Properties

Label 4225.2.a.bg.1.3
Level $4225$
Weight $2$
Character 4225.1
Self dual yes
Analytic conductor $33.737$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4225,2,Mod(1,4225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4225, 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("4225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4225 = 5^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4225.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: \(33.7367948540\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 169)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.24698\) of defining polynomial
Character \(\chi\) \(=\) 4225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.24698 q^{2} +0.554958 q^{3} +3.04892 q^{4} +1.24698 q^{6} -2.04892 q^{7} +2.35690 q^{8} -2.69202 q^{9} +O(q^{10})\) \(q+2.24698 q^{2} +0.554958 q^{3} +3.04892 q^{4} +1.24698 q^{6} -2.04892 q^{7} +2.35690 q^{8} -2.69202 q^{9} -2.55496 q^{11} +1.69202 q^{12} -4.60388 q^{14} -0.801938 q^{16} +5.29590 q^{17} -6.04892 q^{18} -5.85086 q^{19} -1.13706 q^{21} -5.74094 q^{22} +1.89008 q^{23} +1.30798 q^{24} -3.15883 q^{27} -6.24698 q^{28} +2.26875 q^{29} -4.26875 q^{31} -6.51573 q^{32} -1.41789 q^{33} +11.8998 q^{34} -8.20775 q^{36} -5.35690 q^{37} -13.1468 q^{38} +1.27413 q^{41} -2.55496 q^{42} -6.13706 q^{43} -7.78986 q^{44} +4.24698 q^{46} +2.95108 q^{47} -0.445042 q^{48} -2.80194 q^{49} +2.93900 q^{51} -5.52111 q^{53} -7.09783 q^{54} -4.82908 q^{56} -3.24698 q^{57} +5.09783 q^{58} -12.2078 q^{59} +8.56465 q^{61} -9.59179 q^{62} +5.51573 q^{63} -13.0368 q^{64} -3.18598 q^{66} -0.576728 q^{67} +16.1468 q^{68} +1.04892 q^{69} -4.59419 q^{71} -6.34481 q^{72} +10.5526 q^{73} -12.0368 q^{74} -17.8388 q^{76} +5.23490 q^{77} -15.7778 q^{79} +6.32304 q^{81} +2.86294 q^{82} -7.72348 q^{83} -3.46681 q^{84} -13.7899 q^{86} +1.25906 q^{87} -6.02177 q^{88} +6.61356 q^{89} +5.76271 q^{92} -2.36898 q^{93} +6.63102 q^{94} -3.61596 q^{96} -11.9269 q^{97} -6.29590 q^{98} +6.87800 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} + 2 q^{3} - q^{6} + 3 q^{7} + 3 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{2} + 2 q^{3} - q^{6} + 3 q^{7} + 3 q^{8} - 3 q^{9} - 8 q^{11} - 5 q^{14} + 2 q^{16} + 2 q^{17} - 9 q^{18} - 4 q^{19} + 2 q^{21} - 3 q^{22} + 5 q^{23} + 9 q^{24} - q^{27} - 14 q^{28} - q^{29} - 5 q^{31} - 7 q^{32} - 10 q^{33} + 13 q^{34} - 7 q^{36} - 12 q^{37} - 12 q^{38} - 7 q^{41} - 8 q^{42} - 13 q^{43} + 8 q^{46} + 18 q^{47} - q^{48} - 4 q^{49} - q^{51} - q^{53} - 3 q^{54} - 4 q^{56} - 5 q^{57} - 3 q^{58} - 19 q^{59} + 4 q^{61} - q^{62} + 4 q^{63} - 11 q^{64} + 5 q^{66} + q^{67} + 21 q^{68} - 6 q^{69} - 27 q^{71} + 4 q^{72} - 9 q^{73} - 8 q^{74} - 21 q^{76} - 8 q^{77} - 5 q^{79} - q^{81} + 14 q^{82} + 7 q^{83} - 7 q^{84} - 18 q^{86} + 18 q^{87} - 15 q^{88} - 11 q^{89} - 22 q^{93} + 5 q^{94} - 21 q^{96} - 7 q^{97} - 5 q^{98} + q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.24698 1.58885 0.794427 0.607359i \(-0.207771\pi\)
0.794427 + 0.607359i \(0.207771\pi\)
\(3\) 0.554958 0.320405 0.160203 0.987084i \(-0.448785\pi\)
0.160203 + 0.987084i \(0.448785\pi\)
\(4\) 3.04892 1.52446
\(5\) 0 0
\(6\) 1.24698 0.509077
\(7\) −2.04892 −0.774418 −0.387209 0.921992i \(-0.626561\pi\)
−0.387209 + 0.921992i \(0.626561\pi\)
\(8\) 2.35690 0.833289
\(9\) −2.69202 −0.897340
\(10\) 0 0
\(11\) −2.55496 −0.770349 −0.385174 0.922844i \(-0.625859\pi\)
−0.385174 + 0.922844i \(0.625859\pi\)
\(12\) 1.69202 0.488445
\(13\) 0 0
\(14\) −4.60388 −1.23044
\(15\) 0 0
\(16\) −0.801938 −0.200484
\(17\) 5.29590 1.28444 0.642222 0.766519i \(-0.278013\pi\)
0.642222 + 0.766519i \(0.278013\pi\)
\(18\) −6.04892 −1.42574
\(19\) −5.85086 −1.34228 −0.671139 0.741331i \(-0.734195\pi\)
−0.671139 + 0.741331i \(0.734195\pi\)
\(20\) 0 0
\(21\) −1.13706 −0.248128
\(22\) −5.74094 −1.22397
\(23\) 1.89008 0.394110 0.197055 0.980392i \(-0.436862\pi\)
0.197055 + 0.980392i \(0.436862\pi\)
\(24\) 1.30798 0.266990
\(25\) 0 0
\(26\) 0 0
\(27\) −3.15883 −0.607918
\(28\) −6.24698 −1.18057
\(29\) 2.26875 0.421296 0.210648 0.977562i \(-0.432443\pi\)
0.210648 + 0.977562i \(0.432443\pi\)
\(30\) 0 0
\(31\) −4.26875 −0.766690 −0.383345 0.923605i \(-0.625228\pi\)
−0.383345 + 0.923605i \(0.625228\pi\)
\(32\) −6.51573 −1.15183
\(33\) −1.41789 −0.246824
\(34\) 11.8998 2.04079
\(35\) 0 0
\(36\) −8.20775 −1.36796
\(37\) −5.35690 −0.880668 −0.440334 0.897834i \(-0.645140\pi\)
−0.440334 + 0.897834i \(0.645140\pi\)
\(38\) −13.1468 −2.13268
\(39\) 0 0
\(40\) 0 0
\(41\) 1.27413 0.198985 0.0994926 0.995038i \(-0.468278\pi\)
0.0994926 + 0.995038i \(0.468278\pi\)
\(42\) −2.55496 −0.394239
\(43\) −6.13706 −0.935893 −0.467947 0.883757i \(-0.655006\pi\)
−0.467947 + 0.883757i \(0.655006\pi\)
\(44\) −7.78986 −1.17437
\(45\) 0 0
\(46\) 4.24698 0.626183
\(47\) 2.95108 0.430460 0.215230 0.976563i \(-0.430950\pi\)
0.215230 + 0.976563i \(0.430950\pi\)
\(48\) −0.445042 −0.0642363
\(49\) −2.80194 −0.400277
\(50\) 0 0
\(51\) 2.93900 0.411542
\(52\) 0 0
\(53\) −5.52111 −0.758382 −0.379191 0.925318i \(-0.623798\pi\)
−0.379191 + 0.925318i \(0.623798\pi\)
\(54\) −7.09783 −0.965893
\(55\) 0 0
\(56\) −4.82908 −0.645314
\(57\) −3.24698 −0.430073
\(58\) 5.09783 0.669378
\(59\) −12.2078 −1.58931 −0.794657 0.607059i \(-0.792349\pi\)
−0.794657 + 0.607059i \(0.792349\pi\)
\(60\) 0 0
\(61\) 8.56465 1.09659 0.548295 0.836285i \(-0.315277\pi\)
0.548295 + 0.836285i \(0.315277\pi\)
\(62\) −9.59179 −1.21816
\(63\) 5.51573 0.694917
\(64\) −13.0368 −1.62960
\(65\) 0 0
\(66\) −3.18598 −0.392167
\(67\) −0.576728 −0.0704586 −0.0352293 0.999379i \(-0.511216\pi\)
−0.0352293 + 0.999379i \(0.511216\pi\)
\(68\) 16.1468 1.95808
\(69\) 1.04892 0.126275
\(70\) 0 0
\(71\) −4.59419 −0.545230 −0.272615 0.962123i \(-0.587888\pi\)
−0.272615 + 0.962123i \(0.587888\pi\)
\(72\) −6.34481 −0.747744
\(73\) 10.5526 1.23508 0.617542 0.786538i \(-0.288128\pi\)
0.617542 + 0.786538i \(0.288128\pi\)
\(74\) −12.0368 −1.39925
\(75\) 0 0
\(76\) −17.8388 −2.04625
\(77\) 5.23490 0.596572
\(78\) 0 0
\(79\) −15.7778 −1.77514 −0.887569 0.460674i \(-0.847608\pi\)
−0.887569 + 0.460674i \(0.847608\pi\)
\(80\) 0 0
\(81\) 6.32304 0.702560
\(82\) 2.86294 0.316158
\(83\) −7.72348 −0.847762 −0.423881 0.905718i \(-0.639333\pi\)
−0.423881 + 0.905718i \(0.639333\pi\)
\(84\) −3.46681 −0.378260
\(85\) 0 0
\(86\) −13.7899 −1.48700
\(87\) 1.25906 0.134986
\(88\) −6.02177 −0.641923
\(89\) 6.61356 0.701036 0.350518 0.936556i \(-0.386005\pi\)
0.350518 + 0.936556i \(0.386005\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 5.76271 0.600804
\(93\) −2.36898 −0.245652
\(94\) 6.63102 0.683938
\(95\) 0 0
\(96\) −3.61596 −0.369052
\(97\) −11.9269 −1.21100 −0.605498 0.795847i \(-0.707026\pi\)
−0.605498 + 0.795847i \(0.707026\pi\)
\(98\) −6.29590 −0.635982
\(99\) 6.87800 0.691265
\(100\) 0 0
\(101\) 13.0640 1.29991 0.649957 0.759971i \(-0.274787\pi\)
0.649957 + 0.759971i \(0.274787\pi\)
\(102\) 6.60388 0.653881
\(103\) −9.16852 −0.903401 −0.451701 0.892170i \(-0.649182\pi\)
−0.451701 + 0.892170i \(0.649182\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −12.4058 −1.20496
\(107\) 6.89977 0.667026 0.333513 0.942745i \(-0.391766\pi\)
0.333513 + 0.942745i \(0.391766\pi\)
\(108\) −9.63102 −0.926746
\(109\) 0.121998 0.0116853 0.00584264 0.999983i \(-0.498140\pi\)
0.00584264 + 0.999983i \(0.498140\pi\)
\(110\) 0 0
\(111\) −2.97285 −0.282171
\(112\) 1.64310 0.155259
\(113\) −7.30798 −0.687477 −0.343738 0.939065i \(-0.611693\pi\)
−0.343738 + 0.939065i \(0.611693\pi\)
\(114\) −7.29590 −0.683323
\(115\) 0 0
\(116\) 6.91723 0.642249
\(117\) 0 0
\(118\) −27.4306 −2.52519
\(119\) −10.8509 −0.994696
\(120\) 0 0
\(121\) −4.47219 −0.406563
\(122\) 19.2446 1.74232
\(123\) 0.707087 0.0637559
\(124\) −13.0151 −1.16879
\(125\) 0 0
\(126\) 12.3937 1.10412
\(127\) 18.9705 1.68336 0.841678 0.539980i \(-0.181568\pi\)
0.841678 + 0.539980i \(0.181568\pi\)
\(128\) −16.2620 −1.43738
\(129\) −3.40581 −0.299865
\(130\) 0 0
\(131\) 3.25667 0.284536 0.142268 0.989828i \(-0.454560\pi\)
0.142268 + 0.989828i \(0.454560\pi\)
\(132\) −4.32304 −0.376273
\(133\) 11.9879 1.03948
\(134\) −1.29590 −0.111948
\(135\) 0 0
\(136\) 12.4819 1.07031
\(137\) −0.792249 −0.0676864 −0.0338432 0.999427i \(-0.510775\pi\)
−0.0338432 + 0.999427i \(0.510775\pi\)
\(138\) 2.35690 0.200632
\(139\) −11.3394 −0.961799 −0.480899 0.876776i \(-0.659690\pi\)
−0.480899 + 0.876776i \(0.659690\pi\)
\(140\) 0 0
\(141\) 1.63773 0.137922
\(142\) −10.3230 −0.866291
\(143\) 0 0
\(144\) 2.15883 0.179903
\(145\) 0 0
\(146\) 23.7114 1.96237
\(147\) −1.55496 −0.128251
\(148\) −16.3327 −1.34254
\(149\) 8.40581 0.688631 0.344316 0.938854i \(-0.388111\pi\)
0.344316 + 0.938854i \(0.388111\pi\)
\(150\) 0 0
\(151\) 14.1293 1.14983 0.574913 0.818215i \(-0.305036\pi\)
0.574913 + 0.818215i \(0.305036\pi\)
\(152\) −13.7899 −1.11851
\(153\) −14.2567 −1.15258
\(154\) 11.7627 0.947866
\(155\) 0 0
\(156\) 0 0
\(157\) 9.43296 0.752832 0.376416 0.926451i \(-0.377156\pi\)
0.376416 + 0.926451i \(0.377156\pi\)
\(158\) −35.4523 −2.82044
\(159\) −3.06398 −0.242990
\(160\) 0 0
\(161\) −3.87263 −0.305206
\(162\) 14.2078 1.11627
\(163\) −8.70410 −0.681758 −0.340879 0.940107i \(-0.610725\pi\)
−0.340879 + 0.940107i \(0.610725\pi\)
\(164\) 3.88471 0.303345
\(165\) 0 0
\(166\) −17.3545 −1.34697
\(167\) 23.8538 1.84587 0.922933 0.384961i \(-0.125785\pi\)
0.922933 + 0.384961i \(0.125785\pi\)
\(168\) −2.67994 −0.206762
\(169\) 0 0
\(170\) 0 0
\(171\) 15.7506 1.20448
\(172\) −18.7114 −1.42673
\(173\) 18.8552 1.43353 0.716766 0.697314i \(-0.245622\pi\)
0.716766 + 0.697314i \(0.245622\pi\)
\(174\) 2.82908 0.214472
\(175\) 0 0
\(176\) 2.04892 0.154443
\(177\) −6.77479 −0.509224
\(178\) 14.8605 1.11384
\(179\) 6.02177 0.450088 0.225044 0.974349i \(-0.427747\pi\)
0.225044 + 0.974349i \(0.427747\pi\)
\(180\) 0 0
\(181\) −4.77777 −0.355129 −0.177565 0.984109i \(-0.556822\pi\)
−0.177565 + 0.984109i \(0.556822\pi\)
\(182\) 0 0
\(183\) 4.75302 0.351353
\(184\) 4.45473 0.328407
\(185\) 0 0
\(186\) −5.32304 −0.390305
\(187\) −13.5308 −0.989470
\(188\) 8.99761 0.656218
\(189\) 6.47219 0.470782
\(190\) 0 0
\(191\) 18.4306 1.33359 0.666795 0.745242i \(-0.267666\pi\)
0.666795 + 0.745242i \(0.267666\pi\)
\(192\) −7.23490 −0.522134
\(193\) −6.05429 −0.435798 −0.217899 0.975971i \(-0.569920\pi\)
−0.217899 + 0.975971i \(0.569920\pi\)
\(194\) −26.7995 −1.92410
\(195\) 0 0
\(196\) −8.54288 −0.610205
\(197\) 11.4155 0.813321 0.406660 0.913579i \(-0.366693\pi\)
0.406660 + 0.913579i \(0.366693\pi\)
\(198\) 15.4547 1.09832
\(199\) −13.9051 −0.985710 −0.492855 0.870111i \(-0.664047\pi\)
−0.492855 + 0.870111i \(0.664047\pi\)
\(200\) 0 0
\(201\) −0.320060 −0.0225753
\(202\) 29.3545 2.06538
\(203\) −4.64848 −0.326259
\(204\) 8.96077 0.627379
\(205\) 0 0
\(206\) −20.6015 −1.43537
\(207\) −5.08815 −0.353651
\(208\) 0 0
\(209\) 14.9487 1.03402
\(210\) 0 0
\(211\) −13.2446 −0.911795 −0.455897 0.890032i \(-0.650682\pi\)
−0.455897 + 0.890032i \(0.650682\pi\)
\(212\) −16.8334 −1.15612
\(213\) −2.54958 −0.174694
\(214\) 15.5036 1.05981
\(215\) 0 0
\(216\) −7.44504 −0.506571
\(217\) 8.74632 0.593739
\(218\) 0.274127 0.0185662
\(219\) 5.85623 0.395727
\(220\) 0 0
\(221\) 0 0
\(222\) −6.67994 −0.448328
\(223\) 7.33513 0.491196 0.245598 0.969372i \(-0.421016\pi\)
0.245598 + 0.969372i \(0.421016\pi\)
\(224\) 13.3502 0.891997
\(225\) 0 0
\(226\) −16.4209 −1.09230
\(227\) 8.67456 0.575751 0.287875 0.957668i \(-0.407051\pi\)
0.287875 + 0.957668i \(0.407051\pi\)
\(228\) −9.89977 −0.655628
\(229\) −13.6866 −0.904439 −0.452219 0.891907i \(-0.649368\pi\)
−0.452219 + 0.891907i \(0.649368\pi\)
\(230\) 0 0
\(231\) 2.90515 0.191145
\(232\) 5.34721 0.351061
\(233\) 5.08815 0.333336 0.166668 0.986013i \(-0.446699\pi\)
0.166668 + 0.986013i \(0.446699\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −37.2204 −2.42284
\(237\) −8.75600 −0.568764
\(238\) −24.3817 −1.58043
\(239\) −10.9239 −0.706611 −0.353305 0.935508i \(-0.614942\pi\)
−0.353305 + 0.935508i \(0.614942\pi\)
\(240\) 0 0
\(241\) 11.9148 0.767502 0.383751 0.923437i \(-0.374632\pi\)
0.383751 + 0.923437i \(0.374632\pi\)
\(242\) −10.0489 −0.645969
\(243\) 12.9855 0.833022
\(244\) 26.1129 1.67171
\(245\) 0 0
\(246\) 1.58881 0.101299
\(247\) 0 0
\(248\) −10.0610 −0.638874
\(249\) −4.28621 −0.271627
\(250\) 0 0
\(251\) 22.3478 1.41058 0.705290 0.708919i \(-0.250817\pi\)
0.705290 + 0.708919i \(0.250817\pi\)
\(252\) 16.8170 1.05937
\(253\) −4.82908 −0.303602
\(254\) 42.6262 2.67461
\(255\) 0 0
\(256\) −10.4668 −0.654176
\(257\) 18.6601 1.16398 0.581992 0.813194i \(-0.302273\pi\)
0.581992 + 0.813194i \(0.302273\pi\)
\(258\) −7.65279 −0.476442
\(259\) 10.9758 0.682005
\(260\) 0 0
\(261\) −6.10752 −0.378046
\(262\) 7.31767 0.452087
\(263\) −14.3991 −0.887887 −0.443944 0.896055i \(-0.646421\pi\)
−0.443944 + 0.896055i \(0.646421\pi\)
\(264\) −3.34183 −0.205675
\(265\) 0 0
\(266\) 26.9366 1.65159
\(267\) 3.67025 0.224616
\(268\) −1.75840 −0.107411
\(269\) 0.652793 0.0398015 0.0199007 0.999802i \(-0.493665\pi\)
0.0199007 + 0.999802i \(0.493665\pi\)
\(270\) 0 0
\(271\) −1.99569 −0.121229 −0.0606147 0.998161i \(-0.519306\pi\)
−0.0606147 + 0.998161i \(0.519306\pi\)
\(272\) −4.24698 −0.257511
\(273\) 0 0
\(274\) −1.78017 −0.107544
\(275\) 0 0
\(276\) 3.19806 0.192501
\(277\) −11.7845 −0.708061 −0.354030 0.935234i \(-0.615189\pi\)
−0.354030 + 0.935234i \(0.615189\pi\)
\(278\) −25.4795 −1.52816
\(279\) 11.4916 0.687982
\(280\) 0 0
\(281\) −6.47219 −0.386098 −0.193049 0.981189i \(-0.561838\pi\)
−0.193049 + 0.981189i \(0.561838\pi\)
\(282\) 3.67994 0.219137
\(283\) −6.58104 −0.391202 −0.195601 0.980684i \(-0.562666\pi\)
−0.195601 + 0.980684i \(0.562666\pi\)
\(284\) −14.0073 −0.831180
\(285\) 0 0
\(286\) 0 0
\(287\) −2.61058 −0.154098
\(288\) 17.5405 1.03358
\(289\) 11.0465 0.649796
\(290\) 0 0
\(291\) −6.61894 −0.388009
\(292\) 32.1739 1.88284
\(293\) 24.3381 1.42185 0.710924 0.703269i \(-0.248277\pi\)
0.710924 + 0.703269i \(0.248277\pi\)
\(294\) −3.49396 −0.203772
\(295\) 0 0
\(296\) −12.6256 −0.733851
\(297\) 8.07069 0.468309
\(298\) 18.8877 1.09413
\(299\) 0 0
\(300\) 0 0
\(301\) 12.5743 0.724773
\(302\) 31.7482 1.82691
\(303\) 7.24996 0.416500
\(304\) 4.69202 0.269106
\(305\) 0 0
\(306\) −32.0344 −1.83129
\(307\) 14.0737 0.803227 0.401613 0.915809i \(-0.368450\pi\)
0.401613 + 0.915809i \(0.368450\pi\)
\(308\) 15.9608 0.909449
\(309\) −5.08815 −0.289455
\(310\) 0 0
\(311\) −29.7700 −1.68810 −0.844051 0.536263i \(-0.819836\pi\)
−0.844051 + 0.536263i \(0.819836\pi\)
\(312\) 0 0
\(313\) 7.47889 0.422732 0.211366 0.977407i \(-0.432209\pi\)
0.211366 + 0.977407i \(0.432209\pi\)
\(314\) 21.1957 1.19614
\(315\) 0 0
\(316\) −48.1051 −2.70613
\(317\) −30.0301 −1.68666 −0.843330 0.537396i \(-0.819408\pi\)
−0.843330 + 0.537396i \(0.819408\pi\)
\(318\) −6.88471 −0.386075
\(319\) −5.79656 −0.324545
\(320\) 0 0
\(321\) 3.82908 0.213719
\(322\) −8.70171 −0.484927
\(323\) −30.9855 −1.72408
\(324\) 19.2784 1.07102
\(325\) 0 0
\(326\) −19.5579 −1.08321
\(327\) 0.0677037 0.00374402
\(328\) 3.00298 0.165812
\(329\) −6.04652 −0.333356
\(330\) 0 0
\(331\) −15.7168 −0.863872 −0.431936 0.901904i \(-0.642169\pi\)
−0.431936 + 0.901904i \(0.642169\pi\)
\(332\) −23.5483 −1.29238
\(333\) 14.4209 0.790259
\(334\) 53.5991 2.93281
\(335\) 0 0
\(336\) 0.911854 0.0497457
\(337\) −1.95407 −0.106445 −0.0532224 0.998583i \(-0.516949\pi\)
−0.0532224 + 0.998583i \(0.516949\pi\)
\(338\) 0 0
\(339\) −4.05562 −0.220271
\(340\) 0 0
\(341\) 10.9065 0.590619
\(342\) 35.3913 1.91374
\(343\) 20.0834 1.08440
\(344\) −14.4644 −0.779869
\(345\) 0 0
\(346\) 42.3672 2.27767
\(347\) 17.1250 0.919317 0.459659 0.888096i \(-0.347972\pi\)
0.459659 + 0.888096i \(0.347972\pi\)
\(348\) 3.83877 0.205780
\(349\) −10.4668 −0.560276 −0.280138 0.959960i \(-0.590380\pi\)
−0.280138 + 0.959960i \(0.590380\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 16.6474 0.887310
\(353\) −15.5308 −0.826621 −0.413310 0.910590i \(-0.635628\pi\)
−0.413310 + 0.910590i \(0.635628\pi\)
\(354\) −15.2228 −0.809084
\(355\) 0 0
\(356\) 20.1642 1.06870
\(357\) −6.02177 −0.318706
\(358\) 13.5308 0.715125
\(359\) −21.4263 −1.13083 −0.565417 0.824805i \(-0.691285\pi\)
−0.565417 + 0.824805i \(0.691285\pi\)
\(360\) 0 0
\(361\) 15.2325 0.801711
\(362\) −10.7356 −0.564249
\(363\) −2.48188 −0.130265
\(364\) 0 0
\(365\) 0 0
\(366\) 10.6799 0.558249
\(367\) −34.3032 −1.79061 −0.895306 0.445452i \(-0.853043\pi\)
−0.895306 + 0.445452i \(0.853043\pi\)
\(368\) −1.51573 −0.0790129
\(369\) −3.42998 −0.178557
\(370\) 0 0
\(371\) 11.3123 0.587305
\(372\) −7.22282 −0.374486
\(373\) 12.5961 0.652202 0.326101 0.945335i \(-0.394265\pi\)
0.326101 + 0.945335i \(0.394265\pi\)
\(374\) −30.4034 −1.57212
\(375\) 0 0
\(376\) 6.95539 0.358697
\(377\) 0 0
\(378\) 14.5429 0.748005
\(379\) 16.5386 0.849529 0.424765 0.905304i \(-0.360357\pi\)
0.424765 + 0.905304i \(0.360357\pi\)
\(380\) 0 0
\(381\) 10.5278 0.539356
\(382\) 41.4131 2.11888
\(383\) −7.53617 −0.385080 −0.192540 0.981289i \(-0.561673\pi\)
−0.192540 + 0.981289i \(0.561673\pi\)
\(384\) −9.02475 −0.460543
\(385\) 0 0
\(386\) −13.6039 −0.692419
\(387\) 16.5211 0.839815
\(388\) −36.3642 −1.84611
\(389\) 35.5555 1.80274 0.901369 0.433052i \(-0.142563\pi\)
0.901369 + 0.433052i \(0.142563\pi\)
\(390\) 0 0
\(391\) 10.0097 0.506212
\(392\) −6.60388 −0.333546
\(393\) 1.80731 0.0911670
\(394\) 25.6504 1.29225
\(395\) 0 0
\(396\) 20.9705 1.05381
\(397\) −1.35152 −0.0678308 −0.0339154 0.999425i \(-0.510798\pi\)
−0.0339154 + 0.999425i \(0.510798\pi\)
\(398\) −31.2446 −1.56615
\(399\) 6.65279 0.333056
\(400\) 0 0
\(401\) 0.579121 0.0289199 0.0144600 0.999895i \(-0.495397\pi\)
0.0144600 + 0.999895i \(0.495397\pi\)
\(402\) −0.719169 −0.0358689
\(403\) 0 0
\(404\) 39.8310 1.98167
\(405\) 0 0
\(406\) −10.4450 −0.518379
\(407\) 13.6866 0.678422
\(408\) 6.92692 0.342934
\(409\) −15.1575 −0.749490 −0.374745 0.927128i \(-0.622270\pi\)
−0.374745 + 0.927128i \(0.622270\pi\)
\(410\) 0 0
\(411\) −0.439665 −0.0216871
\(412\) −27.9541 −1.37720
\(413\) 25.0127 1.23079
\(414\) −11.4330 −0.561899
\(415\) 0 0
\(416\) 0 0
\(417\) −6.29291 −0.308165
\(418\) 33.5894 1.64291
\(419\) −35.7235 −1.74521 −0.872603 0.488430i \(-0.837570\pi\)
−0.872603 + 0.488430i \(0.837570\pi\)
\(420\) 0 0
\(421\) −35.0465 −1.70806 −0.854032 0.520221i \(-0.825849\pi\)
−0.854032 + 0.520221i \(0.825849\pi\)
\(422\) −29.7603 −1.44871
\(423\) −7.94438 −0.386269
\(424\) −13.0127 −0.631951
\(425\) 0 0
\(426\) −5.72886 −0.277564
\(427\) −17.5483 −0.849220
\(428\) 21.0368 1.01685
\(429\) 0 0
\(430\) 0 0
\(431\) −34.2814 −1.65128 −0.825639 0.564199i \(-0.809185\pi\)
−0.825639 + 0.564199i \(0.809185\pi\)
\(432\) 2.53319 0.121878
\(433\) −13.7385 −0.660232 −0.330116 0.943940i \(-0.607088\pi\)
−0.330116 + 0.943940i \(0.607088\pi\)
\(434\) 19.6528 0.943364
\(435\) 0 0
\(436\) 0.371961 0.0178137
\(437\) −11.0586 −0.529005
\(438\) 13.1588 0.628753
\(439\) 10.2403 0.488742 0.244371 0.969682i \(-0.421419\pi\)
0.244371 + 0.969682i \(0.421419\pi\)
\(440\) 0 0
\(441\) 7.54288 0.359185
\(442\) 0 0
\(443\) −12.1763 −0.578513 −0.289257 0.957252i \(-0.593408\pi\)
−0.289257 + 0.957252i \(0.593408\pi\)
\(444\) −9.06398 −0.430158
\(445\) 0 0
\(446\) 16.4819 0.780440
\(447\) 4.66487 0.220641
\(448\) 26.7114 1.26199
\(449\) −12.9051 −0.609032 −0.304516 0.952507i \(-0.598495\pi\)
−0.304516 + 0.952507i \(0.598495\pi\)
\(450\) 0 0
\(451\) −3.25534 −0.153288
\(452\) −22.2814 −1.04803
\(453\) 7.84117 0.368410
\(454\) 19.4916 0.914785
\(455\) 0 0
\(456\) −7.65279 −0.358375
\(457\) −4.65710 −0.217850 −0.108925 0.994050i \(-0.534741\pi\)
−0.108925 + 0.994050i \(0.534741\pi\)
\(458\) −30.7536 −1.43702
\(459\) −16.7289 −0.780836
\(460\) 0 0
\(461\) −31.5405 −1.46899 −0.734493 0.678616i \(-0.762580\pi\)
−0.734493 + 0.678616i \(0.762580\pi\)
\(462\) 6.52781 0.303701
\(463\) −17.6504 −0.820284 −0.410142 0.912022i \(-0.634521\pi\)
−0.410142 + 0.912022i \(0.634521\pi\)
\(464\) −1.81940 −0.0844633
\(465\) 0 0
\(466\) 11.4330 0.529622
\(467\) 32.1726 1.48877 0.744385 0.667751i \(-0.232743\pi\)
0.744385 + 0.667751i \(0.232743\pi\)
\(468\) 0 0
\(469\) 1.18167 0.0545644
\(470\) 0 0
\(471\) 5.23490 0.241211
\(472\) −28.7724 −1.32436
\(473\) 15.6799 0.720964
\(474\) −19.6746 −0.903683
\(475\) 0 0
\(476\) −33.0834 −1.51637
\(477\) 14.8629 0.680527
\(478\) −24.5459 −1.12270
\(479\) −34.8998 −1.59461 −0.797306 0.603576i \(-0.793742\pi\)
−0.797306 + 0.603576i \(0.793742\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 26.7724 1.21945
\(483\) −2.14914 −0.0977895
\(484\) −13.6353 −0.619788
\(485\) 0 0
\(486\) 29.1782 1.32355
\(487\) −41.8351 −1.89573 −0.947864 0.318676i \(-0.896762\pi\)
−0.947864 + 0.318676i \(0.896762\pi\)
\(488\) 20.1860 0.913776
\(489\) −4.83041 −0.218439
\(490\) 0 0
\(491\) 21.8455 0.985873 0.492936 0.870065i \(-0.335924\pi\)
0.492936 + 0.870065i \(0.335924\pi\)
\(492\) 2.15585 0.0971932
\(493\) 12.0151 0.541131
\(494\) 0 0
\(495\) 0 0
\(496\) 3.42327 0.153709
\(497\) 9.41311 0.422236
\(498\) −9.63102 −0.431576
\(499\) 23.5472 1.05412 0.527058 0.849829i \(-0.323295\pi\)
0.527058 + 0.849829i \(0.323295\pi\)
\(500\) 0 0
\(501\) 13.2379 0.591425
\(502\) 50.2150 2.24121
\(503\) 7.08682 0.315986 0.157993 0.987440i \(-0.449498\pi\)
0.157993 + 0.987440i \(0.449498\pi\)
\(504\) 13.0000 0.579066
\(505\) 0 0
\(506\) −10.8509 −0.482379
\(507\) 0 0
\(508\) 57.8394 2.56621
\(509\) 7.61894 0.337704 0.168852 0.985641i \(-0.445994\pi\)
0.168852 + 0.985641i \(0.445994\pi\)
\(510\) 0 0
\(511\) −21.6213 −0.956471
\(512\) 9.00538 0.397985
\(513\) 18.4819 0.815995
\(514\) 41.9288 1.84940
\(515\) 0 0
\(516\) −10.3840 −0.457132
\(517\) −7.53989 −0.331604
\(518\) 24.6625 1.08361
\(519\) 10.4638 0.459311
\(520\) 0 0
\(521\) −39.5133 −1.73111 −0.865555 0.500813i \(-0.833034\pi\)
−0.865555 + 0.500813i \(0.833034\pi\)
\(522\) −13.7235 −0.600660
\(523\) 15.8194 0.691734 0.345867 0.938284i \(-0.387585\pi\)
0.345867 + 0.938284i \(0.387585\pi\)
\(524\) 9.92931 0.433764
\(525\) 0 0
\(526\) −32.3545 −1.41072
\(527\) −22.6069 −0.984770
\(528\) 1.13706 0.0494843
\(529\) −19.4276 −0.844678
\(530\) 0 0
\(531\) 32.8635 1.42616
\(532\) 36.5502 1.58465
\(533\) 0 0
\(534\) 8.24698 0.356882
\(535\) 0 0
\(536\) −1.35929 −0.0587123
\(537\) 3.34183 0.144211
\(538\) 1.46681 0.0632388
\(539\) 7.15883 0.308353
\(540\) 0 0
\(541\) 34.4819 1.48249 0.741246 0.671234i \(-0.234235\pi\)
0.741246 + 0.671234i \(0.234235\pi\)
\(542\) −4.48427 −0.192616
\(543\) −2.65146 −0.113785
\(544\) −34.5066 −1.47946
\(545\) 0 0
\(546\) 0 0
\(547\) −36.8582 −1.57594 −0.787970 0.615713i \(-0.788868\pi\)
−0.787970 + 0.615713i \(0.788868\pi\)
\(548\) −2.41550 −0.103185
\(549\) −23.0562 −0.984015
\(550\) 0 0
\(551\) −13.2741 −0.565497
\(552\) 2.47219 0.105223
\(553\) 32.3274 1.37470
\(554\) −26.4795 −1.12501
\(555\) 0 0
\(556\) −34.5730 −1.46622
\(557\) −1.27652 −0.0540879 −0.0270439 0.999634i \(-0.508609\pi\)
−0.0270439 + 0.999634i \(0.508609\pi\)
\(558\) 25.8213 1.09310
\(559\) 0 0
\(560\) 0 0
\(561\) −7.50902 −0.317031
\(562\) −14.5429 −0.613454
\(563\) 9.12737 0.384673 0.192336 0.981329i \(-0.438393\pi\)
0.192336 + 0.981329i \(0.438393\pi\)
\(564\) 4.99330 0.210256
\(565\) 0 0
\(566\) −14.7875 −0.621563
\(567\) −12.9554 −0.544075
\(568\) −10.8280 −0.454334
\(569\) −5.72156 −0.239860 −0.119930 0.992782i \(-0.538267\pi\)
−0.119930 + 0.992782i \(0.538267\pi\)
\(570\) 0 0
\(571\) 7.60148 0.318112 0.159056 0.987270i \(-0.449155\pi\)
0.159056 + 0.987270i \(0.449155\pi\)
\(572\) 0 0
\(573\) 10.2282 0.427289
\(574\) −5.86592 −0.244839
\(575\) 0 0
\(576\) 35.0954 1.46231
\(577\) −45.1564 −1.87989 −0.939944 0.341330i \(-0.889123\pi\)
−0.939944 + 0.341330i \(0.889123\pi\)
\(578\) 24.8213 1.03243
\(579\) −3.35988 −0.139632
\(580\) 0 0
\(581\) 15.8248 0.656522
\(582\) −14.8726 −0.616490
\(583\) 14.1062 0.584219
\(584\) 24.8713 1.02918
\(585\) 0 0
\(586\) 54.6872 2.25911
\(587\) 32.4040 1.33746 0.668728 0.743507i \(-0.266839\pi\)
0.668728 + 0.743507i \(0.266839\pi\)
\(588\) −4.74094 −0.195513
\(589\) 24.9758 1.02911
\(590\) 0 0
\(591\) 6.33513 0.260592
\(592\) 4.29590 0.176560
\(593\) −36.6848 −1.50647 −0.753233 0.657754i \(-0.771507\pi\)
−0.753233 + 0.657754i \(0.771507\pi\)
\(594\) 18.1347 0.744075
\(595\) 0 0
\(596\) 25.6286 1.04979
\(597\) −7.71678 −0.315827
\(598\) 0 0
\(599\) −9.99223 −0.408271 −0.204136 0.978943i \(-0.565438\pi\)
−0.204136 + 0.978943i \(0.565438\pi\)
\(600\) 0 0
\(601\) −1.81163 −0.0738978 −0.0369489 0.999317i \(-0.511764\pi\)
−0.0369489 + 0.999317i \(0.511764\pi\)
\(602\) 28.2543 1.15156
\(603\) 1.55257 0.0632253
\(604\) 43.0790 1.75286
\(605\) 0 0
\(606\) 16.2905 0.661757
\(607\) −11.2161 −0.455248 −0.227624 0.973749i \(-0.573096\pi\)
−0.227624 + 0.973749i \(0.573096\pi\)
\(608\) 38.1226 1.54608
\(609\) −2.57971 −0.104535
\(610\) 0 0
\(611\) 0 0
\(612\) −43.4674 −1.75707
\(613\) 20.8944 0.843917 0.421958 0.906615i \(-0.361343\pi\)
0.421958 + 0.906615i \(0.361343\pi\)
\(614\) 31.6233 1.27621
\(615\) 0 0
\(616\) 12.3381 0.497117
\(617\) −12.0992 −0.487094 −0.243547 0.969889i \(-0.578311\pi\)
−0.243547 + 0.969889i \(0.578311\pi\)
\(618\) −11.4330 −0.459901
\(619\) −10.5526 −0.424143 −0.212072 0.977254i \(-0.568021\pi\)
−0.212072 + 0.977254i \(0.568021\pi\)
\(620\) 0 0
\(621\) −5.97046 −0.239586
\(622\) −66.8926 −2.68215
\(623\) −13.5506 −0.542895
\(624\) 0 0
\(625\) 0 0
\(626\) 16.8049 0.671660
\(627\) 8.29590 0.331306
\(628\) 28.7603 1.14766
\(629\) −28.3696 −1.13117
\(630\) 0 0
\(631\) −13.8514 −0.551417 −0.275709 0.961241i \(-0.588913\pi\)
−0.275709 + 0.961241i \(0.588913\pi\)
\(632\) −37.1866 −1.47920
\(633\) −7.35019 −0.292144
\(634\) −67.4771 −2.67986
\(635\) 0 0
\(636\) −9.34183 −0.370428
\(637\) 0 0
\(638\) −13.0248 −0.515655
\(639\) 12.3676 0.489257
\(640\) 0 0
\(641\) 34.9608 1.38087 0.690434 0.723396i \(-0.257420\pi\)
0.690434 + 0.723396i \(0.257420\pi\)
\(642\) 8.60388 0.339568
\(643\) −33.3980 −1.31709 −0.658545 0.752541i \(-0.728828\pi\)
−0.658545 + 0.752541i \(0.728828\pi\)
\(644\) −11.8073 −0.465273
\(645\) 0 0
\(646\) −69.6238 −2.73931
\(647\) −2.32842 −0.0915397 −0.0457698 0.998952i \(-0.514574\pi\)
−0.0457698 + 0.998952i \(0.514574\pi\)
\(648\) 14.9028 0.585436
\(649\) 31.1903 1.22433
\(650\) 0 0
\(651\) 4.85384 0.190237
\(652\) −26.5381 −1.03931
\(653\) −14.5714 −0.570221 −0.285111 0.958495i \(-0.592030\pi\)
−0.285111 + 0.958495i \(0.592030\pi\)
\(654\) 0.152129 0.00594871
\(655\) 0 0
\(656\) −1.02177 −0.0398934
\(657\) −28.4077 −1.10829
\(658\) −13.5864 −0.529654
\(659\) 11.1395 0.433932 0.216966 0.976179i \(-0.430384\pi\)
0.216966 + 0.976179i \(0.430384\pi\)
\(660\) 0 0
\(661\) −13.8498 −0.538694 −0.269347 0.963043i \(-0.586808\pi\)
−0.269347 + 0.963043i \(0.586808\pi\)
\(662\) −35.3153 −1.37257
\(663\) 0 0
\(664\) −18.2034 −0.706430
\(665\) 0 0
\(666\) 32.4034 1.25561
\(667\) 4.28813 0.166037
\(668\) 72.7284 2.81395
\(669\) 4.07069 0.157382
\(670\) 0 0
\(671\) −21.8823 −0.844757
\(672\) 7.40880 0.285801
\(673\) 6.52973 0.251703 0.125851 0.992049i \(-0.459834\pi\)
0.125851 + 0.992049i \(0.459834\pi\)
\(674\) −4.39075 −0.169125
\(675\) 0 0
\(676\) 0 0
\(677\) 11.3104 0.434693 0.217346 0.976095i \(-0.430260\pi\)
0.217346 + 0.976095i \(0.430260\pi\)
\(678\) −9.11290 −0.349979
\(679\) 24.4373 0.937816
\(680\) 0 0
\(681\) 4.81402 0.184474
\(682\) 24.5066 0.938407
\(683\) −14.1793 −0.542555 −0.271277 0.962501i \(-0.587446\pi\)
−0.271277 + 0.962501i \(0.587446\pi\)
\(684\) 48.0224 1.83618
\(685\) 0 0
\(686\) 45.1269 1.72295
\(687\) −7.59551 −0.289787
\(688\) 4.92154 0.187632
\(689\) 0 0
\(690\) 0 0
\(691\) 30.7952 1.17151 0.585753 0.810490i \(-0.300799\pi\)
0.585753 + 0.810490i \(0.300799\pi\)
\(692\) 57.4878 2.18536
\(693\) −14.0925 −0.535328
\(694\) 38.4795 1.46066
\(695\) 0 0
\(696\) 2.96748 0.112482
\(697\) 6.74764 0.255585
\(698\) −23.5187 −0.890196
\(699\) 2.82371 0.106802
\(700\) 0 0
\(701\) 6.73184 0.254258 0.127129 0.991886i \(-0.459424\pi\)
0.127129 + 0.991886i \(0.459424\pi\)
\(702\) 0 0
\(703\) 31.3424 1.18210
\(704\) 33.3086 1.25536
\(705\) 0 0
\(706\) −34.8974 −1.31338
\(707\) −26.7670 −1.00668
\(708\) −20.6558 −0.776292
\(709\) −47.6252 −1.78860 −0.894300 0.447467i \(-0.852326\pi\)
−0.894300 + 0.447467i \(0.852326\pi\)
\(710\) 0 0
\(711\) 42.4741 1.59290
\(712\) 15.5875 0.584166
\(713\) −8.06829 −0.302160
\(714\) −13.5308 −0.506377
\(715\) 0 0
\(716\) 18.3599 0.686141
\(717\) −6.06233 −0.226402
\(718\) −48.1444 −1.79673
\(719\) −5.99330 −0.223512 −0.111756 0.993736i \(-0.535648\pi\)
−0.111756 + 0.993736i \(0.535648\pi\)
\(720\) 0 0
\(721\) 18.7855 0.699610
\(722\) 34.2271 1.27380
\(723\) 6.61224 0.245912
\(724\) −14.5670 −0.541380
\(725\) 0 0
\(726\) −5.57673 −0.206972
\(727\) 24.1226 0.894657 0.447329 0.894370i \(-0.352375\pi\)
0.447329 + 0.894370i \(0.352375\pi\)
\(728\) 0 0
\(729\) −11.7627 −0.435656
\(730\) 0 0
\(731\) −32.5013 −1.20210
\(732\) 14.4916 0.535624
\(733\) −36.0646 −1.33208 −0.666038 0.745918i \(-0.732011\pi\)
−0.666038 + 0.745918i \(0.732011\pi\)
\(734\) −77.0786 −2.84502
\(735\) 0 0
\(736\) −12.3153 −0.453947
\(737\) 1.47352 0.0542777
\(738\) −7.70709 −0.283702
\(739\) 27.5254 1.01254 0.506269 0.862375i \(-0.331024\pi\)
0.506269 + 0.862375i \(0.331024\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 25.4185 0.933142
\(743\) 10.4692 0.384078 0.192039 0.981387i \(-0.438490\pi\)
0.192039 + 0.981387i \(0.438490\pi\)
\(744\) −5.58343 −0.204699
\(745\) 0 0
\(746\) 28.3032 1.03625
\(747\) 20.7918 0.760731
\(748\) −41.2543 −1.50841
\(749\) −14.1371 −0.516557
\(750\) 0 0
\(751\) 4.06770 0.148433 0.0742163 0.997242i \(-0.476354\pi\)
0.0742163 + 0.997242i \(0.476354\pi\)
\(752\) −2.36658 −0.0863005
\(753\) 12.4021 0.451957
\(754\) 0 0
\(755\) 0 0
\(756\) 19.7332 0.717688
\(757\) −20.4336 −0.742670 −0.371335 0.928499i \(-0.621100\pi\)
−0.371335 + 0.928499i \(0.621100\pi\)
\(758\) 37.1618 1.34978
\(759\) −2.67994 −0.0972757
\(760\) 0 0
\(761\) −27.0237 −0.979608 −0.489804 0.871833i \(-0.662932\pi\)
−0.489804 + 0.871833i \(0.662932\pi\)
\(762\) 23.6558 0.856958
\(763\) −0.249964 −0.00904929
\(764\) 56.1933 2.03300
\(765\) 0 0
\(766\) −16.9336 −0.611837
\(767\) 0 0
\(768\) −5.80864 −0.209601
\(769\) −37.9407 −1.36818 −0.684088 0.729400i \(-0.739799\pi\)
−0.684088 + 0.729400i \(0.739799\pi\)
\(770\) 0 0
\(771\) 10.3556 0.372947
\(772\) −18.4590 −0.664355
\(773\) 16.3375 0.587620 0.293810 0.955864i \(-0.405077\pi\)
0.293810 + 0.955864i \(0.405077\pi\)
\(774\) 37.1226 1.33434
\(775\) 0 0
\(776\) −28.1105 −1.00911
\(777\) 6.09113 0.218518
\(778\) 79.8926 2.86429
\(779\) −7.45473 −0.267093
\(780\) 0 0
\(781\) 11.7380 0.420017
\(782\) 22.4916 0.804297
\(783\) −7.16660 −0.256114
\(784\) 2.24698 0.0802493
\(785\) 0 0
\(786\) 4.06100 0.144851
\(787\) 18.6907 0.666251 0.333126 0.942882i \(-0.391897\pi\)
0.333126 + 0.942882i \(0.391897\pi\)
\(788\) 34.8049 1.23987
\(789\) −7.99090 −0.284484
\(790\) 0 0
\(791\) 14.9734 0.532394
\(792\) 16.2107 0.576023
\(793\) 0 0
\(794\) −3.03684 −0.107773
\(795\) 0 0
\(796\) −42.3957 −1.50267
\(797\) −29.2519 −1.03615 −0.518077 0.855334i \(-0.673352\pi\)
−0.518077 + 0.855334i \(0.673352\pi\)
\(798\) 14.9487 0.529178
\(799\) 15.6286 0.552901
\(800\) 0 0
\(801\) −17.8039 −0.629068
\(802\) 1.30127 0.0459496
\(803\) −26.9614 −0.951446
\(804\) −0.975837 −0.0344151
\(805\) 0 0
\(806\) 0 0
\(807\) 0.362273 0.0127526
\(808\) 30.7904 1.08320
\(809\) −6.65087 −0.233832 −0.116916 0.993142i \(-0.537301\pi\)
−0.116916 + 0.993142i \(0.537301\pi\)
\(810\) 0 0
\(811\) 3.89200 0.136667 0.0683333 0.997663i \(-0.478232\pi\)
0.0683333 + 0.997663i \(0.478232\pi\)
\(812\) −14.1728 −0.497369
\(813\) −1.10752 −0.0388425
\(814\) 30.7536 1.07791
\(815\) 0 0
\(816\) −2.35690 −0.0825079
\(817\) 35.9071 1.25623
\(818\) −34.0586 −1.19083
\(819\) 0 0
\(820\) 0 0
\(821\) 45.9982 1.60535 0.802674 0.596418i \(-0.203410\pi\)
0.802674 + 0.596418i \(0.203410\pi\)
\(822\) −0.987918 −0.0344576
\(823\) −7.95300 −0.277224 −0.138612 0.990347i \(-0.544264\pi\)
−0.138612 + 0.990347i \(0.544264\pi\)
\(824\) −21.6093 −0.752794
\(825\) 0 0
\(826\) 56.2030 1.95555
\(827\) −27.9648 −0.972432 −0.486216 0.873839i \(-0.661623\pi\)
−0.486216 + 0.873839i \(0.661623\pi\)
\(828\) −15.5133 −0.539126
\(829\) 27.6310 0.959665 0.479833 0.877360i \(-0.340697\pi\)
0.479833 + 0.877360i \(0.340697\pi\)
\(830\) 0 0
\(831\) −6.53989 −0.226866
\(832\) 0 0
\(833\) −14.8388 −0.514133
\(834\) −14.1400 −0.489630
\(835\) 0 0
\(836\) 45.5773 1.57632
\(837\) 13.4843 0.466085
\(838\) −80.2699 −2.77288
\(839\) 28.6848 0.990311 0.495155 0.868804i \(-0.335111\pi\)
0.495155 + 0.868804i \(0.335111\pi\)
\(840\) 0 0
\(841\) −23.8528 −0.822509
\(842\) −78.7488 −2.71386
\(843\) −3.59179 −0.123708
\(844\) −40.3817 −1.38999
\(845\) 0 0
\(846\) −17.8509 −0.613725
\(847\) 9.16315 0.314849
\(848\) 4.42758 0.152044
\(849\) −3.65220 −0.125343
\(850\) 0 0
\(851\) −10.1250 −0.347080
\(852\) −7.77346 −0.266314
\(853\) 43.2078 1.47941 0.739703 0.672934i \(-0.234966\pi\)
0.739703 + 0.672934i \(0.234966\pi\)
\(854\) −39.4306 −1.34929
\(855\) 0 0
\(856\) 16.2620 0.555825
\(857\) −35.1685 −1.20133 −0.600667 0.799499i \(-0.705098\pi\)
−0.600667 + 0.799499i \(0.705098\pi\)
\(858\) 0 0
\(859\) 27.3793 0.934168 0.467084 0.884213i \(-0.345305\pi\)
0.467084 + 0.884213i \(0.345305\pi\)
\(860\) 0 0
\(861\) −1.44876 −0.0493737
\(862\) −77.0297 −2.62364
\(863\) 41.3913 1.40898 0.704489 0.709715i \(-0.251176\pi\)
0.704489 + 0.709715i \(0.251176\pi\)
\(864\) 20.5821 0.700217
\(865\) 0 0
\(866\) −30.8702 −1.04901
\(867\) 6.13036 0.208198
\(868\) 26.6668 0.905130
\(869\) 40.3116 1.36748
\(870\) 0 0
\(871\) 0 0
\(872\) 0.287536 0.00973721
\(873\) 32.1075 1.08668
\(874\) −24.8485 −0.840512
\(875\) 0 0
\(876\) 17.8552 0.603270
\(877\) 24.7472 0.835653 0.417826 0.908527i \(-0.362792\pi\)
0.417826 + 0.908527i \(0.362792\pi\)
\(878\) 23.0097 0.776539
\(879\) 13.5066 0.455567
\(880\) 0 0
\(881\) −28.5875 −0.963137 −0.481568 0.876409i \(-0.659933\pi\)
−0.481568 + 0.876409i \(0.659933\pi\)
\(882\) 16.9487 0.570692
\(883\) −9.61702 −0.323639 −0.161819 0.986820i \(-0.551736\pi\)
−0.161819 + 0.986820i \(0.551736\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −27.3599 −0.919173
\(887\) −15.9661 −0.536091 −0.268045 0.963406i \(-0.586378\pi\)
−0.268045 + 0.963406i \(0.586378\pi\)
\(888\) −7.00670 −0.235130
\(889\) −38.8689 −1.30362
\(890\) 0 0
\(891\) −16.1551 −0.541217
\(892\) 22.3642 0.748809
\(893\) −17.2664 −0.577797
\(894\) 10.4819 0.350566
\(895\) 0 0
\(896\) 33.3196 1.11313
\(897\) 0 0
\(898\) −28.9976 −0.967663
\(899\) −9.68473 −0.323004
\(900\) 0 0
\(901\) −29.2392 −0.974099
\(902\) −7.31468 −0.243552
\(903\) 6.97823 0.232221
\(904\) −17.2241 −0.572867
\(905\) 0 0
\(906\) 17.6189 0.585350
\(907\) 28.8364 0.957496 0.478748 0.877952i \(-0.341091\pi\)
0.478748 + 0.877952i \(0.341091\pi\)
\(908\) 26.4480 0.877709
\(909\) −35.1685 −1.16647
\(910\) 0 0
\(911\) 38.5633 1.27766 0.638830 0.769348i \(-0.279419\pi\)
0.638830 + 0.769348i \(0.279419\pi\)
\(912\) 2.60388 0.0862229
\(913\) 19.7332 0.653073
\(914\) −10.4644 −0.346132
\(915\) 0 0
\(916\) −41.7294 −1.37878
\(917\) −6.67264 −0.220350
\(918\) −37.5894 −1.24064
\(919\) 8.87502 0.292760 0.146380 0.989228i \(-0.453238\pi\)
0.146380 + 0.989228i \(0.453238\pi\)
\(920\) 0 0
\(921\) 7.81030 0.257358
\(922\) −70.8708 −2.33401
\(923\) 0 0
\(924\) 8.85756 0.291392
\(925\) 0 0
\(926\) −39.6601 −1.30331
\(927\) 24.6819 0.810659
\(928\) −14.7826 −0.485261
\(929\) 24.2295 0.794945 0.397472 0.917614i \(-0.369887\pi\)
0.397472 + 0.917614i \(0.369887\pi\)
\(930\) 0 0
\(931\) 16.3937 0.537283
\(932\) 15.5133 0.508156
\(933\) −16.5211 −0.540877
\(934\) 72.2911 2.36544
\(935\) 0 0
\(936\) 0 0
\(937\) −17.2644 −0.564005 −0.282002 0.959414i \(-0.590999\pi\)
−0.282002 + 0.959414i \(0.590999\pi\)
\(938\) 2.65519 0.0866949
\(939\) 4.15047 0.135446
\(940\) 0 0
\(941\) 4.34050 0.141496 0.0707482 0.997494i \(-0.477461\pi\)
0.0707482 + 0.997494i \(0.477461\pi\)
\(942\) 11.7627 0.383250
\(943\) 2.40821 0.0784220
\(944\) 9.78986 0.318633
\(945\) 0 0
\(946\) 35.2325 1.14551
\(947\) 45.0146 1.46278 0.731389 0.681961i \(-0.238872\pi\)
0.731389 + 0.681961i \(0.238872\pi\)
\(948\) −26.6963 −0.867057
\(949\) 0 0
\(950\) 0 0
\(951\) −16.6655 −0.540415
\(952\) −25.5743 −0.828869
\(953\) 46.8859 1.51878 0.759391 0.650634i \(-0.225497\pi\)
0.759391 + 0.650634i \(0.225497\pi\)
\(954\) 33.3967 1.08126
\(955\) 0 0
\(956\) −33.3062 −1.07720
\(957\) −3.21685 −0.103986
\(958\) −78.4191 −2.53361
\(959\) 1.62325 0.0524176
\(960\) 0 0
\(961\) −12.7778 −0.412186
\(962\) 0 0
\(963\) −18.5743 −0.598550
\(964\) 36.3274 1.17003
\(965\) 0 0
\(966\) −4.82908 −0.155373
\(967\) −6.29457 −0.202420 −0.101210 0.994865i \(-0.532271\pi\)
−0.101210 + 0.994865i \(0.532271\pi\)
\(968\) −10.5405 −0.338784
\(969\) −17.1957 −0.552404
\(970\) 0 0
\(971\) −41.8068 −1.34165 −0.670823 0.741618i \(-0.734059\pi\)
−0.670823 + 0.741618i \(0.734059\pi\)
\(972\) 39.5918 1.26991
\(973\) 23.2336 0.744834
\(974\) −94.0025 −3.01203
\(975\) 0 0
\(976\) −6.86831 −0.219849
\(977\) −23.7530 −0.759926 −0.379963 0.925002i \(-0.624063\pi\)
−0.379963 + 0.925002i \(0.624063\pi\)
\(978\) −10.8538 −0.347067
\(979\) −16.8974 −0.540043
\(980\) 0 0
\(981\) −0.328421 −0.0104857
\(982\) 49.0863 1.56641
\(983\) 55.7251 1.77736 0.888678 0.458532i \(-0.151625\pi\)
0.888678 + 0.458532i \(0.151625\pi\)
\(984\) 1.66653 0.0531270
\(985\) 0 0
\(986\) 26.9976 0.859779
\(987\) −3.35557 −0.106809
\(988\) 0 0
\(989\) −11.5996 −0.368845
\(990\) 0 0
\(991\) −35.5512 −1.12932 −0.564661 0.825323i \(-0.690993\pi\)
−0.564661 + 0.825323i \(0.690993\pi\)
\(992\) 27.8140 0.883096
\(993\) −8.72215 −0.276789
\(994\) 21.1511 0.670871
\(995\) 0 0
\(996\) −13.0683 −0.414085
\(997\) −6.61058 −0.209359 −0.104680 0.994506i \(-0.533382\pi\)
−0.104680 + 0.994506i \(0.533382\pi\)
\(998\) 52.9101 1.67484
\(999\) 16.9215 0.535374
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 4225.2.a.bg.1.3 3
5.4 even 2 169.2.a.b.1.1 3
13.12 even 2 4225.2.a.bb.1.1 3
15.14 odd 2 1521.2.a.r.1.3 3
20.19 odd 2 2704.2.a.z.1.2 3
35.34 odd 2 8281.2.a.bf.1.1 3
65.4 even 6 169.2.c.b.146.1 6
65.9 even 6 169.2.c.c.146.3 6
65.19 odd 12 169.2.e.b.23.6 12
65.24 odd 12 169.2.e.b.147.6 12
65.29 even 6 169.2.c.c.22.3 6
65.34 odd 4 169.2.b.b.168.1 6
65.44 odd 4 169.2.b.b.168.6 6
65.49 even 6 169.2.c.b.22.1 6
65.54 odd 12 169.2.e.b.147.1 12
65.59 odd 12 169.2.e.b.23.1 12
65.64 even 2 169.2.a.c.1.3 yes 3
195.44 even 4 1521.2.b.l.1351.1 6
195.164 even 4 1521.2.b.l.1351.6 6
195.194 odd 2 1521.2.a.o.1.1 3
260.99 even 4 2704.2.f.o.337.3 6
260.239 even 4 2704.2.f.o.337.4 6
260.259 odd 2 2704.2.a.ba.1.2 3
455.454 odd 2 8281.2.a.bj.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.1 3 5.4 even 2
169.2.a.c.1.3 yes 3 65.64 even 2
169.2.b.b.168.1 6 65.34 odd 4
169.2.b.b.168.6 6 65.44 odd 4
169.2.c.b.22.1 6 65.49 even 6
169.2.c.b.146.1 6 65.4 even 6
169.2.c.c.22.3 6 65.29 even 6
169.2.c.c.146.3 6 65.9 even 6
169.2.e.b.23.1 12 65.59 odd 12
169.2.e.b.23.6 12 65.19 odd 12
169.2.e.b.147.1 12 65.54 odd 12
169.2.e.b.147.6 12 65.24 odd 12
1521.2.a.o.1.1 3 195.194 odd 2
1521.2.a.r.1.3 3 15.14 odd 2
1521.2.b.l.1351.1 6 195.44 even 4
1521.2.b.l.1351.6 6 195.164 even 4
2704.2.a.z.1.2 3 20.19 odd 2
2704.2.a.ba.1.2 3 260.259 odd 2
2704.2.f.o.337.3 6 260.99 even 4
2704.2.f.o.337.4 6 260.239 even 4
4225.2.a.bb.1.1 3 13.12 even 2
4225.2.a.bg.1.3 3 1.1 even 1 trivial
8281.2.a.bf.1.1 3 35.34 odd 2
8281.2.a.bj.1.3 3 455.454 odd 2