(0.159644\pi\)
\(702\) −2.54243 −0.0959578
\(703\) 48.9309 1.84547
\(704\) 18.0070 0.678664
\(705\) 0 0
\(706\) −39.4011 −1.48288
\(707\) 28.4090 1.06843
\(708\) −22.9297 −0.861750
\(709\) 48.6331 1.82646 0.913228 0.407450i \(-0.133582\pi\)
0.913228 + 0.407450i \(0.133582\pi\)
\(710\) 0 0
\(711\) 14.5531 0.545782
\(712\) 10.7437 0.402638
\(713\) 11.1421 0.417275
\(714\) −10.3816 −0.388520
\(715\) 0 0
\(716\) 1.99529 0.0745675
\(717\) −0.0427926 −0.00159812
\(718\) −22.5062 −0.839923
\(719\) −21.1954 −0.790456 −0.395228 0.918583i \(-0.629334\pi\)
−0.395228 + 0.918583i \(0.629334\pi\)
\(720\) 0 0
\(721\) 35.0797 1.30644
\(722\) −84.8128 −3.15641
\(723\) 11.7711 0.437771
\(724\) −35.6073 −1.32333
\(725\) 0 0
\(726\) −18.0481 −0.669828
\(727\) −13.5911 −0.504066 −0.252033 0.967719i \(-0.581099\pi\)
−0.252033 + 0.967719i \(0.581099\pi\)
\(728\) −5.58810 −0.207109
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 1.40290 0.0518880
\(732\) −3.49570 −0.129205
\(733\) −18.6531 −0.688967 −0.344484 0.938792i \(-0.611946\pi\)
−0.344484 + 0.938792i \(0.611946\pi\)
\(734\) −24.6565 −0.910088
\(735\) 0 0
\(736\) 18.4452 0.679900
\(737\) −6.10270 −0.224796
\(738\) 15.8584 0.583754
\(739\) 26.4452 0.972803 0.486401 0.873735i \(-0.338309\pi\)
0.486401 + 0.873735i \(0.338309\pi\)
\(740\) 0 0
\(741\) 9.19859 0.337919
\(742\) −34.6475 −1.27195
\(743\) 29.0191 1.06461 0.532304 0.846553i \(-0.321326\pi\)
0.532304 + 0.846553i \(0.321326\pi\)
\(744\) 5.13961 0.188427
\(745\) 0 0
\(746\) −49.0837 −1.79708
\(747\) 8.73603 0.319635
\(748\) −4.44058 −0.162364
\(749\) 41.3065 1.50931
\(750\) 0 0
\(751\) 15.9489 0.581985 0.290992 0.956725i \(-0.406015\pi\)
0.290992 + 0.956725i \(0.406015\pi\)
\(752\) 11.2977 0.411985
\(753\) −17.4764 −0.636875
\(754\) −13.9943 −0.509642
\(755\) 0 0
\(756\) 10.9147 0.396965
\(757\) 16.4183 0.596734 0.298367 0.954451i \(-0.403558\pi\)
0.298367 + 0.954451i \(0.403558\pi\)
\(758\) −52.8274 −1.91878
\(759\) 3.66581 0.133061
\(760\) 0 0
\(761\) −49.2715 −1.78609 −0.893045 0.449967i \(-0.851436\pi\)
−0.893045 + 0.449967i \(0.851436\pi\)
\(762\) 36.0705 1.30670
\(763\) 50.8440 1.84068
\(764\) −25.9688 −0.939518
\(765\) 0 0
\(766\) −33.3676 −1.20562
\(767\) −10.9685 −0.396049
\(768\) −5.20057 −0.187660
\(769\) 49.4236 1.78226 0.891129 0.453749i \(-0.149914\pi\)
0.891129 + 0.453749i \(0.149914\pi\)
\(770\) 0 0
\(771\) 15.4671 0.557036
\(772\) −10.8865 −0.391814
\(773\) −25.9084 −0.931860 −0.465930 0.884822i \(-0.654280\pi\)
−0.465930 + 0.884822i \(0.654280\pi\)
\(774\) −2.65288 −0.0953557
\(775\) 0 0
\(776\) −8.02715 −0.288158
\(777\) 27.7731 0.996355
\(778\) 18.0533 0.647241
\(779\) −57.3761 −2.05571
\(780\) 0 0
\(781\) 16.7551 0.599543
\(782\) −5.52681 −0.197638
\(783\) 5.50430 0.196708
\(784\) −32.8331 −1.17261
\(785\) 0 0
\(786\) −0.697977 −0.0248960
\(787\) −30.7155 −1.09489 −0.547444 0.836842i \(-0.684399\pi\)
−0.547444 + 0.836842i \(0.684399\pi\)
\(788\) −4.70783 −0.167710
\(789\) 1.44759 0.0515357
\(790\) 0 0
\(791\) −65.0508 −2.31294
\(792\) 1.69096 0.0600857
\(793\) −1.67218 −0.0593808
\(794\) 55.7216 1.97749
\(795\) 0 0
\(796\) 10.6690 0.378154
\(797\) 33.7141 1.19421 0.597107 0.802161i \(-0.296317\pi\)
0.597107 + 0.802161i \(0.296317\pi\)
\(798\) −71.0276 −2.51435
\(799\) −4.63260 −0.163890
\(800\) 0 0
\(801\) −10.0381 −0.354680
\(802\) 53.6559 1.89465
\(803\) 7.89434 0.278585
\(804\) −9.67336 −0.341153
\(805\) 0 0
\(806\) 12.2089 0.430040
\(807\) 8.70617 0.306472
\(808\) 6.97639 0.245429
\(809\) −0.923014 −0.0324514 −0.0162257 0.999868i \(-0.505165\pi\)
−0.0162257 + 0.999868i \(0.505165\pi\)
\(810\) 0 0
\(811\) 26.7815 0.940424 0.470212 0.882553i \(-0.344177\pi\)
0.470212 + 0.882553i \(0.344177\pi\)
\(812\) 60.0780 2.10832
\(813\) 28.6338 1.00423
\(814\) 21.3669 0.748910
\(815\) 0 0
\(816\) 3.07192 0.107539
\(817\) 9.59820 0.335799
\(818\) −71.7204 −2.50765
\(819\) 5.22110 0.182440
\(820\) 0 0
\(821\) −36.8489 −1.28604 −0.643018 0.765851i \(-0.722318\pi\)
−0.643018 + 0.765851i \(0.722318\pi\)
\(822\) −9.68359 −0.337754
\(823\) −20.4262 −0.712012 −0.356006 0.934484i \(-0.615862\pi\)
−0.356006 + 0.934484i \(0.615862\pi\)
\(824\) 8.61452 0.300101
\(825\) 0 0
\(826\) 84.6940 2.94688
\(827\) −4.33017 −0.150575 −0.0752874 0.997162i \(-0.523987\pi\)
−0.0752874 + 0.997162i \(0.523987\pi\)
\(828\) 5.81066 0.201934
\(829\) −0.614348 −0.0213372 −0.0106686 0.999943i \(-0.503396\pi\)
−0.0106686 + 0.999943i \(0.503396\pi\)
\(830\) 0 0
\(831\) 5.30210 0.183928
\(832\) 13.6535 0.473351
\(833\) 13.4631 0.466470
\(834\) −10.0471 −0.347904
\(835\) 0 0
\(836\) −30.3811 −1.05075
\(837\) −4.80206 −0.165983
\(838\) −16.8372 −0.581630
\(839\) 5.39981 0.186422 0.0932111 0.995646i \(-0.470287\pi\)
0.0932111 + 0.995646i \(0.470287\pi\)
\(840\) 0 0
\(841\) 1.29731 0.0447349
\(842\) −15.8919 −0.547672
\(843\) 24.0832 0.829469
\(844\) −23.2817 −0.801391
\(845\) 0 0
\(846\) 8.76025 0.301183
\(847\) 37.0634 1.27351
\(848\) 10.2522 0.352063
\(849\) −27.1143 −0.930561
\(850\) 0 0
\(851\) 14.7855 0.506841
\(852\) 26.5583 0.909874
\(853\) 39.7935 1.36250 0.681252 0.732049i \(-0.261436\pi\)
0.681252 + 0.732049i \(0.261436\pi\)
\(854\) 12.9119 0.441835
\(855\) 0 0
\(856\) 10.1436 0.346702
\(857\) 42.8643 1.46422 0.732109 0.681188i \(-0.238536\pi\)
0.732109 + 0.681188i \(0.238536\pi\)
\(858\) 4.01680 0.137131
\(859\) −4.56494 −0.155754 −0.0778769 0.996963i \(-0.524814\pi\)
−0.0778769 + 0.996963i \(0.524814\pi\)
\(860\) 0 0
\(861\) −32.5666 −1.10987
\(862\) −49.5502 −1.68769
\(863\) 37.6273 1.28085 0.640424 0.768022i \(-0.278759\pi\)
0.640424 + 0.768022i \(0.278759\pi\)
\(864\) −7.94959 −0.270450
\(865\) 0 0
\(866\) −7.12419 −0.242090
\(867\) 15.7404 0.534571
\(868\) −52.4133 −1.77902
\(869\) −22.9925 −0.779966
\(870\) 0 0
\(871\) −4.62728 −0.156790
\(872\) 12.4857 0.422821
\(873\) 7.49996 0.253835
\(874\) −37.8128 −1.27904
\(875\) 0 0
\(876\) 12.5133 0.422784
\(877\) −52.4041 −1.76956 −0.884781 0.466008i \(-0.845692\pi\)
−0.884781 + 0.466008i \(0.845692\pi\)
\(878\) 20.1283 0.679296
\(879\) −20.3016 −0.684757
\(880\) 0 0
\(881\) −7.93778 −0.267430 −0.133715 0.991020i \(-0.542691\pi\)
−0.133715 + 0.991020i \(0.542691\pi\)
\(882\) −25.4588 −0.857242
\(883\) −31.6919 −1.06652 −0.533259 0.845952i \(-0.679033\pi\)
−0.533259 + 0.845952i \(0.679033\pi\)
\(884\) −3.36700 −0.113244
\(885\) 0 0
\(886\) 20.4900 0.688374
\(887\) −2.82104 −0.0947213 −0.0473606 0.998878i \(-0.515081\pi\)
−0.0473606 + 0.998878i \(0.515081\pi\)
\(888\) 6.82024 0.228872
\(889\) −74.0739 −2.48436
\(890\) 0 0
\(891\) −1.57991 −0.0529288
\(892\) 8.50856 0.284888
\(893\) −31.6949 −1.06063
\(894\) 9.92497 0.331941
\(895\) 0 0
\(896\) −36.1318 −1.20708
\(897\) 2.77955 0.0928064
\(898\) −66.9087 −2.23277
\(899\) −26.4320 −0.881556
\(900\) 0 0
\(901\) −4.20390 −0.140052
\(902\) −25.0547 −0.834231
\(903\) 5.44792 0.181295
\(904\) −15.9745 −0.531304
\(905\) 0 0
\(906\) −13.8838 −0.461259
\(907\) 44.1799 1.46697 0.733485 0.679705i \(-0.237892\pi\)
0.733485 + 0.679705i \(0.237892\pi\)
\(908\) −67.1085 −2.22707
\(909\) −6.51821 −0.216195
\(910\) 0 0
\(911\) −34.5260 −1.14390 −0.571949 0.820289i \(-0.693812\pi\)
−0.571949 + 0.820289i \(0.693812\pi\)
\(912\) 21.0172 0.695948
\(913\) −13.8021 −0.456783
\(914\) −21.1059 −0.698122
\(915\) 0 0
\(916\) −2.96007 −0.0978036
\(917\) 1.43336 0.0473336
\(918\) 2.38197 0.0786166
\(919\) −0.531075 −0.0175185 −0.00875927 0.999962i \(-0.502788\pi\)
−0.00875927 + 0.999962i \(0.502788\pi\)
\(920\) 0 0
\(921\) −2.51330 −0.0828161
\(922\) 50.2645 1.65537
\(923\) 12.7043 0.418166
\(924\) −17.2443 −0.567295
\(925\) 0 0
\(926\) 58.6833 1.92845
\(927\) −8.04876 −0.264356
\(928\) −43.7569 −1.43639
\(929\) 46.1961 1.51565 0.757823 0.652460i \(-0.226263\pi\)
0.757823 + 0.652460i \(0.226263\pi\)
\(930\) 0 0
\(931\) 92.1108 3.01881
\(932\) −17.7676 −0.581997
\(933\) −4.82417 −0.157936
\(934\) 9.24998 0.302669
\(935\) 0 0
\(936\) 1.28215 0.0419082
\(937\) 30.1220 0.984042 0.492021 0.870583i \(-0.336258\pi\)
0.492021 + 0.870583i \(0.336258\pi\)
\(938\) 35.7299 1.16662
\(939\) −5.98812 −0.195415
\(940\) 0 0
\(941\) 57.0783 1.86070 0.930350 0.366672i \(-0.119503\pi\)
0.930350 + 0.366672i \(0.119503\pi\)
\(942\) −8.47668 −0.276185
\(943\) −17.3374 −0.564583
\(944\) −25.0611 −0.815668
\(945\) 0 0
\(946\) 4.19130 0.136271
\(947\) 41.1250 1.33638 0.668192 0.743989i \(-0.267069\pi\)
0.668192 + 0.743989i \(0.267069\pi\)
\(948\) −36.4452 −1.18369
\(949\) 5.98576 0.194306
\(950\) 0 0
\(951\) 17.3531 0.562712
\(952\) 5.23542 0.169681
\(953\) −34.8143 −1.12775 −0.563873 0.825861i \(-0.690689\pi\)
−0.563873 + 0.825861i \(0.690689\pi\)
\(954\) 7.94959 0.257377
\(955\) 0 0
\(956\) 0.107165 0.00346598
\(957\) −8.69627 −0.281111
\(958\) −27.1962 −0.878668
\(959\) 19.8861 0.642156
\(960\) 0 0
\(961\) −7.94022 −0.256136
\(962\) 16.2012 0.522346
\(963\) −9.47745 −0.305407
\(964\) −29.4783 −0.949432
\(965\) 0 0
\(966\) −21.4625 −0.690544
\(967\) 37.2697 1.19851 0.599257 0.800557i \(-0.295463\pi\)
0.599257 + 0.800557i \(0.295463\pi\)
\(968\) 9.10165 0.292538
\(969\) −8.61803 −0.276851
\(970\) 0 0
\(971\) −26.0092 −0.834677 −0.417338 0.908751i \(-0.637037\pi\)
−0.417338 + 0.908751i \(0.637037\pi\)
\(972\) −2.50430 −0.0803254
\(973\) 20.6327 0.661453
\(974\) −14.6910 −0.470729
\(975\) 0 0
\(976\) −3.82064 −0.122296
\(977\) −0.458579 −0.0146712 −0.00733561 0.999973i \(-0.502335\pi\)
−0.00733561 + 0.999973i \(0.502335\pi\)
\(978\) −10.1902 −0.325846
\(979\) 15.8593 0.506865
\(980\) 0 0
\(981\) −11.6657 −0.372458
\(982\) −56.9128 −1.81616
\(983\) −8.94565 −0.285322 −0.142661 0.989772i \(-0.545566\pi\)
−0.142661 + 0.989772i \(0.545566\pi\)
\(984\) −7.99737 −0.254947
\(985\) 0 0
\(986\) 13.1111 0.417541
\(987\) −17.9899 −0.572626
\(988\) −23.0360 −0.732874
\(989\) 2.90030 0.0922241
\(990\) 0 0
\(991\) −56.5337 −1.79585 −0.897926 0.440147i \(-0.854926\pi\)
−0.897926 + 0.440147i \(0.854926\pi\)
\(992\) 38.1744 1.21204
\(993\) −20.3355 −0.645327
\(994\) −98.0970 −3.11145
\(995\) 0 0
\(996\) −21.8776 −0.693219
\(997\) −39.6114 −1.25451 −0.627253 0.778816i \(-0.715821\pi\)
−0.627253 + 0.778816i \(0.715821\pi\)
\(998\) −5.84619 −0.185058
\(999\) −6.37232 −0.201611
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 1875.2.a.e.1.2 4
3.2 odd 2 5625.2.a.n.1.3 4
5.2 odd 4 1875.2.b.c.1249.2 8
5.3 odd 4 1875.2.b.c.1249.7 8
5.4 even 2 1875.2.a.h.1.3 4
15.14 odd 2 5625.2.a.i.1.2 4
25.2 odd 20 375.2.i.b.274.1 16
25.9 even 10 75.2.g.b.31.2 8
25.11 even 5 375.2.g.b.226.1 8
25.12 odd 20 375.2.i.b.349.4 16
25.13 odd 20 375.2.i.b.349.1 16
25.14 even 10 75.2.g.b.46.2 yes 8
25.16 even 5 375.2.g.b.151.1 8
25.23 odd 20 375.2.i.b.274.4 16
75.14 odd 10 225.2.h.c.46.1 8
75.59 odd 10 225.2.h.c.181.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.g.b.31.2 8 25.9 even 10
75.2.g.b.46.2 yes 8 25.14 even 10
225.2.h.c.46.1 8 75.14 odd 10
225.2.h.c.181.1 8 75.59 odd 10
375.2.g.b.151.1 8 25.16 even 5
375.2.g.b.226.1 8 25.11 even 5
375.2.i.b.274.1 16 25.2 odd 20
375.2.i.b.274.4 16 25.23 odd 20
375.2.i.b.349.1 16 25.13 odd 20
375.2.i.b.349.4 16 25.12 odd 20
1875.2.a.e.1.2 4 1.1 even 1 trivial
1875.2.a.h.1.3 4 5.4 even 2
1875.2.b.c.1249.2 8 5.2 odd 4
1875.2.b.c.1249.7 8 5.3 odd 4
5625.2.a.i.1.2 4 15.14 odd 2
5625.2.a.n.1.3 4 3.2 odd 2