Properties

Label 1875.2.b.c.1249.4
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 1249.4
Root \(-1.12233i\) of defining polynomial
Character \(\chi\) \(=\) 1875.1249
Dual form 1875.2.b.c.1249.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.12233i q^{2} -1.00000i q^{3} +0.740367 q^{4} -1.12233 q^{6} +1.11373i q^{7} -3.07561i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.12233i q^{2} -1.00000i q^{3} +0.740367 q^{4} -1.12233 q^{6} +1.11373i q^{7} -3.07561i q^{8} -1.00000 q^{9} +3.67008 q^{11} -0.740367i q^{12} +4.05204i q^{13} +1.24998 q^{14} -1.97112 q^{16} +2.12233i q^{17} +1.12233i q^{18} +4.06064 q^{19} +1.11373 q^{21} -4.11905i q^{22} -6.17438i q^{23} -3.07561 q^{24} +4.54774 q^{26} +1.00000i q^{27} +0.824573i q^{28} +2.25963 q^{29} +10.0520 q^{31} -3.93896i q^{32} -3.67008i q^{33} +2.38197 q^{34} -0.740367 q^{36} +7.37232i q^{37} -4.55739i q^{38} +4.05204 q^{39} -7.47214 q^{41} -1.24998i q^{42} -9.24998i q^{43} +2.71720 q^{44} -6.92971 q^{46} -3.12765i q^{47} +1.97112i q^{48} +5.75960 q^{49} +2.12233 q^{51} +3.00000i q^{52} +3.50961i q^{53} +1.12233 q^{54} +3.42541 q^{56} -4.06064i q^{57} -2.53606i q^{58} +6.59382 q^{59} -9.10408 q^{61} -11.2817i q^{62} -1.11373i q^{63} -8.36307 q^{64} -4.11905 q^{66} +2.62663i q^{67} +1.57131i q^{68} -6.17438 q^{69} +0.660827 q^{71} +3.07561i q^{72} -7.47542i q^{73} +8.27420 q^{74} +3.00637 q^{76} +4.08749i q^{77} -4.54774i q^{78} -8.53711 q^{79} +1.00000 q^{81} +8.38623i q^{82} -12.2639i q^{83} +0.824573 q^{84} -10.3816 q^{86} -2.25963i q^{87} -11.2877i q^{88} +15.2881 q^{89} -4.51290 q^{91} -4.57131i q^{92} -10.0520i q^{93} -3.51026 q^{94} -3.93896 q^{96} +13.5000i q^{97} -6.46419i q^{98} -3.67008 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{4} + 4 q^{6} - 8 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.12233i − 0.793610i −0.917903 0.396805i \(-0.870119\pi\)
0.917903 0.396805i \(-0.129881\pi\)
\(3\) − 1.00000i − 0.577350i
\(4\) 0.740367 0.370184
\(5\) 0 0
\(6\) −1.12233 −0.458191
\(7\) 1.11373i 0.420952i 0.977599 + 0.210476i \(0.0675014\pi\)
−0.977599 + 0.210476i \(0.932499\pi\)
\(8\) − 3.07561i − 1.08739i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 3.67008 1.10657 0.553285 0.832992i \(-0.313374\pi\)
0.553285 + 0.832992i \(0.313374\pi\)
\(12\) − 0.740367i − 0.213726i
\(13\) 4.05204i 1.12383i 0.827194 + 0.561917i \(0.189936\pi\)
−0.827194 + 0.561917i \(0.810064\pi\)
\(14\) 1.24998 0.334072
\(15\) 0 0
\(16\) −1.97112 −0.492780
\(17\) 2.12233i 0.514741i 0.966313 + 0.257371i \(0.0828562\pi\)
−0.966313 + 0.257371i \(0.917144\pi\)
\(18\) 1.12233i 0.264537i
\(19\) 4.06064 0.931575 0.465787 0.884897i \(-0.345771\pi\)
0.465787 + 0.884897i \(0.345771\pi\)
\(20\) 0 0
\(21\) 1.11373 0.243037
\(22\) − 4.11905i − 0.878184i
\(23\) − 6.17438i − 1.28745i −0.765258 0.643723i \(-0.777389\pi\)
0.765258 0.643723i \(-0.222611\pi\)
\(24\) −3.07561 −0.627806
\(25\) 0 0
\(26\) 4.54774 0.891886
\(27\) 1.00000i 0.192450i
\(28\) 0.824573i 0.155830i
\(29\) 2.25963 0.419603 0.209802 0.977744i \(-0.432718\pi\)
0.209802 + 0.977744i \(0.432718\pi\)
\(30\) 0 0
\(31\) 10.0520 1.80540 0.902700 0.430271i \(-0.141582\pi\)
0.902700 + 0.430271i \(0.141582\pi\)
\(32\) − 3.93896i − 0.696316i
\(33\) − 3.67008i − 0.638878i
\(34\) 2.38197 0.408504
\(35\) 0 0
\(36\) −0.740367 −0.123395
\(37\) 7.37232i 1.21200i 0.795464 + 0.606001i \(0.207227\pi\)
−0.795464 + 0.606001i \(0.792773\pi\)
\(38\) − 4.55739i − 0.739307i
\(39\) 4.05204 0.648846
\(40\) 0 0
\(41\) −7.47214 −1.16695 −0.583476 0.812131i \(-0.698308\pi\)
−0.583476 + 0.812131i \(0.698308\pi\)
\(42\) − 1.24998i − 0.192876i
\(43\) − 9.24998i − 1.41061i −0.708905 0.705304i \(-0.750810\pi\)
0.708905 0.705304i \(-0.249190\pi\)
\(44\) 2.71720 0.409634
\(45\) 0 0
\(46\) −6.92971 −1.02173
\(47\) − 3.12765i − 0.456214i −0.973636 0.228107i \(-0.926746\pi\)
0.973636 0.228107i \(-0.0732537\pi\)
\(48\) 1.97112i 0.284507i
\(49\) 5.75960 0.822799
\(50\) 0 0
\(51\) 2.12233 0.297186
\(52\) 3.00000i 0.416025i
\(53\) 3.50961i 0.482083i 0.970515 + 0.241041i \(0.0774889\pi\)
−0.970515 + 0.241041i \(0.922511\pi\)
\(54\) 1.12233 0.152730
\(55\) 0 0
\(56\) 3.42541 0.457739
\(57\) − 4.06064i − 0.537845i
\(58\) − 2.53606i − 0.333001i
\(59\) 6.59382 0.858442 0.429221 0.903199i \(-0.358788\pi\)
0.429221 + 0.903199i \(0.358788\pi\)
\(60\) 0 0
\(61\) −9.10408 −1.16566 −0.582829 0.812595i \(-0.698054\pi\)
−0.582829 + 0.812595i \(0.698054\pi\)
\(62\) − 11.2817i − 1.43278i
\(63\) − 1.11373i − 0.140317i
\(64\) −8.36307 −1.04538
\(65\) 0 0
\(66\) −4.11905 −0.507020
\(67\) 2.62663i 0.320894i 0.987044 + 0.160447i \(0.0512936\pi\)
−0.987044 + 0.160447i \(0.948706\pi\)
\(68\) 1.57131i 0.190549i
\(69\) −6.17438 −0.743307
\(70\) 0 0
\(71\) 0.660827 0.0784257 0.0392128 0.999231i \(-0.487515\pi\)
0.0392128 + 0.999231i \(0.487515\pi\)
\(72\) 3.07561i 0.362464i
\(73\) − 7.47542i − 0.874932i −0.899235 0.437466i \(-0.855876\pi\)
0.899235 0.437466i \(-0.144124\pi\)
\(74\) 8.27420 0.961856
\(75\) 0 0
\(76\) 3.00637 0.344854
\(77\) 4.08749i 0.465813i
\(78\) − 4.54774i − 0.514930i
\(79\) −8.53711 −0.960500 −0.480250 0.877132i \(-0.659454\pi\)
−0.480250 + 0.877132i \(0.659454\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 8.38623i 0.926104i
\(83\) − 12.2639i − 1.34614i −0.739580 0.673069i \(-0.764976\pi\)
0.739580 0.673069i \(-0.235024\pi\)
\(84\) 0.824573 0.0899683
\(85\) 0 0
\(86\) −10.3816 −1.11947
\(87\) − 2.25963i − 0.242258i
\(88\) − 11.2877i − 1.20327i
\(89\) 15.2881 1.62054 0.810268 0.586059i \(-0.199321\pi\)
0.810268 + 0.586059i \(0.199321\pi\)
\(90\) 0 0
\(91\) −4.51290 −0.473080
\(92\) − 4.57131i − 0.476592i
\(93\) − 10.0520i − 1.04235i
\(94\) −3.51026 −0.362056
\(95\) 0 0
\(96\) −3.93896 −0.402018
\(97\) 13.5000i 1.37071i 0.728207 + 0.685357i \(0.240354\pi\)
−0.728207 + 0.685357i \(0.759646\pi\)
\(98\) − 6.46419i − 0.652982i
\(99\) −3.67008 −0.368856
\(100\) 0 0
\(101\) 7.22642 0.719055 0.359528 0.933134i \(-0.382938\pi\)
0.359528 + 0.933134i \(0.382938\pi\)
\(102\) − 2.38197i − 0.235850i
\(103\) − 5.27748i − 0.520006i −0.965608 0.260003i \(-0.916277\pi\)
0.965608 0.260003i \(-0.0837235\pi\)
\(104\) 12.4625 1.22205
\(105\) 0 0
\(106\) 3.93896 0.382585
\(107\) 5.46682i 0.528498i 0.964455 + 0.264249i \(0.0851240\pi\)
−0.964455 + 0.264249i \(0.914876\pi\)
\(108\) 0.740367i 0.0712419i
\(109\) −11.3395 −1.08613 −0.543064 0.839692i \(-0.682736\pi\)
−0.543064 + 0.839692i \(0.682736\pi\)
\(110\) 0 0
\(111\) 7.37232 0.699749
\(112\) − 2.19531i − 0.207437i
\(113\) − 4.83520i − 0.454858i −0.973795 0.227429i \(-0.926968\pi\)
0.973795 0.227429i \(-0.0730319\pi\)
\(114\) −4.55739 −0.426839
\(115\) 0 0
\(116\) 1.67296 0.155330
\(117\) − 4.05204i − 0.374611i
\(118\) − 7.40046i − 0.681268i
\(119\) −2.36372 −0.216681
\(120\) 0 0
\(121\) 2.46946 0.224496
\(122\) 10.2178i 0.925078i
\(123\) 7.47214i 0.673740i
\(124\) 7.44220 0.668330
\(125\) 0 0
\(126\) −1.24998 −0.111357
\(127\) 0.759596i 0.0674032i 0.999432 + 0.0337016i \(0.0107296\pi\)
−0.999432 + 0.0337016i \(0.989270\pi\)
\(128\) 1.50823i 0.133310i
\(129\) −9.24998 −0.814415
\(130\) 0 0
\(131\) 14.6551 1.28042 0.640211 0.768199i \(-0.278847\pi\)
0.640211 + 0.768199i \(0.278847\pi\)
\(132\) − 2.71720i − 0.236502i
\(133\) 4.52248i 0.392148i
\(134\) 2.94796 0.254665
\(135\) 0 0
\(136\) 6.52746 0.559725
\(137\) 12.2914i 1.05012i 0.851064 + 0.525062i \(0.175958\pi\)
−0.851064 + 0.525062i \(0.824042\pi\)
\(138\) 6.92971i 0.589896i
\(139\) −14.7340 −1.24972 −0.624861 0.780736i \(-0.714844\pi\)
−0.624861 + 0.780736i \(0.714844\pi\)
\(140\) 0 0
\(141\) −3.12765 −0.263395
\(142\) − 0.741668i − 0.0622394i
\(143\) 14.8713i 1.24360i
\(144\) 1.97112 0.164260
\(145\) 0 0
\(146\) −8.38991 −0.694354
\(147\) − 5.75960i − 0.475043i
\(148\) 5.45822i 0.448663i
\(149\) −15.6498 −1.28208 −0.641041 0.767507i \(-0.721497\pi\)
−0.641041 + 0.767507i \(0.721497\pi\)
\(150\) 0 0
\(151\) 3.95819 0.322113 0.161056 0.986945i \(-0.448510\pi\)
0.161056 + 0.986945i \(0.448510\pi\)
\(152\) − 12.4889i − 1.01299i
\(153\) − 2.12233i − 0.171580i
\(154\) 4.58753 0.369673
\(155\) 0 0
\(156\) 3.00000 0.240192
\(157\) − 4.50061i − 0.359188i −0.983741 0.179594i \(-0.942522\pi\)
0.983741 0.179594i \(-0.0574784\pi\)
\(158\) 9.58149i 0.762262i
\(159\) 3.50961 0.278330
\(160\) 0 0
\(161\) 6.87661 0.541953
\(162\) − 1.12233i − 0.0881788i
\(163\) 2.45389i 0.192203i 0.995372 + 0.0961016i \(0.0306374\pi\)
−0.995372 + 0.0961016i \(0.969363\pi\)
\(164\) −5.53213 −0.431987
\(165\) 0 0
\(166\) −13.7642 −1.06831
\(167\) − 3.06328i − 0.237043i −0.992951 0.118522i \(-0.962184\pi\)
0.992951 0.118522i \(-0.0378155\pi\)
\(168\) − 3.42541i − 0.264276i
\(169\) −3.41904 −0.263003
\(170\) 0 0
\(171\) −4.06064 −0.310525
\(172\) − 6.84839i − 0.522185i
\(173\) − 11.2596i − 0.856054i −0.903766 0.428027i \(-0.859209\pi\)
0.903766 0.428027i \(-0.140791\pi\)
\(174\) −2.53606 −0.192258
\(175\) 0 0
\(176\) −7.23416 −0.545296
\(177\) − 6.59382i − 0.495622i
\(178\) − 17.1584i − 1.28607i
\(179\) −10.0574 −0.751722 −0.375861 0.926676i \(-0.622653\pi\)
−0.375861 + 0.926676i \(0.622653\pi\)
\(180\) 0 0
\(181\) −19.4684 −1.44708 −0.723539 0.690283i \(-0.757486\pi\)
−0.723539 + 0.690283i \(0.757486\pi\)
\(182\) 5.06498i 0.375441i
\(183\) 9.10408i 0.672993i
\(184\) −18.9899 −1.39996
\(185\) 0 0
\(186\) −11.2817 −0.827218
\(187\) 7.78912i 0.569597i
\(188\) − 2.31561i − 0.168883i
\(189\) −1.11373 −0.0810123
\(190\) 0 0
\(191\) −11.6090 −0.840000 −0.420000 0.907524i \(-0.637970\pi\)
−0.420000 + 0.907524i \(0.637970\pi\)
\(192\) 8.36307i 0.603552i
\(193\) 3.38156i 0.243410i 0.992566 + 0.121705i \(0.0388362\pi\)
−0.992566 + 0.121705i \(0.961164\pi\)
\(194\) 15.1515 1.08781
\(195\) 0 0
\(196\) 4.26422 0.304587
\(197\) − 17.5881i − 1.25310i −0.779381 0.626550i \(-0.784466\pi\)
0.779381 0.626550i \(-0.215534\pi\)
\(198\) 4.11905i 0.292728i
\(199\) −20.0102 −1.41849 −0.709244 0.704963i \(-0.750963\pi\)
−0.709244 + 0.704963i \(0.750963\pi\)
\(200\) 0 0
\(201\) 2.62663 0.185268
\(202\) − 8.11045i − 0.570649i
\(203\) 2.51663i 0.176633i
\(204\) 1.57131 0.110013
\(205\) 0 0
\(206\) −5.92309 −0.412681
\(207\) 6.17438i 0.429149i
\(208\) − 7.98706i − 0.553803i
\(209\) 14.9029 1.03085
\(210\) 0 0
\(211\) 2.44261 0.168156 0.0840780 0.996459i \(-0.473206\pi\)
0.0840780 + 0.996459i \(0.473206\pi\)
\(212\) 2.59840i 0.178459i
\(213\) − 0.660827i − 0.0452791i
\(214\) 6.13560 0.419421
\(215\) 0 0
\(216\) 3.07561 0.209269
\(217\) 11.1953i 0.759987i
\(218\) 12.7267i 0.861961i
\(219\) −7.47542 −0.505142
\(220\) 0 0
\(221\) −8.59978 −0.578484
\(222\) − 8.27420i − 0.555328i
\(223\) 25.6369i 1.71677i 0.513005 + 0.858386i \(0.328532\pi\)
−0.513005 + 0.858386i \(0.671468\pi\)
\(224\) 4.38695 0.293116
\(225\) 0 0
\(226\) −5.42671 −0.360979
\(227\) 6.56336i 0.435625i 0.975991 + 0.217813i \(0.0698922\pi\)
−0.975991 + 0.217813i \(0.930108\pi\)
\(228\) − 3.00637i − 0.199101i
\(229\) 7.96390 0.526269 0.263135 0.964759i \(-0.415244\pi\)
0.263135 + 0.964759i \(0.415244\pi\)
\(230\) 0 0
\(231\) 4.08749 0.268937
\(232\) − 6.94974i − 0.456273i
\(233\) 9.42107i 0.617195i 0.951193 + 0.308597i \(0.0998596\pi\)
−0.951193 + 0.308597i \(0.900140\pi\)
\(234\) −4.54774 −0.297295
\(235\) 0 0
\(236\) 4.88185 0.317781
\(237\) 8.53711i 0.554545i
\(238\) 2.65288i 0.171961i
\(239\) 11.2231 0.725964 0.362982 0.931796i \(-0.381759\pi\)
0.362982 + 0.931796i \(0.381759\pi\)
\(240\) 0 0
\(241\) 22.6809 1.46101 0.730503 0.682910i \(-0.239286\pi\)
0.730503 + 0.682910i \(0.239286\pi\)
\(242\) − 2.77155i − 0.178162i
\(243\) − 1.00000i − 0.0641500i
\(244\) −6.74037 −0.431508
\(245\) 0 0
\(246\) 8.38623 0.534686
\(247\) 16.4539i 1.04694i
\(248\) − 30.9161i − 1.96318i
\(249\) −12.2639 −0.777193
\(250\) 0 0
\(251\) −6.76819 −0.427205 −0.213602 0.976921i \(-0.568520\pi\)
−0.213602 + 0.976921i \(0.568520\pi\)
\(252\) − 0.824573i − 0.0519432i
\(253\) − 22.6604i − 1.42465i
\(254\) 0.852520 0.0534918
\(255\) 0 0
\(256\) −15.0334 −0.939587
\(257\) − 14.7934i − 0.922786i −0.887196 0.461393i \(-0.847350\pi\)
0.887196 0.461393i \(-0.152650\pi\)
\(258\) 10.3816i 0.646328i
\(259\) −8.21080 −0.510194
\(260\) 0 0
\(261\) −2.25963 −0.139868
\(262\) − 16.4479i − 1.01616i
\(263\) − 19.9688i − 1.23133i −0.788008 0.615665i \(-0.788887\pi\)
0.788008 0.615665i \(-0.211113\pi\)
\(264\) −11.2877 −0.694710
\(265\) 0 0
\(266\) 5.07573 0.311213
\(267\) − 15.2881i − 0.935617i
\(268\) 1.94467i 0.118790i
\(269\) 10.2381 0.624228 0.312114 0.950045i \(-0.398963\pi\)
0.312114 + 0.950045i \(0.398963\pi\)
\(270\) 0 0
\(271\) 12.3075 0.747630 0.373815 0.927503i \(-0.378050\pi\)
0.373815 + 0.927503i \(0.378050\pi\)
\(272\) − 4.18338i − 0.253654i
\(273\) 4.51290i 0.273133i
\(274\) 13.7950 0.833389
\(275\) 0 0
\(276\) −4.57131 −0.275160
\(277\) 31.5520i 1.89578i 0.318603 + 0.947888i \(0.396786\pi\)
−0.318603 + 0.947888i \(0.603214\pi\)
\(278\) 16.5365i 0.991791i
\(279\) −10.0520 −0.601800
\(280\) 0 0
\(281\) −21.3119 −1.27136 −0.635681 0.771952i \(-0.719281\pi\)
−0.635681 + 0.771952i \(0.719281\pi\)
\(282\) 3.51026i 0.209033i
\(283\) 0.864403i 0.0513834i 0.999670 + 0.0256917i \(0.00817883\pi\)
−0.999670 + 0.0256917i \(0.991821\pi\)
\(284\) 0.489254 0.0290319
\(285\) 0 0
\(286\) 16.6906 0.986933
\(287\) − 8.32198i − 0.491231i
\(288\) 3.93896i 0.232105i
\(289\) 12.4957 0.735041
\(290\) 0 0
\(291\) 13.5000 0.791382
\(292\) − 5.53456i − 0.323886i
\(293\) − 3.17701i − 0.185603i −0.995685 0.0928014i \(-0.970418\pi\)
0.995685 0.0928014i \(-0.0295822\pi\)
\(294\) −6.46419 −0.376999
\(295\) 0 0
\(296\) 22.6743 1.31792
\(297\) 3.67008i 0.212959i
\(298\) 17.5643i 1.01747i
\(299\) 25.0188 1.44688
\(300\) 0 0
\(301\) 10.3020 0.593799
\(302\) − 4.44240i − 0.255632i
\(303\) − 7.22642i − 0.415147i
\(304\) −8.00401 −0.459062
\(305\) 0 0
\(306\) −2.38197 −0.136168
\(307\) − 0.507986i − 0.0289923i −0.999895 0.0144961i \(-0.995386\pi\)
0.999895 0.0144961i \(-0.00461443\pi\)
\(308\) 3.02624i 0.172436i
\(309\) −5.27748 −0.300225
\(310\) 0 0
\(311\) −14.6438 −0.830375 −0.415188 0.909736i \(-0.636284\pi\)
−0.415188 + 0.909736i \(0.636284\pi\)
\(312\) − 12.4625i − 0.705549i
\(313\) 14.9562i 0.845372i 0.906276 + 0.422686i \(0.138913\pi\)
−0.906276 + 0.422686i \(0.861087\pi\)
\(314\) −5.05119 −0.285055
\(315\) 0 0
\(316\) −6.32060 −0.355562
\(317\) 18.1190i 1.01767i 0.860865 + 0.508834i \(0.169923\pi\)
−0.860865 + 0.508834i \(0.830077\pi\)
\(318\) − 3.93896i − 0.220886i
\(319\) 8.29302 0.464320
\(320\) 0 0
\(321\) 5.46682 0.305128
\(322\) − 7.71785i − 0.430099i
\(323\) 8.61803i 0.479520i
\(324\) 0.740367 0.0411315
\(325\) 0 0
\(326\) 2.75408 0.152534
\(327\) 11.3395i 0.627076i
\(328\) 22.9813i 1.26893i
\(329\) 3.48337 0.192044
\(330\) 0 0
\(331\) 26.3514 1.44841 0.724203 0.689587i \(-0.242208\pi\)
0.724203 + 0.689587i \(0.242208\pi\)
\(332\) − 9.07979i − 0.498318i
\(333\) − 7.37232i − 0.404000i
\(334\) −3.43802 −0.188120
\(335\) 0 0
\(336\) −2.19531 −0.119764
\(337\) − 10.6948i − 0.582584i −0.956634 0.291292i \(-0.905915\pi\)
0.956634 0.291292i \(-0.0940851\pi\)
\(338\) 3.83731i 0.208722i
\(339\) −4.83520 −0.262612
\(340\) 0 0
\(341\) 36.8918 1.99780
\(342\) 4.55739i 0.246436i
\(343\) 14.2108i 0.767311i
\(344\) −28.4493 −1.53388
\(345\) 0 0
\(346\) −12.6371 −0.679373
\(347\) 16.1740i 0.868264i 0.900849 + 0.434132i \(0.142945\pi\)
−0.900849 + 0.434132i \(0.857055\pi\)
\(348\) − 1.67296i − 0.0896800i
\(349\) −15.2383 −0.815688 −0.407844 0.913052i \(-0.633719\pi\)
−0.407844 + 0.913052i \(0.633719\pi\)
\(350\) 0 0
\(351\) −4.05204 −0.216282
\(352\) − 14.4563i − 0.770522i
\(353\) 29.5383i 1.57217i 0.618120 + 0.786084i \(0.287894\pi\)
−0.618120 + 0.786084i \(0.712106\pi\)
\(354\) −7.40046 −0.393330
\(355\) 0 0
\(356\) 11.3188 0.599896
\(357\) 2.36372i 0.125101i
\(358\) 11.2877i 0.596574i
\(359\) 13.1668 0.694915 0.347457 0.937696i \(-0.387045\pi\)
0.347457 + 0.937696i \(0.387045\pi\)
\(360\) 0 0
\(361\) −2.51120 −0.132168
\(362\) 21.8501i 1.14842i
\(363\) − 2.46946i − 0.129613i
\(364\) −3.34120 −0.175127
\(365\) 0 0
\(366\) 10.2178 0.534094
\(367\) − 3.88895i − 0.203001i −0.994835 0.101501i \(-0.967636\pi\)
0.994835 0.101501i \(-0.0323644\pi\)
\(368\) 12.1704i 0.634428i
\(369\) 7.47214 0.388984
\(370\) 0 0
\(371\) −3.90878 −0.202934
\(372\) − 7.44220i − 0.385860i
\(373\) 8.14326i 0.421642i 0.977525 + 0.210821i \(0.0676138\pi\)
−0.977525 + 0.210821i \(0.932386\pi\)
\(374\) 8.74200 0.452038
\(375\) 0 0
\(376\) −9.61941 −0.496083
\(377\) 9.15613i 0.471564i
\(378\) 1.24998i 0.0642921i
\(379\) −9.90720 −0.508898 −0.254449 0.967086i \(-0.581894\pi\)
−0.254449 + 0.967086i \(0.581894\pi\)
\(380\) 0 0
\(381\) 0.759596 0.0389153
\(382\) 13.0292i 0.666632i
\(383\) 5.22215i 0.266840i 0.991060 + 0.133420i \(0.0425959\pi\)
−0.991060 + 0.133420i \(0.957404\pi\)
\(384\) 1.50823 0.0769668
\(385\) 0 0
\(386\) 3.79524 0.193173
\(387\) 9.24998i 0.470203i
\(388\) 9.99493i 0.507416i
\(389\) 3.72974 0.189105 0.0945526 0.995520i \(-0.469858\pi\)
0.0945526 + 0.995520i \(0.469858\pi\)
\(390\) 0 0
\(391\) 13.1041 0.662702
\(392\) − 17.7142i − 0.894705i
\(393\) − 14.6551i − 0.739253i
\(394\) −19.7397 −0.994473
\(395\) 0 0
\(396\) −2.71720 −0.136545
\(397\) 4.01562i 0.201538i 0.994910 + 0.100769i \(0.0321303\pi\)
−0.994910 + 0.100769i \(0.967870\pi\)
\(398\) 22.4581i 1.12573i
\(399\) 4.52248 0.226407
\(400\) 0 0
\(401\) −24.9890 −1.24789 −0.623945 0.781468i \(-0.714471\pi\)
−0.623945 + 0.781468i \(0.714471\pi\)
\(402\) − 2.94796i − 0.147031i
\(403\) 40.7313i 2.02897i
\(404\) 5.35020 0.266183
\(405\) 0 0
\(406\) 2.82450 0.140178
\(407\) 27.0570i 1.34116i
\(408\) − 6.52746i − 0.323158i
\(409\) 25.3768 1.25480 0.627401 0.778697i \(-0.284119\pi\)
0.627401 + 0.778697i \(0.284119\pi\)
\(410\) 0 0
\(411\) 12.2914 0.606290
\(412\) − 3.90728i − 0.192498i
\(413\) 7.34376i 0.361363i
\(414\) 6.92971 0.340577
\(415\) 0 0
\(416\) 15.9608 0.782544
\(417\) 14.7340i 0.721527i
\(418\) − 16.7260i − 0.818094i
\(419\) 30.8219 1.50575 0.752873 0.658165i \(-0.228667\pi\)
0.752873 + 0.658165i \(0.228667\pi\)
\(420\) 0 0
\(421\) 8.54649 0.416530 0.208265 0.978072i \(-0.433218\pi\)
0.208265 + 0.978072i \(0.433218\pi\)
\(422\) − 2.74142i − 0.133450i
\(423\) 3.12765i 0.152071i
\(424\) 10.7942 0.524212
\(425\) 0 0
\(426\) −0.741668 −0.0359339
\(427\) − 10.1395i − 0.490686i
\(428\) 4.04746i 0.195641i
\(429\) 14.8713 0.717993
\(430\) 0 0
\(431\) −26.3815 −1.27075 −0.635376 0.772203i \(-0.719155\pi\)
−0.635376 + 0.772203i \(0.719155\pi\)
\(432\) − 1.97112i − 0.0948356i
\(433\) 9.37272i 0.450424i 0.974310 + 0.225212i \(0.0723075\pi\)
−0.974310 + 0.225212i \(0.927693\pi\)
\(434\) 12.5649 0.603133
\(435\) 0 0
\(436\) −8.39540 −0.402067
\(437\) − 25.0719i − 1.19935i
\(438\) 8.38991i 0.400886i
\(439\) 0.515980 0.0246264 0.0123132 0.999924i \(-0.496080\pi\)
0.0123132 + 0.999924i \(0.496080\pi\)
\(440\) 0 0
\(441\) −5.75960 −0.274266
\(442\) 9.65183i 0.459091i
\(443\) 17.8348i 0.847357i 0.905813 + 0.423678i \(0.139261\pi\)
−0.905813 + 0.423678i \(0.860739\pi\)
\(444\) 5.45822 0.259036
\(445\) 0 0
\(446\) 28.7731 1.36245
\(447\) 15.6498i 0.740210i
\(448\) − 9.31423i − 0.440056i
\(449\) 4.16533 0.196574 0.0982870 0.995158i \(-0.468664\pi\)
0.0982870 + 0.995158i \(0.468664\pi\)
\(450\) 0 0
\(451\) −27.4233 −1.29131
\(452\) − 3.57983i − 0.168381i
\(453\) − 3.95819i − 0.185972i
\(454\) 7.36628 0.345716
\(455\) 0 0
\(456\) −12.4889 −0.584848
\(457\) − 17.6734i − 0.826725i −0.910567 0.413362i \(-0.864354\pi\)
0.910567 0.413362i \(-0.135646\pi\)
\(458\) − 8.93815i − 0.417652i
\(459\) −2.12233 −0.0990620
\(460\) 0 0
\(461\) −0.204956 −0.00954577 −0.00477288 0.999989i \(-0.501519\pi\)
−0.00477288 + 0.999989i \(0.501519\pi\)
\(462\) − 4.58753i − 0.213431i
\(463\) 12.8176i 0.595682i 0.954615 + 0.297841i \(0.0962666\pi\)
−0.954615 + 0.297841i \(0.903733\pi\)
\(464\) −4.45401 −0.206772
\(465\) 0 0
\(466\) 10.5736 0.489812
\(467\) 1.11373i 0.0515375i 0.999668 + 0.0257687i \(0.00820335\pi\)
−0.999668 + 0.0257687i \(0.991797\pi\)
\(468\) − 3.00000i − 0.138675i
\(469\) −2.92537 −0.135081
\(470\) 0 0
\(471\) −4.50061 −0.207377
\(472\) − 20.2800i − 0.933462i
\(473\) − 33.9481i − 1.56094i
\(474\) 9.58149 0.440092
\(475\) 0 0
\(476\) −1.75002 −0.0802120
\(477\) − 3.50961i − 0.160694i
\(478\) − 12.5961i − 0.576132i
\(479\) 25.1750 1.15027 0.575136 0.818057i \(-0.304949\pi\)
0.575136 + 0.818057i \(0.304949\pi\)
\(480\) 0 0
\(481\) −29.8729 −1.36209
\(482\) − 25.4555i − 1.15947i
\(483\) − 6.87661i − 0.312897i
\(484\) 1.82830 0.0831048
\(485\) 0 0
\(486\) −1.12233 −0.0509101
\(487\) 10.3598i 0.469447i 0.972062 + 0.234723i \(0.0754184\pi\)
−0.972062 + 0.234723i \(0.924582\pi\)
\(488\) 28.0006i 1.26753i
\(489\) 2.45389 0.110969
\(490\) 0 0
\(491\) −15.6571 −0.706595 −0.353297 0.935511i \(-0.614940\pi\)
−0.353297 + 0.935511i \(0.614940\pi\)
\(492\) 5.53213i 0.249408i
\(493\) 4.79569i 0.215987i
\(494\) 18.4667 0.830858
\(495\) 0 0
\(496\) −19.8138 −0.889665
\(497\) 0.735985i 0.0330135i
\(498\) 13.7642i 0.616788i
\(499\) −35.7864 −1.60202 −0.801010 0.598651i \(-0.795704\pi\)
−0.801010 + 0.598651i \(0.795704\pi\)
\(500\) 0 0
\(501\) −3.06328 −0.136857
\(502\) 7.59617i 0.339034i
\(503\) 11.7791i 0.525203i 0.964904 + 0.262601i \(0.0845804\pi\)
−0.964904 + 0.262601i \(0.915420\pi\)
\(504\) −3.42541 −0.152580
\(505\) 0 0
\(506\) −25.4326 −1.13061
\(507\) 3.41904i 0.151845i
\(508\) 0.562380i 0.0249516i
\(509\) −33.6507 −1.49154 −0.745771 0.666203i \(-0.767918\pi\)
−0.745771 + 0.666203i \(0.767918\pi\)
\(510\) 0 0
\(511\) 8.32563 0.368304
\(512\) 19.8889i 0.878976i
\(513\) 4.06064i 0.179282i
\(514\) −16.6031 −0.732332
\(515\) 0 0
\(516\) −6.84839 −0.301483
\(517\) − 11.4787i − 0.504833i
\(518\) 9.21526i 0.404895i
\(519\) −11.2596 −0.494243
\(520\) 0 0
\(521\) −11.8448 −0.518929 −0.259465 0.965753i \(-0.583546\pi\)
−0.259465 + 0.965753i \(0.583546\pi\)
\(522\) 2.53606i 0.111000i
\(523\) − 4.09694i − 0.179147i −0.995980 0.0895733i \(-0.971450\pi\)
0.995980 0.0895733i \(-0.0285503\pi\)
\(524\) 10.8502 0.473992
\(525\) 0 0
\(526\) −22.4117 −0.977195
\(527\) 21.3338i 0.929314i
\(528\) 7.23416i 0.314827i
\(529\) −15.1229 −0.657518
\(530\) 0 0
\(531\) −6.59382 −0.286147
\(532\) 3.34829i 0.145167i
\(533\) − 30.2774i − 1.31146i
\(534\) −17.1584 −0.742515
\(535\) 0 0
\(536\) 8.07849 0.348938
\(537\) 10.0574i 0.434007i
\(538\) − 11.4906i − 0.495393i
\(539\) 21.1382 0.910485
\(540\) 0 0
\(541\) 13.9582 0.600109 0.300055 0.953922i \(-0.402995\pi\)
0.300055 + 0.953922i \(0.402995\pi\)
\(542\) − 13.8132i − 0.593326i
\(543\) 19.4684i 0.835471i
\(544\) 8.35978 0.358423
\(545\) 0 0
\(546\) 5.06498 0.216761
\(547\) − 26.5045i − 1.13325i −0.823976 0.566625i \(-0.808249\pi\)
0.823976 0.566625i \(-0.191751\pi\)
\(548\) 9.10015i 0.388739i
\(549\) 9.10408 0.388553
\(550\) 0 0
\(551\) 9.17556 0.390892
\(552\) 18.9899i 0.808266i
\(553\) − 9.50808i − 0.404325i
\(554\) 35.4119 1.50451
\(555\) 0 0
\(556\) −10.9086 −0.462627
\(557\) 10.6860i 0.452781i 0.974037 + 0.226391i \(0.0726926\pi\)
−0.974037 + 0.226391i \(0.927307\pi\)
\(558\) 11.2817i 0.477594i
\(559\) 37.4813 1.58529
\(560\) 0 0
\(561\) 7.78912 0.328857
\(562\) 23.9191i 1.00897i
\(563\) 25.5750i 1.07786i 0.842352 + 0.538928i \(0.181171\pi\)
−0.842352 + 0.538928i \(0.818829\pi\)
\(564\) −2.31561 −0.0975047
\(565\) 0 0
\(566\) 0.970149 0.0407784
\(567\) 1.11373i 0.0467725i
\(568\) − 2.03244i − 0.0852794i
\(569\) −6.10210 −0.255813 −0.127907 0.991786i \(-0.540826\pi\)
−0.127907 + 0.991786i \(0.540826\pi\)
\(570\) 0 0
\(571\) 23.7396 0.993473 0.496737 0.867901i \(-0.334532\pi\)
0.496737 + 0.867901i \(0.334532\pi\)
\(572\) 11.0102i 0.460361i
\(573\) 11.6090i 0.484974i
\(574\) −9.34003 −0.389845
\(575\) 0 0
\(576\) 8.36307 0.348461
\(577\) − 44.1639i − 1.83857i −0.393595 0.919284i \(-0.628769\pi\)
0.393595 0.919284i \(-0.371231\pi\)
\(578\) − 14.0243i − 0.583336i
\(579\) 3.38156 0.140533
\(580\) 0 0
\(581\) 13.6587 0.566659
\(582\) − 15.1515i − 0.628048i
\(583\) 12.8805i 0.533458i
\(584\) −22.9915 −0.951393
\(585\) 0 0
\(586\) −3.56566 −0.147296
\(587\) 0.511966i 0.0211311i 0.999944 + 0.0105656i \(0.00336318\pi\)
−0.999944 + 0.0105656i \(0.996637\pi\)
\(588\) − 4.26422i − 0.175853i
\(589\) 40.8177 1.68187
\(590\) 0 0
\(591\) −17.5881 −0.723478
\(592\) − 14.5317i − 0.597250i
\(593\) 18.8405i 0.773687i 0.922145 + 0.386844i \(0.126435\pi\)
−0.922145 + 0.386844i \(0.873565\pi\)
\(594\) 4.11905 0.169007
\(595\) 0 0
\(596\) −11.5866 −0.474606
\(597\) 20.0102i 0.818964i
\(598\) − 28.0795i − 1.14825i
\(599\) −6.20712 −0.253616 −0.126808 0.991927i \(-0.540473\pi\)
−0.126808 + 0.991927i \(0.540473\pi\)
\(600\) 0 0
\(601\) 34.0303 1.38813 0.694063 0.719915i \(-0.255819\pi\)
0.694063 + 0.719915i \(0.255819\pi\)
\(602\) − 11.5623i − 0.471244i
\(603\) − 2.62663i − 0.106965i
\(604\) 2.93051 0.119241
\(605\) 0 0
\(606\) −8.11045 −0.329465
\(607\) 16.2488i 0.659518i 0.944065 + 0.329759i \(0.106968\pi\)
−0.944065 + 0.329759i \(0.893032\pi\)
\(608\) − 15.9947i − 0.648670i
\(609\) 2.51663 0.101979
\(610\) 0 0
\(611\) 12.6734 0.512709
\(612\) − 1.57131i − 0.0635163i
\(613\) 0.0679852i 0.00274590i 0.999999 + 0.00137295i \(0.000437023\pi\)
−0.999999 + 0.00137295i \(0.999563\pi\)
\(614\) −0.570130 −0.0230086
\(615\) 0 0
\(616\) 12.5715 0.506521
\(617\) 32.8108i 1.32091i 0.750864 + 0.660456i \(0.229637\pi\)
−0.750864 + 0.660456i \(0.770363\pi\)
\(618\) 5.92309i 0.238262i
\(619\) 6.77712 0.272395 0.136198 0.990682i \(-0.456512\pi\)
0.136198 + 0.990682i \(0.456512\pi\)
\(620\) 0 0
\(621\) 6.17438 0.247769
\(622\) 16.4353i 0.658994i
\(623\) 17.0269i 0.682168i
\(624\) −7.98706 −0.319738
\(625\) 0 0
\(626\) 16.7858 0.670895
\(627\) − 14.9029i − 0.595163i
\(628\) − 3.33211i − 0.132966i
\(629\) −15.6465 −0.623867
\(630\) 0 0
\(631\) −7.66797 −0.305257 −0.152629 0.988284i \(-0.548774\pi\)
−0.152629 + 0.988284i \(0.548774\pi\)
\(632\) 26.2568i 1.04444i
\(633\) − 2.44261i − 0.0970849i
\(634\) 20.3356 0.807630
\(635\) 0 0
\(636\) 2.59840 0.103033
\(637\) 23.3381i 0.924690i
\(638\) − 9.30754i − 0.368489i
\(639\) −0.660827 −0.0261419
\(640\) 0 0
\(641\) −23.6911 −0.935744 −0.467872 0.883796i \(-0.654979\pi\)
−0.467872 + 0.883796i \(0.654979\pi\)
\(642\) − 6.13560i − 0.242153i
\(643\) − 2.16861i − 0.0855218i −0.999085 0.0427609i \(-0.986385\pi\)
0.999085 0.0427609i \(-0.0136154\pi\)
\(644\) 5.09122 0.200622
\(645\) 0 0
\(646\) 9.67231 0.380552
\(647\) 2.55541i 0.100463i 0.998738 + 0.0502317i \(0.0159960\pi\)
−0.998738 + 0.0502317i \(0.984004\pi\)
\(648\) − 3.07561i − 0.120821i
\(649\) 24.1998 0.949926
\(650\) 0 0
\(651\) 11.1953 0.438779
\(652\) 1.81678i 0.0711505i
\(653\) − 32.4137i − 1.26845i −0.773150 0.634224i \(-0.781320\pi\)
0.773150 0.634224i \(-0.218680\pi\)
\(654\) 12.7267 0.497653
\(655\) 0 0
\(656\) 14.7285 0.575051
\(657\) 7.47542i 0.291644i
\(658\) − 3.90950i − 0.152408i
\(659\) −0.449951 −0.0175276 −0.00876381 0.999962i \(-0.502790\pi\)
−0.00876381 + 0.999962i \(0.502790\pi\)
\(660\) 0 0
\(661\) −26.9827 −1.04951 −0.524753 0.851254i \(-0.675842\pi\)
−0.524753 + 0.851254i \(0.675842\pi\)
\(662\) − 29.5751i − 1.14947i
\(663\) 8.59978i 0.333988i
\(664\) −37.7189 −1.46378
\(665\) 0 0
\(666\) −8.27420 −0.320619
\(667\) − 13.9518i − 0.540217i
\(668\) − 2.26795i − 0.0877496i
\(669\) 25.6369 0.991178
\(670\) 0 0
\(671\) −33.4127 −1.28988
\(672\) − 4.38695i − 0.169230i
\(673\) − 17.6224i − 0.679292i −0.940553 0.339646i \(-0.889693\pi\)
0.940553 0.339646i \(-0.110307\pi\)
\(674\) −12.0032 −0.462344
\(675\) 0 0
\(676\) −2.53135 −0.0973595
\(677\) − 19.3484i − 0.743621i −0.928309 0.371810i \(-0.878737\pi\)
0.928309 0.371810i \(-0.121263\pi\)
\(678\) 5.42671i 0.208412i
\(679\) −15.0354 −0.577005
\(680\) 0 0
\(681\) 6.56336 0.251508
\(682\) − 41.4049i − 1.58547i
\(683\) 29.9460i 1.14585i 0.819607 + 0.572926i \(0.194192\pi\)
−0.819607 + 0.572926i \(0.805808\pi\)
\(684\) −3.00637 −0.114951
\(685\) 0 0
\(686\) 15.9493 0.608946
\(687\) − 7.96390i − 0.303842i
\(688\) 18.2328i 0.695120i
\(689\) −14.2211 −0.541781
\(690\) 0 0
\(691\) −19.9349 −0.758361 −0.379181 0.925323i \(-0.623794\pi\)
−0.379181 + 0.925323i \(0.623794\pi\)
\(692\) − 8.33627i − 0.316897i
\(693\) − 4.08749i − 0.155271i
\(694\) 18.1526 0.689063
\(695\) 0 0
\(696\) −6.94974 −0.263429
\(697\) − 15.8584i − 0.600678i
\(698\) 17.1025i 0.647337i
\(699\) 9.42107 0.356338
\(700\) 0 0
\(701\) −49.0150 −1.85127 −0.925636 0.378415i \(-0.876469\pi\)
−0.925636 + 0.378415i \(0.876469\pi\)
\(702\) 4.54774i 0.171643i
\(703\) 29.9363i 1.12907i
\(704\) −30.6931 −1.15679
\(705\) 0 0
\(706\) 33.1519 1.24769
\(707\) 8.04831i 0.302688i
\(708\) − 4.88185i − 0.183471i
\(709\) 4.81347 0.180774 0.0903868 0.995907i \(-0.471190\pi\)
0.0903868 + 0.995907i \(0.471190\pi\)
\(710\) 0 0
\(711\) 8.53711 0.320167
\(712\) − 47.0202i − 1.76216i
\(713\) − 62.0651i − 2.32436i
\(714\) 2.65288 0.0992815
\(715\) 0 0
\(716\) −7.44614 −0.278275
\(717\) − 11.2231i − 0.419136i
\(718\) − 14.7775i − 0.551491i
\(719\) −32.2512 −1.20277 −0.601383 0.798961i \(-0.705383\pi\)
−0.601383 + 0.798961i \(0.705383\pi\)
\(720\) 0 0
\(721\) 5.87771 0.218897
\(722\) 2.81840i 0.104890i
\(723\) − 22.6809i − 0.843512i
\(724\) −14.4138 −0.535685
\(725\) 0 0
\(726\) −2.77155 −0.102862
\(727\) − 29.1747i − 1.08203i −0.841013 0.541015i \(-0.818040\pi\)
0.841013 0.541015i \(-0.181960\pi\)
\(728\) 13.8799i 0.514423i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 19.6315 0.726099
\(732\) 6.74037i 0.249131i
\(733\) − 5.20102i − 0.192104i −0.995376 0.0960521i \(-0.969378\pi\)
0.995376 0.0960521i \(-0.0306215\pi\)
\(734\) −4.36469 −0.161104
\(735\) 0 0
\(736\) −24.3206 −0.896469
\(737\) 9.63994i 0.355092i
\(738\) − 8.38623i − 0.308701i
\(739\) 16.3206 0.600363 0.300182 0.953882i \(-0.402953\pi\)
0.300182 + 0.953882i \(0.402953\pi\)
\(740\) 0 0
\(741\) 16.4539 0.604449
\(742\) 4.38695i 0.161050i
\(743\) − 53.4487i − 1.96084i −0.196914 0.980421i \(-0.563092\pi\)
0.196914 0.980421i \(-0.436908\pi\)
\(744\) −30.9161 −1.13344
\(745\) 0 0
\(746\) 9.13946 0.334619
\(747\) 12.2639i 0.448712i
\(748\) 5.76681i 0.210856i
\(749\) −6.08859 −0.222472
\(750\) 0 0
\(751\) −0.566970 −0.0206890 −0.0103445 0.999946i \(-0.503293\pi\)
−0.0103445 + 0.999946i \(0.503293\pi\)
\(752\) 6.16497i 0.224813i
\(753\) 6.76819i 0.246647i
\(754\) 10.2762 0.374238
\(755\) 0 0
\(756\) −0.824573 −0.0299894
\(757\) 53.0708i 1.92889i 0.264282 + 0.964445i \(0.414865\pi\)
−0.264282 + 0.964445i \(0.585135\pi\)
\(758\) 11.1192i 0.403867i
\(759\) −22.6604 −0.822521
\(760\) 0 0
\(761\) −27.5056 −0.997077 −0.498539 0.866867i \(-0.666130\pi\)
−0.498539 + 0.866867i \(0.666130\pi\)
\(762\) − 0.852520i − 0.0308835i
\(763\) − 12.6292i − 0.457208i
\(764\) −8.59495 −0.310954
\(765\) 0 0
\(766\) 5.86100 0.211766
\(767\) 26.7184i 0.964747i
\(768\) 15.0334i 0.542471i
\(769\) −14.9716 −0.539889 −0.269944 0.962876i \(-0.587005\pi\)
−0.269944 + 0.962876i \(0.587005\pi\)
\(770\) 0 0
\(771\) −14.7934 −0.532771
\(772\) 2.50360i 0.0901065i
\(773\) 2.05427i 0.0738871i 0.999317 + 0.0369436i \(0.0117622\pi\)
−0.999317 + 0.0369436i \(0.988238\pi\)
\(774\) 10.3816 0.373158
\(775\) 0 0
\(776\) 41.5206 1.49050
\(777\) 8.21080i 0.294561i
\(778\) − 4.18601i − 0.150076i
\(779\) −30.3417 −1.08710
\(780\) 0 0
\(781\) 2.42528 0.0867835
\(782\) − 14.7072i − 0.525927i
\(783\) 2.25963i 0.0807527i
\(784\) −11.3529 −0.405459
\(785\) 0 0
\(786\) −16.4479 −0.586678
\(787\) − 16.8269i − 0.599815i −0.953968 0.299908i \(-0.903044\pi\)
0.953968 0.299908i \(-0.0969559\pi\)
\(788\) − 13.0217i − 0.463877i
\(789\) −19.9688 −0.710909
\(790\) 0 0
\(791\) 5.38513 0.191473
\(792\) 11.2877i 0.401091i
\(793\) − 36.8901i − 1.31001i
\(794\) 4.50686 0.159942
\(795\) 0 0
\(796\) −14.8149 −0.525101
\(797\) − 0.208891i − 0.00739930i −0.999993 0.00369965i \(-0.998822\pi\)
0.999993 0.00369965i \(-0.00117764\pi\)
\(798\) − 5.07573i − 0.179679i
\(799\) 6.63791 0.234832
\(800\) 0 0
\(801\) −15.2881 −0.540179
\(802\) 28.0460i 0.990337i
\(803\) − 27.4354i − 0.968173i
\(804\) 1.94467 0.0685834
\(805\) 0 0
\(806\) 45.7141 1.61021
\(807\) − 10.2381i − 0.360398i
\(808\) − 22.2256i − 0.781894i
\(809\) 16.9655 0.596476 0.298238 0.954491i \(-0.403601\pi\)
0.298238 + 0.954491i \(0.403601\pi\)
\(810\) 0 0
\(811\) −20.4684 −0.718742 −0.359371 0.933195i \(-0.617009\pi\)
−0.359371 + 0.933195i \(0.617009\pi\)
\(812\) 1.86323i 0.0653866i
\(813\) − 12.3075i − 0.431644i
\(814\) 30.3669 1.06436
\(815\) 0 0
\(816\) −4.18338 −0.146447
\(817\) − 37.5609i − 1.31409i
\(818\) − 28.4812i − 0.995822i
\(819\) 4.51290 0.157693
\(820\) 0 0
\(821\) 38.3636 1.33890 0.669449 0.742858i \(-0.266530\pi\)
0.669449 + 0.742858i \(0.266530\pi\)
\(822\) − 13.7950i − 0.481157i
\(823\) 29.0098i 1.01122i 0.862763 + 0.505609i \(0.168732\pi\)
−0.862763 + 0.505609i \(0.831268\pi\)
\(824\) −16.2315 −0.565449
\(825\) 0 0
\(826\) 8.24215 0.286781
\(827\) 14.3567i 0.499233i 0.968345 + 0.249616i \(0.0803045\pi\)
−0.968345 + 0.249616i \(0.919695\pi\)
\(828\) 4.57131i 0.158864i
\(829\) −4.63563 −0.161002 −0.0805011 0.996755i \(-0.525652\pi\)
−0.0805011 + 0.996755i \(0.525652\pi\)
\(830\) 0 0
\(831\) 31.5520 1.09453
\(832\) − 33.8875i − 1.17484i
\(833\) 12.2238i 0.423529i
\(834\) 16.5365 0.572611
\(835\) 0 0
\(836\) 11.0336 0.381605
\(837\) 10.0520i 0.347449i
\(838\) − 34.5924i − 1.19498i
\(839\) −4.92642 −0.170079 −0.0850395 0.996378i \(-0.527102\pi\)
−0.0850395 + 0.996378i \(0.527102\pi\)
\(840\) 0 0
\(841\) −23.8941 −0.823933
\(842\) − 9.59201i − 0.330562i
\(843\) 21.3119i 0.734022i
\(844\) 1.80843 0.0622486
\(845\) 0 0
\(846\) 3.51026 0.120685
\(847\) 2.75032i 0.0945020i
\(848\) − 6.91787i − 0.237561i
\(849\) 0.864403 0.0296662
\(850\) 0 0
\(851\) 45.5194 1.56039
\(852\) − 0.489254i − 0.0167616i
\(853\) − 53.6476i − 1.83686i −0.395584 0.918430i \(-0.629458\pi\)
0.395584 0.918430i \(-0.370542\pi\)
\(854\) −11.3799 −0.389413
\(855\) 0 0
\(856\) 16.8138 0.574684
\(857\) − 27.1144i − 0.926210i −0.886303 0.463105i \(-0.846735\pi\)
0.886303 0.463105i \(-0.153265\pi\)
\(858\) − 16.6906i − 0.569806i
\(859\) −10.4190 −0.355493 −0.177747 0.984076i \(-0.556881\pi\)
−0.177747 + 0.984076i \(0.556881\pi\)
\(860\) 0 0
\(861\) −8.32198 −0.283612
\(862\) 29.6088i 1.00848i
\(863\) 43.6432i 1.48563i 0.669495 + 0.742816i \(0.266510\pi\)
−0.669495 + 0.742816i \(0.733490\pi\)
\(864\) 3.93896 0.134006
\(865\) 0 0
\(866\) 10.5193 0.357461
\(867\) − 12.4957i − 0.424376i
\(868\) 8.28864i 0.281335i
\(869\) −31.3319 −1.06286
\(870\) 0 0
\(871\) −10.6432 −0.360632
\(872\) 34.8758i 1.18105i
\(873\) − 13.5000i − 0.456905i
\(874\) −28.1391 −0.951818
\(875\) 0 0
\(876\) −5.53456 −0.186995
\(877\) 53.6435i 1.81141i 0.423907 + 0.905706i \(0.360658\pi\)
−0.423907 + 0.905706i \(0.639342\pi\)
\(878\) − 0.579102i − 0.0195438i
\(879\) −3.17701 −0.107158
\(880\) 0 0
\(881\) −54.8950 −1.84946 −0.924730 0.380623i \(-0.875710\pi\)
−0.924730 + 0.380623i \(0.875710\pi\)
\(882\) 6.46419i 0.217661i
\(883\) 19.0952i 0.642603i 0.946977 + 0.321302i \(0.104120\pi\)
−0.946977 + 0.321302i \(0.895880\pi\)
\(884\) −6.36700 −0.214145
\(885\) 0 0
\(886\) 20.0166 0.672471
\(887\) − 11.2162i − 0.376602i −0.982111 0.188301i \(-0.939702\pi\)
0.982111 0.188301i \(-0.0602980\pi\)
\(888\) − 22.6743i − 0.760901i
\(889\) −0.845988 −0.0283735
\(890\) 0 0
\(891\) 3.67008 0.122952
\(892\) 18.9807i 0.635521i
\(893\) − 12.7003i − 0.424998i
\(894\) 17.5643 0.587438
\(895\) 0 0
\(896\) −1.67977 −0.0561173
\(897\) − 25.0188i − 0.835354i
\(898\) − 4.67489i − 0.156003i
\(899\) 22.7139 0.757552
\(900\) 0 0
\(901\) −7.44857 −0.248148
\(902\) 30.7781i 1.02480i
\(903\) − 10.3020i − 0.342830i
\(904\) −14.8712 −0.494608
\(905\) 0 0
\(906\) −4.44240 −0.147589
\(907\) − 36.4513i − 1.21034i −0.796095 0.605172i \(-0.793104\pi\)
0.796095 0.605172i \(-0.206896\pi\)
\(908\) 4.85930i 0.161261i
\(909\) −7.22642 −0.239685
\(910\) 0 0
\(911\) 1.16533 0.0386091 0.0193045 0.999814i \(-0.493855\pi\)
0.0193045 + 0.999814i \(0.493855\pi\)
\(912\) 8.00401i 0.265039i
\(913\) − 45.0094i − 1.48959i
\(914\) −19.8354 −0.656097
\(915\) 0 0
\(916\) 5.89621 0.194816
\(917\) 16.3219i 0.538997i
\(918\) 2.38197i 0.0786166i
\(919\) −43.9475 −1.44969 −0.724847 0.688910i \(-0.758089\pi\)
−0.724847 + 0.688910i \(0.758089\pi\)
\(920\) 0 0
\(921\) −0.507986 −0.0167387
\(922\) 0.230029i 0.00757561i
\(923\) 2.67770i 0.0881375i
\(924\) 3.02624 0.0995561
\(925\) 0 0
\(926\) 14.3856 0.472739
\(927\) 5.27748i 0.173335i
\(928\) − 8.90060i − 0.292176i
\(929\) −31.2121 −1.02404 −0.512019 0.858974i \(-0.671102\pi\)
−0.512019 + 0.858974i \(0.671102\pi\)
\(930\) 0 0
\(931\) 23.3876 0.766499
\(932\) 6.97506i 0.228476i
\(933\) 14.6438i 0.479418i
\(934\) 1.24998 0.0409006
\(935\) 0 0
\(936\) −12.4625 −0.407349
\(937\) − 45.1060i − 1.47355i −0.676139 0.736774i \(-0.736348\pi\)
0.676139 0.736774i \(-0.263652\pi\)
\(938\) 3.28324i 0.107202i
\(939\) 14.9562 0.488076
\(940\) 0 0
\(941\) −39.1341 −1.27573 −0.637867 0.770146i \(-0.720183\pi\)
−0.637867 + 0.770146i \(0.720183\pi\)
\(942\) 5.05119i 0.164577i
\(943\) 46.1358i 1.50239i
\(944\) −12.9972 −0.423023
\(945\) 0 0
\(946\) −38.1011 −1.23877
\(947\) − 48.8537i − 1.58753i −0.608223 0.793766i \(-0.708117\pi\)
0.608223 0.793766i \(-0.291883\pi\)
\(948\) 6.32060i 0.205284i
\(949\) 30.2907 0.983278
\(950\) 0 0
\(951\) 18.1190 0.587550
\(952\) 7.26986i 0.235618i
\(953\) − 22.7824i − 0.737995i −0.929430 0.368998i \(-0.879701\pi\)
0.929430 0.368998i \(-0.120299\pi\)
\(954\) −3.93896 −0.127528
\(955\) 0 0
\(956\) 8.30924 0.268740
\(957\) − 8.29302i − 0.268075i
\(958\) − 28.2547i − 0.912868i
\(959\) −13.6893 −0.442052
\(960\) 0 0
\(961\) 70.0435 2.25947
\(962\) 33.5274i 1.08097i
\(963\) − 5.46682i − 0.176166i
\(964\) 16.7922 0.540841
\(965\) 0 0
\(966\) −7.71785 −0.248318
\(967\) 43.1927i 1.38898i 0.719500 + 0.694492i \(0.244371\pi\)
−0.719500 + 0.694492i \(0.755629\pi\)
\(968\) − 7.59507i − 0.244115i
\(969\) 8.61803 0.276851
\(970\) 0 0
\(971\) −32.0252 −1.02774 −0.513869 0.857869i \(-0.671788\pi\)
−0.513869 + 0.857869i \(0.671788\pi\)
\(972\) − 0.740367i − 0.0237473i
\(973\) − 16.4098i − 0.526073i
\(974\) 11.6271 0.372557
\(975\) 0 0
\(976\) 17.9452 0.574413
\(977\) − 7.27011i − 0.232591i −0.993215 0.116296i \(-0.962898\pi\)
0.993215 0.116296i \(-0.0371020\pi\)
\(978\) − 2.75408i − 0.0880657i
\(979\) 56.1085 1.79324
\(980\) 0 0
\(981\) 11.3395 0.362042
\(982\) 17.5725i 0.560760i
\(983\) 5.09155i 0.162395i 0.996698 + 0.0811976i \(0.0258745\pi\)
−0.996698 + 0.0811976i \(0.974126\pi\)
\(984\) 22.9813 0.732619
\(985\) 0 0
\(986\) 5.38237 0.171410
\(987\) − 3.48337i − 0.110877i
\(988\) 12.1819i 0.387559i
\(989\) −57.1129 −1.81608
\(990\) 0 0
\(991\) −62.0762 −1.97192 −0.985958 0.166992i \(-0.946594\pi\)
−0.985958 + 0.166992i \(0.946594\pi\)
\(992\) − 39.5946i − 1.25713i
\(993\) − 26.3514i − 0.836237i
\(994\) 0.826021 0.0261998
\(995\) 0 0
\(996\) −9.07979 −0.287704
\(997\) 2.56895i 0.0813594i 0.999172 + 0.0406797i \(0.0129523\pi\)
−0.999172 + 0.0406797i \(0.987048\pi\)
\(998\) 40.1643i 1.27138i
\(999\) −7.37232 −0.233250
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.1249.4 8
5.2 odd 4 1875.2.a.e.1.3 4
5.3 odd 4 1875.2.a.h.1.2 4
5.4 even 2 inner 1875.2.b.c.1249.5 8
15.2 even 4 5625.2.a.n.1.2 4
15.8 even 4 5625.2.a.i.1.3 4
25.2 odd 20 375.2.g.b.226.2 8
25.9 even 10 375.2.i.b.349.2 16
25.11 even 5 375.2.i.b.274.2 16
25.12 odd 20 375.2.g.b.151.2 8
25.13 odd 20 75.2.g.b.31.1 8
25.14 even 10 375.2.i.b.274.3 16
25.16 even 5 375.2.i.b.349.3 16
25.23 odd 20 75.2.g.b.46.1 yes 8
75.23 even 20 225.2.h.c.46.2 8
75.38 even 20 225.2.h.c.181.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.g.b.31.1 8 25.13 odd 20
75.2.g.b.46.1 yes 8 25.23 odd 20
225.2.h.c.46.2 8 75.23 even 20
225.2.h.c.181.2 8 75.38 even 20
375.2.g.b.151.2 8 25.12 odd 20
375.2.g.b.226.2 8 25.2 odd 20
375.2.i.b.274.2 16 25.11 even 5
375.2.i.b.274.3 16 25.14 even 10
375.2.i.b.349.2 16 25.9 even 10
375.2.i.b.349.3 16 25.16 even 5
1875.2.a.e.1.3 4 5.2 odd 4
1875.2.a.h.1.2 4 5.3 odd 4
1875.2.b.c.1249.4 8 1.1 even 1 trivial
1875.2.b.c.1249.5 8 5.4 even 2 inner
5625.2.a.i.1.3 4 15.8 even 4
5625.2.a.n.1.2 4 15.2 even 4