Properties

Label 1875.2.b.c.1249.2
Level $1875$
Weight $2$
Character 1875.1249
Analytic conductor $14.972$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1875,2,Mod(1249,1875)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1875.1249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1875 = 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1875.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(14.9719503790\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.6724000000.12
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 16x^{6} + 86x^{4} + 181x^{2} + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 75)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1249.2
Root \(-2.12233i\) of defining polynomial
Character \(\chi\) \(=\) 1875.1249
Dual form 1875.2.b.c.1249.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.12233i q^{2} +1.00000i q^{3} -2.50430 q^{4} +2.12233 q^{6} -4.35840i q^{7} +1.07029i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-2.12233i q^{2} +1.00000i q^{3} -2.50430 q^{4} +2.12233 q^{6} -4.35840i q^{7} +1.07029i q^{8} -1.00000 q^{9} -1.57991 q^{11} -2.50430i q^{12} +1.19794i q^{13} -9.24998 q^{14} -2.73708 q^{16} +1.12233i q^{17} +2.12233i q^{18} -7.67867 q^{19} +4.35840 q^{21} +3.35309i q^{22} -2.32027i q^{23} -1.07029 q^{24} +2.54243 q^{26} -1.00000i q^{27} +10.9147i q^{28} +5.50430 q^{29} +4.80206 q^{31} +7.94959i q^{32} -1.57991i q^{33} +2.38197 q^{34} +2.50430 q^{36} +6.37232i q^{37} +16.2967i q^{38} -1.19794 q^{39} -7.47214 q^{41} -9.24998i q^{42} -1.24998i q^{43} +3.95656 q^{44} -4.92439 q^{46} -4.12765i q^{47} -2.73708i q^{48} -11.9957 q^{49} -1.12233 q^{51} -3.00000i q^{52} +3.74568i q^{53} -2.12233 q^{54} +4.66476 q^{56} -7.67867i q^{57} -11.6820i q^{58} -9.15613 q^{59} +1.39588 q^{61} -10.1916i q^{62} +4.35840i q^{63} +11.3975 q^{64} -3.35309 q^{66} +3.86270i q^{67} -2.81066i q^{68} +2.32027 q^{69} -10.6051 q^{71} -1.07029i q^{72} +4.99672i q^{73} +13.5242 q^{74} +19.2297 q^{76} +6.88586i q^{77} +2.54243i q^{78} -14.5531 q^{79} +1.00000 q^{81} +15.8584i q^{82} -8.73603i q^{83} -10.9147 q^{84} -2.65288 q^{86} +5.50430i q^{87} -1.69096i q^{88} +10.0381 q^{89} +5.22110 q^{91} +5.81066i q^{92} +4.80206i q^{93} -8.76025 q^{94} -7.94959 q^{96} +7.49996i q^{97} +25.4588i q^{98} +1.57991 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{4} + 4 q^{6} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{4} + 4 q^{6} - 8 q^{9} - 14 q^{11} - 32 q^{14} + 8 q^{16} - 10 q^{19} + 4 q^{21} - 30 q^{24} + 6 q^{26} + 40 q^{29} + 46 q^{31} + 28 q^{34} + 16 q^{36} - 2 q^{39} - 24 q^{41} + 58 q^{44} - 34 q^{46} - 16 q^{49} + 4 q^{51} - 4 q^{54} + 10 q^{56} + 30 q^{59} - 4 q^{61} - 46 q^{64} - 12 q^{66} - 2 q^{69} - 4 q^{71} + 38 q^{74} + 80 q^{76} - 70 q^{79} + 8 q^{81} - 18 q^{84} + 6 q^{86} + 70 q^{89} - 24 q^{91} + 18 q^{94} + 24 q^{96} + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1875\mathbb{Z}\right)^\times\).

\(n\) \(626\) \(1252\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.12233i − 1.50072i −0.661031 0.750358i \(-0.729881\pi\)
0.661031 0.750358i \(-0.270119\pi\)
\(3\) 1.00000i 0.577350i
\(4\) −2.50430 −1.25215
\(5\) 0 0
\(6\) 2.12233 0.866439
\(7\) − 4.35840i − 1.64732i −0.567083 0.823660i \(-0.691928\pi\)
0.567083 0.823660i \(-0.308072\pi\)
\(8\) 1.07029i 0.378405i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −1.57991 −0.476360 −0.238180 0.971221i \(-0.576551\pi\)
−0.238180 + 0.971221i \(0.576551\pi\)
\(12\) − 2.50430i − 0.722929i
\(13\) 1.19794i 0.332249i 0.986105 + 0.166124i \(0.0531253\pi\)
−0.986105 + 0.166124i \(0.946875\pi\)
\(14\) −9.24998 −2.47216
\(15\) 0 0
\(16\) −2.73708 −0.684271
\(17\) 1.12233i 0.272206i 0.990695 + 0.136103i \(0.0434578\pi\)
−0.990695 + 0.136103i \(0.956542\pi\)
\(18\) 2.12233i 0.500239i
\(19\) −7.67867 −1.76161 −0.880804 0.473480i \(-0.842998\pi\)
−0.880804 + 0.473480i \(0.842998\pi\)
\(20\) 0 0
\(21\) 4.35840 0.951081
\(22\) 3.35309i 0.714881i
\(23\) − 2.32027i − 0.483810i −0.970300 0.241905i \(-0.922228\pi\)
0.970300 0.241905i \(-0.0777723\pi\)
\(24\) −1.07029 −0.218472
\(25\) 0 0
\(26\) 2.54243 0.498611
\(27\) − 1.00000i − 0.192450i
\(28\) 10.9147i 2.06269i
\(29\) 5.50430 1.02212 0.511061 0.859544i \(-0.329252\pi\)
0.511061 + 0.859544i \(0.329252\pi\)
\(30\) 0 0
\(31\) 4.80206 0.862475 0.431238 0.902238i \(-0.358077\pi\)
0.431238 + 0.902238i \(0.358077\pi\)
\(32\) 7.94959i 1.40530i
\(33\) − 1.57991i − 0.275026i
\(34\) 2.38197 0.408504
\(35\) 0 0
\(36\) 2.50430 0.417383
\(37\) 6.37232i 1.04760i 0.851841 + 0.523801i \(0.175486\pi\)
−0.851841 + 0.523801i \(0.824514\pi\)
\(38\) 16.2967i 2.64368i
\(39\) −1.19794 −0.191824
\(40\) 0 0
\(41\) −7.47214 −1.16695 −0.583476 0.812131i \(-0.698308\pi\)
−0.583476 + 0.812131i \(0.698308\pi\)
\(42\) − 9.24998i − 1.42730i
\(43\) − 1.24998i − 0.190620i −0.995448 0.0953102i \(-0.969616\pi\)
0.995448 0.0953102i \(-0.0303843\pi\)
\(44\) 3.95656 0.596473
\(45\) 0 0
\(46\) −4.92439 −0.726062
\(47\) − 4.12765i − 0.602079i −0.953612 0.301040i \(-0.902666\pi\)
0.953612 0.301040i \(-0.0973337\pi\)
\(48\) − 2.73708i − 0.395064i
\(49\) −11.9957 −1.71367
\(50\) 0 0
\(51\) −1.12233 −0.157158
\(52\) − 3.00000i − 0.416025i
\(53\) 3.74568i 0.514509i 0.966344 + 0.257254i \(0.0828179\pi\)
−0.966344 + 0.257254i \(0.917182\pi\)
\(54\) −2.12233 −0.288813
\(55\) 0 0
\(56\) 4.66476 0.623355
\(57\) − 7.67867i − 1.01707i
\(58\) − 11.6820i − 1.53392i
\(59\) −9.15613 −1.19203 −0.596013 0.802975i \(-0.703249\pi\)
−0.596013 + 0.802975i \(0.703249\pi\)
\(60\) 0 0
\(61\) 1.39588 0.178724 0.0893620 0.995999i \(-0.471517\pi\)
0.0893620 + 0.995999i \(0.471517\pi\)
\(62\) − 10.1916i − 1.29433i
\(63\) 4.35840i 0.549107i
\(64\) 11.3975 1.42469
\(65\) 0 0
\(66\) −3.35309 −0.412736
\(67\) 3.86270i 0.471904i 0.971765 + 0.235952i \(0.0758208\pi\)
−0.971765 + 0.235952i \(0.924179\pi\)
\(68\) − 2.81066i − 0.340843i
\(69\) 2.32027 0.279328
\(70\) 0 0
\(71\) −10.6051 −1.25859 −0.629297 0.777165i \(-0.716657\pi\)
−0.629297 + 0.777165i \(0.716657\pi\)
\(72\) − 1.07029i − 0.126135i
\(73\) 4.99672i 0.584821i 0.956293 + 0.292411i \(0.0944574\pi\)
−0.956293 + 0.292411i \(0.905543\pi\)
\(74\) 13.5242 1.57215
\(75\) 0 0
\(76\) 19.2297 2.20580
\(77\) 6.88586i 0.784717i
\(78\) 2.54243i 0.287873i
\(79\) −14.5531 −1.63735 −0.818673 0.574259i \(-0.805290\pi\)
−0.818673 + 0.574259i \(0.805290\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 15.8584i 1.75126i
\(83\) − 8.73603i − 0.958904i −0.877568 0.479452i \(-0.840835\pi\)
0.877568 0.479452i \(-0.159165\pi\)
\(84\) −10.9147 −1.19090
\(85\) 0 0
\(86\) −2.65288 −0.286067
\(87\) 5.50430i 0.590123i
\(88\) − 1.69096i − 0.180257i
\(89\) 10.0381 1.06404 0.532020 0.846732i \(-0.321433\pi\)
0.532020 + 0.846732i \(0.321433\pi\)
\(90\) 0 0
\(91\) 5.22110 0.547320
\(92\) 5.81066i 0.605803i
\(93\) 4.80206i 0.497950i
\(94\) −8.76025 −0.903550
\(95\) 0 0
\(96\) −7.94959 −0.811351
\(97\) 7.49996i 0.761506i 0.924677 + 0.380753i \(0.124335\pi\)
−0.924677 + 0.380753i \(0.875665\pi\)
\(98\) 25.4588i 2.57173i
\(99\) 1.57991 0.158787
\(100\) 0 0
\(101\) −6.51821 −0.648586 −0.324293 0.945957i \(-0.605126\pi\)
−0.324293 + 0.945957i \(0.605126\pi\)
\(102\) 2.38197i 0.235850i
\(103\) 8.04876i 0.793068i 0.918020 + 0.396534i \(0.129787\pi\)
−0.918020 + 0.396534i \(0.870213\pi\)
\(104\) −1.28215 −0.125725
\(105\) 0 0
\(106\) 7.94959 0.772132
\(107\) − 9.47745i − 0.916220i −0.888896 0.458110i \(-0.848527\pi\)
0.888896 0.458110i \(-0.151473\pi\)
\(108\) 2.50430i 0.240976i
\(109\) 11.6657 1.11738 0.558688 0.829378i \(-0.311305\pi\)
0.558688 + 0.829378i \(0.311305\pi\)
\(110\) 0 0
\(111\) −6.37232 −0.604833
\(112\) 11.9293i 1.12721i
\(113\) − 14.9254i − 1.40406i −0.712147 0.702030i \(-0.752277\pi\)
0.712147 0.702030i \(-0.247723\pi\)
\(114\) −16.2967 −1.52633
\(115\) 0 0
\(116\) −13.7844 −1.27985
\(117\) − 1.19794i − 0.110750i
\(118\) 19.4324i 1.78889i
\(119\) 4.89158 0.448410
\(120\) 0 0
\(121\) −8.50390 −0.773082
\(122\) − 2.96252i − 0.268214i
\(123\) − 7.47214i − 0.673740i
\(124\) −12.0258 −1.07995
\(125\) 0 0
\(126\) 9.24998 0.824054
\(127\) 16.9957i 1.50812i 0.656805 + 0.754061i \(0.271908\pi\)
−0.656805 + 0.754061i \(0.728092\pi\)
\(128\) − 8.29014i − 0.732752i
\(129\) 1.24998 0.110055
\(130\) 0 0
\(131\) −0.328872 −0.0287337 −0.0143669 0.999897i \(-0.504573\pi\)
−0.0143669 + 0.999897i \(0.504573\pi\)
\(132\) 3.95656i 0.344374i
\(133\) 33.4667i 2.90194i
\(134\) 8.19794 0.708194
\(135\) 0 0
\(136\) −1.20122 −0.103004
\(137\) − 4.56271i − 0.389818i −0.980821 0.194909i \(-0.937559\pi\)
0.980821 0.194909i \(-0.0624412\pi\)
\(138\) − 4.92439i − 0.419192i
\(139\) 4.73400 0.401533 0.200766 0.979639i \(-0.435657\pi\)
0.200766 + 0.979639i \(0.435657\pi\)
\(140\) 0 0
\(141\) 4.12765 0.347611
\(142\) 22.5076i 1.88879i
\(143\) − 1.89263i − 0.158270i
\(144\) 2.73708 0.228090
\(145\) 0 0
\(146\) 10.6047 0.877651
\(147\) − 11.9957i − 0.989386i
\(148\) − 15.9582i − 1.31175i
\(149\) −4.67644 −0.383109 −0.191555 0.981482i \(-0.561353\pi\)
−0.191555 + 0.981482i \(0.561353\pi\)
\(150\) 0 0
\(151\) −6.54178 −0.532362 −0.266181 0.963923i \(-0.585762\pi\)
−0.266181 + 0.963923i \(0.585762\pi\)
\(152\) − 8.21842i − 0.666602i
\(153\) − 1.12233i − 0.0907353i
\(154\) 14.6141 1.17764
\(155\) 0 0
\(156\) 3.00000 0.240192
\(157\) − 3.99404i − 0.318759i −0.987217 0.159379i \(-0.949051\pi\)
0.987217 0.159379i \(-0.0509493\pi\)
\(158\) 30.8864i 2.45719i
\(159\) −3.74568 −0.297052
\(160\) 0 0
\(161\) −10.1127 −0.796991
\(162\) − 2.12233i − 0.166746i
\(163\) 4.80141i 0.376075i 0.982162 + 0.188038i \(0.0602127\pi\)
−0.982162 + 0.188038i \(0.939787\pi\)
\(164\) 18.7125 1.46120
\(165\) 0 0
\(166\) −18.5408 −1.43904
\(167\) − 23.6600i − 1.83087i −0.402469 0.915434i \(-0.631848\pi\)
0.402469 0.915434i \(-0.368152\pi\)
\(168\) 4.66476i 0.359894i
\(169\) 11.5649 0.889611
\(170\) 0 0
\(171\) 7.67867 0.587203
\(172\) 3.13033i 0.238685i
\(173\) 14.5043i 1.10274i 0.834260 + 0.551371i \(0.185895\pi\)
−0.834260 + 0.551371i \(0.814105\pi\)
\(174\) 11.6820 0.885607
\(175\) 0 0
\(176\) 4.32433 0.325959
\(177\) − 9.15613i − 0.688217i
\(178\) − 21.3043i − 1.59682i
\(179\) −0.796746 −0.0595516 −0.0297758 0.999557i \(-0.509479\pi\)
−0.0297758 + 0.999557i \(0.509479\pi\)
\(180\) 0 0
\(181\) −14.2185 −1.05685 −0.528425 0.848980i \(-0.677217\pi\)
−0.528425 + 0.848980i \(0.677217\pi\)
\(182\) − 11.0809i − 0.821372i
\(183\) 1.39588i 0.103186i
\(184\) 2.48337 0.183076
\(185\) 0 0
\(186\) 10.1916 0.747282
\(187\) − 1.77318i − 0.129668i
\(188\) 10.3369i 0.753894i
\(189\) −4.35840 −0.317027
\(190\) 0 0
\(191\) −10.3697 −0.750324 −0.375162 0.926959i \(-0.622413\pi\)
−0.375162 + 0.926959i \(0.622413\pi\)
\(192\) 11.3975i 0.822544i
\(193\) 4.34712i 0.312913i 0.987685 + 0.156456i \(0.0500071\pi\)
−0.987685 + 0.156456i \(0.949993\pi\)
\(194\) 15.9174 1.14280
\(195\) 0 0
\(196\) 30.0407 2.14577
\(197\) − 1.87990i − 0.133937i −0.997755 0.0669686i \(-0.978667\pi\)
0.997755 0.0669686i \(-0.0213327\pi\)
\(198\) − 3.35309i − 0.238294i
\(199\) −4.26028 −0.302003 −0.151002 0.988534i \(-0.548250\pi\)
−0.151002 + 0.988534i \(0.548250\pi\)
\(200\) 0 0
\(201\) −3.86270 −0.272454
\(202\) 13.8338i 0.973344i
\(203\) − 23.9899i − 1.68376i
\(204\) 2.81066 0.196786
\(205\) 0 0
\(206\) 17.0821 1.19017
\(207\) 2.32027i 0.161270i
\(208\) − 3.27886i − 0.227348i
\(209\) 12.1316 0.839159
\(210\) 0 0
\(211\) −9.29671 −0.640012 −0.320006 0.947416i \(-0.603685\pi\)
−0.320006 + 0.947416i \(0.603685\pi\)
\(212\) − 9.38031i − 0.644242i
\(213\) − 10.6051i − 0.726649i
\(214\) −20.1143 −1.37499
\(215\) 0 0
\(216\) 1.07029 0.0728241
\(217\) − 20.9293i − 1.42077i
\(218\) − 24.7586i − 1.67686i
\(219\) −4.99672 −0.337647
\(220\) 0 0
\(221\) −1.34449 −0.0904400
\(222\) 13.5242i 0.907683i
\(223\) − 3.39758i − 0.227519i −0.993508 0.113759i \(-0.963711\pi\)
0.993508 0.113759i \(-0.0362893\pi\)
\(224\) 34.6475 2.31498
\(225\) 0 0
\(226\) −31.6766 −2.10710
\(227\) − 26.7973i − 1.77860i −0.457324 0.889300i \(-0.651192\pi\)
0.457324 0.889300i \(-0.348808\pi\)
\(228\) 19.2297i 1.27352i
\(229\) 1.18200 0.0781085 0.0390543 0.999237i \(-0.487565\pi\)
0.0390543 + 0.999237i \(0.487565\pi\)
\(230\) 0 0
\(231\) −6.88586 −0.453057
\(232\) 5.89121i 0.386777i
\(233\) 7.09483i 0.464798i 0.972621 + 0.232399i \(0.0746575\pi\)
−0.972621 + 0.232399i \(0.925342\pi\)
\(234\) −2.54243 −0.166204
\(235\) 0 0
\(236\) 22.9297 1.49260
\(237\) − 14.5531i − 0.945323i
\(238\) − 10.3816i − 0.672937i
\(239\) −0.0427926 −0.00276802 −0.00138401 0.999999i \(-0.500441\pi\)
−0.00138401 + 0.999999i \(0.500441\pi\)
\(240\) 0 0
\(241\) −11.7711 −0.758242 −0.379121 0.925347i \(-0.623774\pi\)
−0.379121 + 0.925347i \(0.623774\pi\)
\(242\) 18.0481i 1.16018i
\(243\) 1.00000i 0.0641500i
\(244\) −3.49570 −0.223789
\(245\) 0 0
\(246\) −15.8584 −1.01109
\(247\) − 9.19859i − 0.585292i
\(248\) 5.13961i 0.326365i
\(249\) 8.73603 0.553623
\(250\) 0 0
\(251\) 17.4764 1.10310 0.551550 0.834142i \(-0.314036\pi\)
0.551550 + 0.834142i \(0.314036\pi\)
\(252\) − 10.9147i − 0.687564i
\(253\) 3.66581i 0.230468i
\(254\) 36.0705 2.26326
\(255\) 0 0
\(256\) 5.20057 0.325036
\(257\) − 15.4671i − 0.964814i −0.875947 0.482407i \(-0.839763\pi\)
0.875947 0.482407i \(-0.160237\pi\)
\(258\) − 2.65288i − 0.165161i
\(259\) 27.7731 1.72574
\(260\) 0 0
\(261\) −5.50430 −0.340708
\(262\) 0.697977i 0.0431211i
\(263\) 1.44759i 0.0892624i 0.999004 + 0.0446312i \(0.0142113\pi\)
−0.999004 + 0.0446312i \(0.985789\pi\)
\(264\) 1.69096 0.104071
\(265\) 0 0
\(266\) 71.0276 4.35498
\(267\) 10.0381i 0.614323i
\(268\) − 9.67336i − 0.590895i
\(269\) 8.70617 0.530825 0.265412 0.964135i \(-0.414492\pi\)
0.265412 + 0.964135i \(0.414492\pi\)
\(270\) 0 0
\(271\) −28.6338 −1.73938 −0.869689 0.493600i \(-0.835681\pi\)
−0.869689 + 0.493600i \(0.835681\pi\)
\(272\) − 3.07192i − 0.186263i
\(273\) 5.22110i 0.315995i
\(274\) −9.68359 −0.585007
\(275\) 0 0
\(276\) −5.81066 −0.349761
\(277\) − 5.30210i − 0.318572i −0.987232 0.159286i \(-0.949081\pi\)
0.987232 0.159286i \(-0.0509192\pi\)
\(278\) − 10.0471i − 0.602587i
\(279\) −4.80206 −0.287492
\(280\) 0 0
\(281\) −24.0832 −1.43668 −0.718342 0.695691i \(-0.755098\pi\)
−0.718342 + 0.695691i \(0.755098\pi\)
\(282\) − 8.76025i − 0.521665i
\(283\) − 27.1143i − 1.61178i −0.592066 0.805889i \(-0.701688\pi\)
0.592066 0.805889i \(-0.298312\pi\)
\(284\) 26.5583 1.57595
\(285\) 0 0
\(286\) −4.01680 −0.237518
\(287\) 32.5666i 1.92234i
\(288\) − 7.94959i − 0.468434i
\(289\) 15.7404 0.925904
\(290\) 0 0
\(291\) −7.49996 −0.439656
\(292\) − 12.5133i − 0.732284i
\(293\) − 20.3016i − 1.18603i −0.805190 0.593017i \(-0.797937\pi\)
0.805190 0.593017i \(-0.202063\pi\)
\(294\) −25.4588 −1.48479
\(295\) 0 0
\(296\) −6.82024 −0.396418
\(297\) 1.57991i 0.0916754i
\(298\) 9.92497i 0.574938i
\(299\) 2.77955 0.160745
\(300\) 0 0
\(301\) −5.44792 −0.314013
\(302\) 13.8838i 0.798925i
\(303\) − 6.51821i − 0.374462i
\(304\) 21.0172 1.20542
\(305\) 0 0
\(306\) −2.38197 −0.136168
\(307\) 2.51330i 0.143442i 0.997425 + 0.0717208i \(0.0228491\pi\)
−0.997425 + 0.0717208i \(0.977151\pi\)
\(308\) − 17.2443i − 0.982583i
\(309\) −8.04876 −0.457878
\(310\) 0 0
\(311\) 4.82417 0.273554 0.136777 0.990602i \(-0.456326\pi\)
0.136777 + 0.990602i \(0.456326\pi\)
\(312\) − 1.28215i − 0.0725872i
\(313\) − 5.98812i − 0.338468i −0.985576 0.169234i \(-0.945871\pi\)
0.985576 0.169234i \(-0.0541294\pi\)
\(314\) −8.47668 −0.478366
\(315\) 0 0
\(316\) 36.4452 2.05020
\(317\) − 17.3531i − 0.974646i −0.873222 0.487323i \(-0.837973\pi\)
0.873222 0.487323i \(-0.162027\pi\)
\(318\) 7.94959i 0.445791i
\(319\) −8.69627 −0.486898
\(320\) 0 0
\(321\) 9.47745 0.528980
\(322\) 21.4625i 1.19606i
\(323\) − 8.61803i − 0.479520i
\(324\) −2.50430 −0.139128
\(325\) 0 0
\(326\) 10.1902 0.564383
\(327\) 11.6657i 0.645117i
\(328\) − 7.99737i − 0.441581i
\(329\) −17.9899 −0.991818
\(330\) 0 0
\(331\) 20.3355 1.11774 0.558870 0.829255i \(-0.311235\pi\)
0.558870 + 0.829255i \(0.311235\pi\)
\(332\) 21.8776i 1.20069i
\(333\) − 6.37232i − 0.349201i
\(334\) −50.2145 −2.74761
\(335\) 0 0
\(336\) −11.9293 −0.650797
\(337\) − 34.2571i − 1.86610i −0.359741 0.933052i \(-0.617135\pi\)
0.359741 0.933052i \(-0.382865\pi\)
\(338\) − 24.5447i − 1.33505i
\(339\) 14.9254 0.810635
\(340\) 0 0
\(341\) −7.58680 −0.410848
\(342\) − 16.2967i − 0.881225i
\(343\) 21.7731i 1.17564i
\(344\) 1.33784 0.0721318
\(345\) 0 0
\(346\) 30.7830 1.65490
\(347\) 0.0493616i 0.00264987i 0.999999 + 0.00132494i \(0.000421740\pi\)
−0.999999 + 0.00132494i \(0.999578\pi\)
\(348\) − 13.7844i − 0.738922i
\(349\) 7.47437 0.400094 0.200047 0.979786i \(-0.435891\pi\)
0.200047 + 0.979786i \(0.435891\pi\)
\(350\) 0 0
\(351\) 1.19794 0.0639413
\(352\) − 12.5596i − 0.669429i
\(353\) − 18.5650i − 0.988115i −0.869429 0.494057i \(-0.835513\pi\)
0.869429 0.494057i \(-0.164487\pi\)
\(354\) −19.4324 −1.03282
\(355\) 0 0
\(356\) −25.1385 −1.33234
\(357\) 4.89158i 0.258890i
\(358\) 1.69096i 0.0893700i
\(359\) −10.6044 −0.559681 −0.279841 0.960046i \(-0.590282\pi\)
−0.279841 + 0.960046i \(0.590282\pi\)
\(360\) 0 0
\(361\) 39.9620 2.10327
\(362\) 30.1763i 1.58603i
\(363\) − 8.50390i − 0.446339i
\(364\) −13.0752 −0.685327
\(365\) 0 0
\(366\) 2.96252 0.154854
\(367\) 11.6176i 0.606435i 0.952921 + 0.303218i \(0.0980609\pi\)
−0.952921 + 0.303218i \(0.901939\pi\)
\(368\) 6.35078i 0.331057i
\(369\) 7.47214 0.388984
\(370\) 0 0
\(371\) 16.3252 0.847561
\(372\) − 12.0258i − 0.623509i
\(373\) − 23.1272i − 1.19748i −0.800942 0.598742i \(-0.795668\pi\)
0.800942 0.598742i \(-0.204332\pi\)
\(374\) −3.76328 −0.194595
\(375\) 0 0
\(376\) 4.41779 0.227830
\(377\) 6.59382i 0.339599i
\(378\) 9.24998i 0.475768i
\(379\) −24.8912 −1.27857 −0.639287 0.768968i \(-0.720770\pi\)
−0.639287 + 0.768968i \(0.720770\pi\)
\(380\) 0 0
\(381\) −16.9957 −0.870714
\(382\) 22.0079i 1.12602i
\(383\) − 15.7221i − 0.803363i −0.915780 0.401681i \(-0.868426\pi\)
0.915780 0.401681i \(-0.131574\pi\)
\(384\) 8.29014 0.423054
\(385\) 0 0
\(386\) 9.22604 0.469593
\(387\) 1.24998i 0.0635401i
\(388\) − 18.7822i − 0.953519i
\(389\) 8.50633 0.431288 0.215644 0.976472i \(-0.430815\pi\)
0.215644 + 0.976472i \(0.430815\pi\)
\(390\) 0 0
\(391\) 2.60412 0.131696
\(392\) − 12.8389i − 0.648460i
\(393\) − 0.328872i − 0.0165894i
\(394\) −3.98977 −0.201002
\(395\) 0 0
\(396\) −3.95656 −0.198824
\(397\) − 26.2549i − 1.31770i −0.752276 0.658848i \(-0.771044\pi\)
0.752276 0.658848i \(-0.228956\pi\)
\(398\) 9.04174i 0.453222i
\(399\) −33.4667 −1.67543
\(400\) 0 0
\(401\) −25.2815 −1.26250 −0.631250 0.775579i \(-0.717458\pi\)
−0.631250 + 0.775579i \(0.717458\pi\)
\(402\) 8.19794i 0.408876i
\(403\) 5.75258i 0.286556i
\(404\) 16.3236 0.812127
\(405\) 0 0
\(406\) −50.9147 −2.52685
\(407\) − 10.0677i − 0.499035i
\(408\) − 1.20122i − 0.0594695i
\(409\) −33.7932 −1.67097 −0.835483 0.549517i \(-0.814812\pi\)
−0.835483 + 0.549517i \(0.814812\pi\)
\(410\) 0 0
\(411\) 4.56271 0.225062
\(412\) − 20.1565i − 0.993039i
\(413\) 39.9061i 1.96365i
\(414\) 4.92439 0.242021
\(415\) 0 0
\(416\) −9.52313 −0.466910
\(417\) 4.73400i 0.231825i
\(418\) − 25.7473i − 1.25934i
\(419\) −7.93332 −0.387568 −0.193784 0.981044i \(-0.562076\pi\)
−0.193784 + 0.981044i \(0.562076\pi\)
\(420\) 0 0
\(421\) 7.48795 0.364941 0.182470 0.983211i \(-0.441591\pi\)
0.182470 + 0.983211i \(0.441591\pi\)
\(422\) 19.7307i 0.960476i
\(423\) 4.12765i 0.200693i
\(424\) −4.00897 −0.194693
\(425\) 0 0
\(426\) −22.5076 −1.09049
\(427\) − 6.08380i − 0.294416i
\(428\) 23.7344i 1.14724i
\(429\) 1.89263 0.0913771
\(430\) 0 0
\(431\) 23.3470 1.12459 0.562294 0.826937i \(-0.309919\pi\)
0.562294 + 0.826937i \(0.309919\pi\)
\(432\) 2.73708i 0.131688i
\(433\) − 3.35677i − 0.161316i −0.996742 0.0806581i \(-0.974298\pi\)
0.996742 0.0806581i \(-0.0257022\pi\)
\(434\) −44.4190 −2.13218
\(435\) 0 0
\(436\) −29.2145 −1.39912
\(437\) 17.8166i 0.852285i
\(438\) 10.6047i 0.506712i
\(439\) 9.48402 0.452648 0.226324 0.974052i \(-0.427329\pi\)
0.226324 + 0.974052i \(0.427329\pi\)
\(440\) 0 0
\(441\) 11.9957 0.571222
\(442\) 2.85345i 0.135725i
\(443\) 9.65446i 0.458697i 0.973344 + 0.229349i \(0.0736596\pi\)
−0.973344 + 0.229349i \(0.926340\pi\)
\(444\) 15.9582 0.757342
\(445\) 0 0
\(446\) −7.21080 −0.341441
\(447\) − 4.67644i − 0.221188i
\(448\) − 49.6749i − 2.34692i
\(449\) −31.5260 −1.48780 −0.743902 0.668289i \(-0.767027\pi\)
−0.743902 + 0.668289i \(0.767027\pi\)
\(450\) 0 0
\(451\) 11.8053 0.555888
\(452\) 37.3776i 1.75809i
\(453\) − 6.54178i − 0.307360i
\(454\) −56.8729 −2.66918
\(455\) 0 0
\(456\) 8.21842 0.384863
\(457\) 9.94467i 0.465192i 0.972573 + 0.232596i \(0.0747220\pi\)
−0.972573 + 0.232596i \(0.925278\pi\)
\(458\) − 2.50859i − 0.117219i
\(459\) 1.12233 0.0523860
\(460\) 0 0
\(461\) −23.6836 −1.10305 −0.551527 0.834157i \(-0.685955\pi\)
−0.551527 + 0.834157i \(0.685955\pi\)
\(462\) 14.6141i 0.679909i
\(463\) 27.6504i 1.28502i 0.766277 + 0.642511i \(0.222107\pi\)
−0.766277 + 0.642511i \(0.777893\pi\)
\(464\) −15.0657 −0.699409
\(465\) 0 0
\(466\) 15.0576 0.697530
\(467\) − 4.35840i − 0.201683i −0.994903 0.100841i \(-0.967847\pi\)
0.994903 0.100841i \(-0.0321535\pi\)
\(468\) 3.00000i 0.138675i
\(469\) 16.8352 0.777377
\(470\) 0 0
\(471\) 3.99404 0.184035
\(472\) − 9.79973i − 0.451069i
\(473\) 1.97485i 0.0908038i
\(474\) −30.8864 −1.41866
\(475\) 0 0
\(476\) −12.2500 −0.561477
\(477\) − 3.74568i − 0.171503i
\(478\) 0.0908201i 0.00415401i
\(479\) −12.8143 −0.585499 −0.292750 0.956189i \(-0.594570\pi\)
−0.292750 + 0.956189i \(0.594570\pi\)
\(480\) 0 0
\(481\) −7.63365 −0.348064
\(482\) 24.9822i 1.13791i
\(483\) − 10.1127i − 0.460143i
\(484\) 21.2963 0.968014
\(485\) 0 0
\(486\) 2.12233 0.0962710
\(487\) 6.92209i 0.313670i 0.987625 + 0.156835i \(0.0501290\pi\)
−0.987625 + 0.156835i \(0.949871\pi\)
\(488\) 1.49400i 0.0676301i
\(489\) −4.80141 −0.217127
\(490\) 0 0
\(491\) 26.8161 1.21020 0.605098 0.796151i \(-0.293134\pi\)
0.605098 + 0.796151i \(0.293134\pi\)
\(492\) 18.7125i 0.843623i
\(493\) 6.17766i 0.278228i
\(494\) −19.5225 −0.878358
\(495\) 0 0
\(496\) −13.1436 −0.590167
\(497\) 46.2213i 2.07331i
\(498\) − 18.5408i − 0.830832i
\(499\) −2.75460 −0.123313 −0.0616565 0.998097i \(-0.519638\pi\)
−0.0616565 + 0.998097i \(0.519638\pi\)
\(500\) 0 0
\(501\) 23.6600 1.05705
\(502\) − 37.0907i − 1.65544i
\(503\) 15.7102i 0.700483i 0.936659 + 0.350241i \(0.113900\pi\)
−0.936659 + 0.350241i \(0.886100\pi\)
\(504\) −4.66476 −0.207785
\(505\) 0 0
\(506\) 7.78008 0.345867
\(507\) 11.5649i 0.513617i
\(508\) − 42.5622i − 1.88839i
\(509\) 41.0884 1.82121 0.910606 0.413277i \(-0.135616\pi\)
0.910606 + 0.413277i \(0.135616\pi\)
\(510\) 0 0
\(511\) 21.7777 0.963388
\(512\) − 27.6176i − 1.22054i
\(513\) 7.67867i 0.339022i
\(514\) −32.8264 −1.44791
\(515\) 0 0
\(516\) −3.13033 −0.137805
\(517\) 6.52129i 0.286806i
\(518\) − 58.9438i − 2.58984i
\(519\) −14.5043 −0.636668
\(520\) 0 0
\(521\) 36.1710 1.58468 0.792341 0.610079i \(-0.208862\pi\)
0.792341 + 0.610079i \(0.208862\pi\)
\(522\) 11.6820i 0.511305i
\(523\) 28.0490i 1.22650i 0.789891 + 0.613248i \(0.210137\pi\)
−0.789891 + 0.613248i \(0.789863\pi\)
\(524\) 0.823595 0.0359789
\(525\) 0 0
\(526\) 3.07228 0.133958
\(527\) 5.38951i 0.234771i
\(528\) 4.32433i 0.188192i
\(529\) 17.6163 0.765927
\(530\) 0 0
\(531\) 9.15613 0.397342
\(532\) − 83.8108i − 3.63366i
\(533\) − 8.95117i − 0.387718i
\(534\) 21.3043 0.921925
\(535\) 0 0
\(536\) −4.13422 −0.178571
\(537\) − 0.796746i − 0.0343821i
\(538\) − 18.4774i − 0.796617i
\(539\) 18.9520 0.816321
\(540\) 0 0
\(541\) 3.45822 0.148681 0.0743403 0.997233i \(-0.476315\pi\)
0.0743403 + 0.997233i \(0.476315\pi\)
\(542\) 60.7704i 2.61031i
\(543\) − 14.2185i − 0.610173i
\(544\) −8.92209 −0.382531
\(545\) 0 0
\(546\) 11.0809 0.474220
\(547\) − 13.9635i − 0.597034i −0.954404 0.298517i \(-0.903508\pi\)
0.954404 0.298517i \(-0.0964920\pi\)
\(548\) 11.4264i 0.488111i
\(549\) −1.39588 −0.0595747
\(550\) 0 0
\(551\) −42.2657 −1.80058
\(552\) 2.48337i 0.105699i
\(553\) 63.4281i 2.69724i
\(554\) −11.2528 −0.478086
\(555\) 0 0
\(556\) −11.8554 −0.502779
\(557\) 6.59585i 0.279475i 0.990189 + 0.139738i \(0.0446259\pi\)
−0.990189 + 0.139738i \(0.955374\pi\)
\(558\) 10.1916i 0.431444i
\(559\) 1.49740 0.0633334
\(560\) 0 0
\(561\) 1.77318 0.0748638
\(562\) 51.1126i 2.15605i
\(563\) − 16.0218i − 0.675238i −0.941283 0.337619i \(-0.890379\pi\)
0.941283 0.337619i \(-0.109621\pi\)
\(564\) −10.3369 −0.435261
\(565\) 0 0
\(566\) −57.5456 −2.41882
\(567\) − 4.35840i − 0.183036i
\(568\) − 11.3505i − 0.476258i
\(569\) −23.0914 −0.968042 −0.484021 0.875056i \(-0.660824\pi\)
−0.484021 + 0.875056i \(0.660824\pi\)
\(570\) 0 0
\(571\) −34.0102 −1.42328 −0.711640 0.702544i \(-0.752047\pi\)
−0.711640 + 0.702544i \(0.752047\pi\)
\(572\) 4.73972i 0.198178i
\(573\) − 10.3697i − 0.433200i
\(574\) 69.1171 2.88489
\(575\) 0 0
\(576\) −11.3975 −0.474896
\(577\) 6.46727i 0.269236i 0.990898 + 0.134618i \(0.0429807\pi\)
−0.990898 + 0.134618i \(0.957019\pi\)
\(578\) − 33.4063i − 1.38952i
\(579\) −4.34712 −0.180660
\(580\) 0 0
\(581\) −38.0751 −1.57962
\(582\) 15.9174i 0.659798i
\(583\) − 5.91782i − 0.245091i
\(584\) −5.34794 −0.221299
\(585\) 0 0
\(586\) −43.0868 −1.77990
\(587\) − 38.5012i − 1.58911i −0.607189 0.794557i \(-0.707703\pi\)
0.607189 0.794557i \(-0.292297\pi\)
\(588\) 30.0407i 1.23886i
\(589\) −36.8735 −1.51934
\(590\) 0 0
\(591\) 1.87990 0.0773287
\(592\) − 17.4416i − 0.716843i
\(593\) 4.93069i 0.202479i 0.994862 + 0.101240i \(0.0322808\pi\)
−0.994862 + 0.101240i \(0.967719\pi\)
\(594\) 3.35309 0.137579
\(595\) 0 0
\(596\) 11.7112 0.479710
\(597\) − 4.26028i − 0.174362i
\(598\) − 5.89913i − 0.241233i
\(599\) 35.0268 1.43116 0.715578 0.698533i \(-0.246164\pi\)
0.715578 + 0.698533i \(0.246164\pi\)
\(600\) 0 0
\(601\) −4.90570 −0.200108 −0.100054 0.994982i \(-0.531901\pi\)
−0.100054 + 0.994982i \(0.531901\pi\)
\(602\) 11.5623i 0.471244i
\(603\) − 3.86270i − 0.157301i
\(604\) 16.3826 0.666597
\(605\) 0 0
\(606\) −13.8338 −0.561961
\(607\) − 48.6955i − 1.97649i −0.152884 0.988244i \(-0.548856\pi\)
0.152884 0.988244i \(-0.451144\pi\)
\(608\) − 61.0423i − 2.47559i
\(609\) 23.9899 0.972122
\(610\) 0 0
\(611\) 4.94467 0.200040
\(612\) 2.81066i 0.113614i
\(613\) 17.2139i 0.695262i 0.937631 + 0.347631i \(0.113014\pi\)
−0.937631 + 0.347631i \(0.886986\pi\)
\(614\) 5.33406 0.215265
\(615\) 0 0
\(616\) −7.36988 −0.296941
\(617\) 31.6092i 1.27254i 0.771468 + 0.636268i \(0.219523\pi\)
−0.771468 + 0.636268i \(0.780477\pi\)
\(618\) 17.0821i 0.687145i
\(619\) 22.8196 0.917198 0.458599 0.888643i \(-0.348351\pi\)
0.458599 + 0.888643i \(0.348351\pi\)
\(620\) 0 0
\(621\) −2.32027 −0.0931094
\(622\) − 10.2385i − 0.410526i
\(623\) − 43.7502i − 1.75281i
\(624\) 3.27886 0.131259
\(625\) 0 0
\(626\) −12.7088 −0.507945
\(627\) 12.1316i 0.484489i
\(628\) 10.0023i 0.399134i
\(629\) −7.15186 −0.285163
\(630\) 0 0
\(631\) 38.0499 1.51474 0.757372 0.652984i \(-0.226483\pi\)
0.757372 + 0.652984i \(0.226483\pi\)
\(632\) − 15.5760i − 0.619581i
\(633\) − 9.29671i − 0.369511i
\(634\) −36.8290 −1.46267
\(635\) 0 0
\(636\) 9.38031 0.371953
\(637\) − 14.3701i − 0.569363i
\(638\) 18.4564i 0.730696i
\(639\) 10.6051 0.419531
\(640\) 0 0
\(641\) 26.5108 1.04711 0.523557 0.851991i \(-0.324605\pi\)
0.523557 + 0.851991i \(0.324605\pi\)
\(642\) − 20.1143i − 0.793849i
\(643\) − 36.0014i − 1.41976i −0.704324 0.709879i \(-0.748750\pi\)
0.704324 0.709879i \(-0.251250\pi\)
\(644\) 25.3252 0.997952
\(645\) 0 0
\(646\) −18.2903 −0.719624
\(647\) − 41.7840i − 1.64270i −0.570426 0.821349i \(-0.693222\pi\)
0.570426 0.821349i \(-0.306778\pi\)
\(648\) 1.07029i 0.0420450i
\(649\) 14.4658 0.567833
\(650\) 0 0
\(651\) 20.9293 0.820284
\(652\) − 12.0242i − 0.470903i
\(653\) 21.4404i 0.839027i 0.907749 + 0.419513i \(0.137799\pi\)
−0.907749 + 0.419513i \(0.862201\pi\)
\(654\) 24.7586 0.968137
\(655\) 0 0
\(656\) 20.4519 0.798511
\(657\) − 4.99672i − 0.194940i
\(658\) 38.1807i 1.48844i
\(659\) 16.8319 0.655678 0.327839 0.944734i \(-0.393680\pi\)
0.327839 + 0.944734i \(0.393680\pi\)
\(660\) 0 0
\(661\) 2.03846 0.0792868 0.0396434 0.999214i \(-0.487378\pi\)
0.0396434 + 0.999214i \(0.487378\pi\)
\(662\) − 43.1587i − 1.67741i
\(663\) − 1.34449i − 0.0522156i
\(664\) 9.35010 0.362854
\(665\) 0 0
\(666\) −13.5242 −0.524051
\(667\) − 12.7715i − 0.494514i
\(668\) 59.2518i 2.29252i
\(669\) 3.39758 0.131358
\(670\) 0 0
\(671\) −2.20536 −0.0851369
\(672\) 34.6475i 1.33656i
\(673\) 35.3776i 1.36371i 0.731489 + 0.681854i \(0.238826\pi\)
−0.731489 + 0.681854i \(0.761174\pi\)
\(674\) −72.7050 −2.80049
\(675\) 0 0
\(676\) −28.9621 −1.11393
\(677\) − 44.4173i − 1.70710i −0.521014 0.853548i \(-0.674446\pi\)
0.521014 0.853548i \(-0.325554\pi\)
\(678\) − 31.6766i − 1.21653i
\(679\) 32.6879 1.25444
\(680\) 0 0
\(681\) 26.7973 1.02688
\(682\) 16.1017i 0.616567i
\(683\) 42.4952i 1.62603i 0.582241 + 0.813016i \(0.302176\pi\)
−0.582241 + 0.813016i \(0.697824\pi\)
\(684\) −19.2297 −0.735266
\(685\) 0 0
\(686\) 46.2098 1.76430
\(687\) 1.18200i 0.0450960i
\(688\) 3.42130i 0.130436i
\(689\) −4.48710 −0.170945
\(690\) 0 0
\(691\) 28.0808 1.06825 0.534123 0.845407i \(-0.320642\pi\)
0.534123 + 0.845407i \(0.320642\pi\)
\(692\) − 36.3231i − 1.38080i
\(693\) − 6.88586i − 0.261572i
\(694\) 0.104762 0.00397671
\(695\) 0 0
\(696\) −5.89121 −0.223306
\(697\) − 8.38623i − 0.317651i
\(698\) − 15.8631i − 0.600427i
\(699\) −7.09483 −0.268351
\(700\) 0 0
\(701\) 46.4314 1.75369 0.876845 0.480772i \(-0.159644\pi\)
0.876845 + 0.480772i \(0.159644\pi\)
\(702\) − 2.54243i − 0.0959578i
\(703\) − 48.9309i − 1.84547i
\(704\) −18.0070 −0.678664
\(705\) 0 0
\(706\) −39.4011 −1.48288
\(707\) 28.4090i 1.06843i
\(708\) 22.9297i 0.861750i
\(709\) −48.6331 −1.82646 −0.913228 0.407450i \(-0.866418\pi\)
−0.913228 + 0.407450i \(0.866418\pi\)
\(710\) 0 0
\(711\) 14.5531 0.545782
\(712\) 10.7437i 0.402638i
\(713\) − 11.1421i − 0.417275i
\(714\) 10.3816 0.388520
\(715\) 0 0
\(716\) 1.99529 0.0745675
\(717\) − 0.0427926i − 0.00159812i
\(718\) 22.5062i 0.839923i
\(719\) 21.1954 0.790456 0.395228 0.918583i \(-0.370666\pi\)
0.395228 + 0.918583i \(0.370666\pi\)
\(720\) 0 0
\(721\) 35.0797 1.30644
\(722\) − 84.8128i − 3.15641i
\(723\) − 11.7711i − 0.437771i
\(724\) 35.6073 1.32333
\(725\) 0 0
\(726\) −18.0481 −0.669828
\(727\) − 13.5911i − 0.504066i −0.967719 0.252033i \(-0.918901\pi\)
0.967719 0.252033i \(-0.0810992\pi\)
\(728\) 5.58810i 0.207109i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 1.40290 0.0518880
\(732\) − 3.49570i − 0.129205i
\(733\) 18.6531i 0.688967i 0.938792 + 0.344484i \(0.111946\pi\)
−0.938792 + 0.344484i \(0.888054\pi\)
\(734\) 24.6565 0.910088
\(735\) 0 0
\(736\) 18.4452 0.679900
\(737\) − 6.10270i − 0.224796i
\(738\) − 15.8584i − 0.583754i
\(739\) −26.4452 −0.972803 −0.486401 0.873735i \(-0.661691\pi\)
−0.486401 + 0.873735i \(0.661691\pi\)
\(740\) 0 0
\(741\) 9.19859 0.337919
\(742\) − 34.6475i − 1.27195i
\(743\) − 29.0191i − 1.06461i −0.846553 0.532304i \(-0.821326\pi\)
0.846553 0.532304i \(-0.178674\pi\)
\(744\) −5.13961 −0.188427
\(745\) 0 0
\(746\) −49.0837 −1.79708
\(747\) 8.73603i 0.319635i
\(748\) 4.44058i 0.162364i
\(749\) −41.3065 −1.50931
\(750\) 0 0
\(751\) 15.9489 0.581985 0.290992 0.956725i \(-0.406015\pi\)
0.290992 + 0.956725i \(0.406015\pi\)
\(752\) 11.2977i 0.411985i
\(753\) 17.4764i 0.636875i
\(754\) 13.9943 0.509642
\(755\) 0 0
\(756\) 10.9147 0.396965
\(757\) 16.4183i 0.596734i 0.954451 + 0.298367i \(0.0964419\pi\)
−0.954451 + 0.298367i \(0.903558\pi\)
\(758\) 52.8274i 1.91878i
\(759\) −3.66581 −0.133061
\(760\) 0 0
\(761\) −49.2715 −1.78609 −0.893045 0.449967i \(-0.851436\pi\)
−0.893045 + 0.449967i \(0.851436\pi\)
\(762\) 36.0705i 1.30670i
\(763\) − 50.8440i − 1.84068i
\(764\) 25.9688 0.939518
\(765\) 0 0
\(766\) −33.3676 −1.20562
\(767\) − 10.9685i − 0.396049i
\(768\) 5.20057i 0.187660i
\(769\) −49.4236 −1.78226 −0.891129 0.453749i \(-0.850086\pi\)
−0.891129 + 0.453749i \(0.850086\pi\)
\(770\) 0 0
\(771\) 15.4671 0.557036
\(772\) − 10.8865i − 0.391814i
\(773\) 25.9084i 0.931860i 0.884822 + 0.465930i \(0.154280\pi\)
−0.884822 + 0.465930i \(0.845720\pi\)
\(774\) 2.65288 0.0953557
\(775\) 0 0
\(776\) −8.02715 −0.288158
\(777\) 27.7731i 0.996355i
\(778\) − 18.0533i − 0.647241i
\(779\) 57.3761 2.05571
\(780\) 0 0
\(781\) 16.7551 0.599543
\(782\) − 5.52681i − 0.197638i
\(783\) − 5.50430i − 0.196708i
\(784\) 32.8331 1.17261
\(785\) 0 0
\(786\) −0.697977 −0.0248960
\(787\) − 30.7155i − 1.09489i −0.836842 0.547444i \(-0.815601\pi\)
0.836842 0.547444i \(-0.184399\pi\)
\(788\) 4.70783i 0.167710i
\(789\) −1.44759 −0.0515357
\(790\) 0 0
\(791\) −65.0508 −2.31294
\(792\) 1.69096i 0.0600857i
\(793\) 1.67218i 0.0593808i
\(794\) −55.7216 −1.97749
\(795\) 0 0
\(796\) 10.6690 0.378154
\(797\) 33.7141i 1.19421i 0.802161 + 0.597107i \(0.203683\pi\)
−0.802161 + 0.597107i \(0.796317\pi\)
\(798\) 71.0276i 2.51435i
\(799\) 4.63260 0.163890
\(800\) 0 0
\(801\) −10.0381 −0.354680
\(802\) 53.6559i 1.89465i
\(803\) − 7.89434i − 0.278585i
\(804\) 9.67336 0.341153
\(805\) 0 0
\(806\) 12.2089 0.430040
\(807\) 8.70617i 0.306472i
\(808\) − 6.97639i − 0.245429i
\(809\) 0.923014 0.0324514 0.0162257 0.999868i \(-0.494835\pi\)
0.0162257 + 0.999868i \(0.494835\pi\)
\(810\) 0 0
\(811\) 26.7815 0.940424 0.470212 0.882553i \(-0.344177\pi\)
0.470212 + 0.882553i \(0.344177\pi\)
\(812\) 60.0780i 2.10832i
\(813\) − 28.6338i − 1.00423i
\(814\) −21.3669 −0.748910
\(815\) 0 0
\(816\) 3.07192 0.107539
\(817\) 9.59820i 0.335799i
\(818\) 71.7204i 2.50765i
\(819\) −5.22110 −0.182440
\(820\) 0 0
\(821\) −36.8489 −1.28604 −0.643018 0.765851i \(-0.722318\pi\)
−0.643018 + 0.765851i \(0.722318\pi\)
\(822\) − 9.68359i − 0.337754i
\(823\) 20.4262i 0.712012i 0.934484 + 0.356006i \(0.115862\pi\)
−0.934484 + 0.356006i \(0.884138\pi\)
\(824\) −8.61452 −0.300101
\(825\) 0 0
\(826\) 84.6940 2.94688
\(827\) − 4.33017i − 0.150575i −0.997162 0.0752874i \(-0.976013\pi\)
0.997162 0.0752874i \(-0.0239874\pi\)
\(828\) − 5.81066i − 0.201934i
\(829\) 0.614348 0.0213372 0.0106686 0.999943i \(-0.496604\pi\)
0.0106686 + 0.999943i \(0.496604\pi\)
\(830\) 0 0
\(831\) 5.30210 0.183928
\(832\) 13.6535i 0.473351i
\(833\) − 13.4631i − 0.466470i
\(834\) 10.0471 0.347904
\(835\) 0 0
\(836\) −30.3811 −1.05075
\(837\) − 4.80206i − 0.165983i
\(838\) 16.8372i 0.581630i
\(839\) −5.39981 −0.186422 −0.0932111 0.995646i \(-0.529713\pi\)
−0.0932111 + 0.995646i \(0.529713\pi\)
\(840\) 0 0
\(841\) 1.29731 0.0447349
\(842\) − 15.8919i − 0.547672i
\(843\) − 24.0832i − 0.829469i
\(844\) 23.2817 0.801391
\(845\) 0 0
\(846\) 8.76025 0.301183
\(847\) 37.0634i 1.27351i
\(848\) − 10.2522i − 0.352063i
\(849\) 27.1143 0.930561
\(850\) 0 0
\(851\) 14.7855 0.506841
\(852\) 26.5583i 0.909874i
\(853\) − 39.7935i − 1.36250i −0.732049 0.681252i \(-0.761436\pi\)
0.732049 0.681252i \(-0.238564\pi\)
\(854\) −12.9119 −0.441835
\(855\) 0 0
\(856\) 10.1436 0.346702
\(857\) 42.8643i 1.46422i 0.681188 + 0.732109i \(0.261464\pi\)
−0.681188 + 0.732109i \(0.738536\pi\)
\(858\) − 4.01680i − 0.137131i
\(859\) 4.56494 0.155754 0.0778769 0.996963i \(-0.475186\pi\)
0.0778769 + 0.996963i \(0.475186\pi\)
\(860\) 0 0
\(861\) −32.5666 −1.10987
\(862\) − 49.5502i − 1.68769i
\(863\) − 37.6273i − 1.28085i −0.768022 0.640424i \(-0.778759\pi\)
0.768022 0.640424i \(-0.221241\pi\)
\(864\) 7.94959 0.270450
\(865\) 0 0
\(866\) −7.12419 −0.242090
\(867\) 15.7404i 0.534571i
\(868\) 52.4133i 1.77902i
\(869\) 22.9925 0.779966
\(870\) 0 0
\(871\) −4.62728 −0.156790
\(872\) 12.4857i 0.422821i
\(873\) − 7.49996i − 0.253835i
\(874\) 37.8128 1.27904
\(875\) 0 0
\(876\) 12.5133 0.422784
\(877\) − 52.4041i − 1.76956i −0.466008 0.884781i \(-0.654308\pi\)
0.466008 0.884781i \(-0.345692\pi\)
\(878\) − 20.1283i − 0.679296i
\(879\) 20.3016 0.684757
\(880\) 0 0
\(881\) −7.93778 −0.267430 −0.133715 0.991020i \(-0.542691\pi\)
−0.133715 + 0.991020i \(0.542691\pi\)
\(882\) − 25.4588i − 0.857242i
\(883\) 31.6919i 1.06652i 0.845952 + 0.533259i \(0.179033\pi\)
−0.845952 + 0.533259i \(0.820967\pi\)
\(884\) 3.36700 0.113244
\(885\) 0 0
\(886\) 20.4900 0.688374
\(887\) − 2.82104i − 0.0947213i −0.998878 0.0473606i \(-0.984919\pi\)
0.998878 0.0473606i \(-0.0150810\pi\)
\(888\) − 6.82024i − 0.228872i
\(889\) 74.0739 2.48436
\(890\) 0 0
\(891\) −1.57991 −0.0529288
\(892\) 8.50856i 0.284888i
\(893\) 31.6949i 1.06063i
\(894\) −9.92497 −0.331941
\(895\) 0 0
\(896\) −36.1318 −1.20708
\(897\) 2.77955i 0.0928064i
\(898\) 66.9087i 2.23277i
\(899\) 26.4320 0.881556
\(900\) 0 0
\(901\) −4.20390 −0.140052
\(902\) − 25.0547i − 0.834231i
\(903\) − 5.44792i − 0.181295i
\(904\) 15.9745 0.531304
\(905\) 0 0
\(906\) −13.8838 −0.461259
\(907\) 44.1799i 1.46697i 0.679705 + 0.733485i \(0.262108\pi\)
−0.679705 + 0.733485i \(0.737892\pi\)
\(908\) 67.1085i 2.22707i
\(909\) 6.51821 0.216195
\(910\) 0 0
\(911\) −34.5260 −1.14390 −0.571949 0.820289i \(-0.693812\pi\)
−0.571949 + 0.820289i \(0.693812\pi\)
\(912\) 21.0172i 0.695948i
\(913\) 13.8021i 0.456783i
\(914\) 21.1059 0.698122
\(915\) 0 0
\(916\) −2.96007 −0.0978036
\(917\) 1.43336i 0.0473336i
\(918\) − 2.38197i − 0.0786166i
\(919\) 0.531075 0.0175185 0.00875927 0.999962i \(-0.497212\pi\)
0.00875927 + 0.999962i \(0.497212\pi\)
\(920\) 0 0
\(921\) −2.51330 −0.0828161
\(922\) 50.2645i 1.65537i
\(923\) − 12.7043i − 0.418166i
\(924\) 17.2443 0.567295
\(925\) 0 0
\(926\) 58.6833 1.92845
\(927\) − 8.04876i − 0.264356i
\(928\) 43.7569i 1.43639i
\(929\) −46.1961 −1.51565 −0.757823 0.652460i \(-0.773737\pi\)
−0.757823 + 0.652460i \(0.773737\pi\)
\(930\) 0 0
\(931\) 92.1108 3.01881
\(932\) − 17.7676i − 0.581997i
\(933\) 4.82417i 0.157936i
\(934\) −9.24998 −0.302669
\(935\) 0 0
\(936\) 1.28215 0.0419082
\(937\) 30.1220i 0.984042i 0.870583 + 0.492021i \(0.163742\pi\)
−0.870583 + 0.492021i \(0.836258\pi\)
\(938\) − 35.7299i − 1.16662i
\(939\) 5.98812 0.195415
\(940\) 0 0
\(941\) 57.0783 1.86070 0.930350 0.366672i \(-0.119503\pi\)
0.930350 + 0.366672i \(0.119503\pi\)
\(942\) − 8.47668i − 0.276185i
\(943\) 17.3374i 0.564583i
\(944\) 25.0611 0.815668
\(945\) 0 0
\(946\) 4.19130 0.136271
\(947\) 41.1250i 1.33638i 0.743989 + 0.668192i \(0.232931\pi\)
−0.743989 + 0.668192i \(0.767069\pi\)
\(948\) 36.4452i 1.18369i
\(949\) −5.98576 −0.194306
\(950\) 0 0
\(951\) 17.3531 0.562712
\(952\) 5.23542i 0.169681i
\(953\) 34.8143i 1.12775i 0.825861 + 0.563873i \(0.190689\pi\)
−0.825861 + 0.563873i \(0.809311\pi\)
\(954\) −7.94959 −0.257377
\(955\) 0 0
\(956\) 0.107165 0.00346598
\(957\) − 8.69627i − 0.281111i
\(958\) 27.1962i 0.878668i
\(959\) −19.8861 −0.642156
\(960\) 0 0
\(961\) −7.94022 −0.256136
\(962\) 16.2012i 0.522346i
\(963\) 9.47745i 0.305407i
\(964\) 29.4783 0.949432
\(965\) 0 0
\(966\) −21.4625 −0.690544
\(967\) 37.2697i 1.19851i 0.800557 + 0.599257i \(0.204537\pi\)
−0.800557 + 0.599257i \(0.795463\pi\)
\(968\) − 9.10165i − 0.292538i
\(969\) 8.61803 0.276851
\(970\) 0 0
\(971\) −26.0092 −0.834677 −0.417338 0.908751i \(-0.637037\pi\)
−0.417338 + 0.908751i \(0.637037\pi\)
\(972\) − 2.50430i − 0.0803254i
\(973\) − 20.6327i − 0.661453i
\(974\) 14.6910 0.470729
\(975\) 0 0
\(976\) −3.82064 −0.122296
\(977\) − 0.458579i − 0.0146712i −0.999973 0.00733561i \(-0.997665\pi\)
0.999973 0.00733561i \(-0.00233502\pi\)
\(978\) 10.1902i 0.325846i
\(979\) −15.8593 −0.506865
\(980\) 0 0
\(981\) −11.6657 −0.372458
\(982\) − 56.9128i − 1.81616i
\(983\) 8.94565i 0.285322i 0.989772 + 0.142661i \(0.0455659\pi\)
−0.989772 + 0.142661i \(0.954434\pi\)
\(984\) 7.99737 0.254947
\(985\) 0 0
\(986\) 13.1111 0.417541
\(987\) − 17.9899i − 0.572626i
\(988\) 23.0360i 0.732874i
\(989\) −2.90030 −0.0922241
\(990\) 0 0
\(991\) −56.5337 −1.79585 −0.897926 0.440147i \(-0.854926\pi\)
−0.897926 + 0.440147i \(0.854926\pi\)
\(992\) 38.1744i 1.21204i
\(993\) 20.3355i 0.645327i
\(994\) 98.0970 3.11145
\(995\) 0 0
\(996\) −21.8776 −0.693219
\(997\) − 39.6114i − 1.25451i −0.778816 0.627253i \(-0.784179\pi\)
0.778816 0.627253i \(-0.215821\pi\)
\(998\) 5.84619i 0.185058i
\(999\) 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.b.c.1249.2 8
5.2 odd 4 1875.2.a.h.1.3 4
5.3 odd 4 1875.2.a.e.1.2 4
5.4 even 2 inner 1875.2.b.c.1249.7 8
15.2 even 4 5625.2.a.i.1.2 4
15.8 even 4 5625.2.a.n.1.3 4
25.2 odd 20 75.2.g.b.46.2 yes 8
25.9 even 10 375.2.i.b.349.1 16
25.11 even 5 375.2.i.b.274.1 16
25.12 odd 20 75.2.g.b.31.2 8
25.13 odd 20 375.2.g.b.151.1 8
25.14 even 10 375.2.i.b.274.4 16
25.16 even 5 375.2.i.b.349.4 16
25.23 odd 20 375.2.g.b.226.1 8
75.2 even 20 225.2.h.c.46.1 8
75.62 even 20 225.2.h.c.181.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.g.b.31.2 8 25.12 odd 20
75.2.g.b.46.2 yes 8 25.2 odd 20
225.2.h.c.46.1 8 75.2 even 20
225.2.h.c.181.1 8 75.62 even 20
375.2.g.b.151.1 8 25.13 odd 20
375.2.g.b.226.1 8 25.23 odd 20
375.2.i.b.274.1 16 25.11 even 5
375.2.i.b.274.4 16 25.14 even 10
375.2.i.b.349.1 16 25.9 even 10
375.2.i.b.349.4 16 25.16 even 5
1875.2.a.e.1.2 4 5.3 odd 4
1875.2.a.h.1.3 4 5.2 odd 4
1875.2.b.c.1249.2 8 1.1 even 1 trivial
1875.2.b.c.1249.7 8 5.4 even 2 inner
5625.2.a.i.1.2 4 15.2 even 4
5625.2.a.n.1.3 4 15.8 even 4