Properties

Label 4225.2.a.bg.1.2
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 1.2
Root \(0.445042\) 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)
 
\(f(q)\) \(=\) \(q+0.554958 q^{2} -0.801938 q^{3} -1.69202 q^{4} -0.445042 q^{6} +2.69202 q^{7} -2.04892 q^{8} -2.35690 q^{9} +O(q^{10})\) \(q+0.554958 q^{2} -0.801938 q^{3} -1.69202 q^{4} -0.445042 q^{6} +2.69202 q^{7} -2.04892 q^{8} -2.35690 q^{9} -1.19806 q^{11} +1.35690 q^{12} +1.49396 q^{14} +2.24698 q^{16} -1.13706 q^{17} -1.30798 q^{18} +1.93900 q^{19} -2.15883 q^{21} -0.664874 q^{22} +4.60388 q^{23} +1.64310 q^{24} +4.29590 q^{27} -4.55496 q^{28} -7.89977 q^{29} +5.89977 q^{31} +5.34481 q^{32} +0.960771 q^{33} -0.631023 q^{34} +3.98792 q^{36} -0.951083 q^{37} +1.07606 q^{38} +3.31767 q^{41} -1.19806 q^{42} -7.15883 q^{43} +2.02715 q^{44} +2.55496 q^{46} +7.69202 q^{47} -1.80194 q^{48} +0.246980 q^{49} +0.911854 q^{51} -5.87263 q^{53} +2.38404 q^{54} -5.51573 q^{56} -1.55496 q^{57} -4.38404 q^{58} -0.0120816 q^{59} -8.03684 q^{61} +3.27413 q^{62} -6.34481 q^{63} -1.52781 q^{64} +0.533188 q^{66} +9.25667 q^{67} +1.92394 q^{68} -3.69202 q^{69} -13.7409 q^{71} +4.82908 q^{72} -12.8170 q^{73} -0.527811 q^{74} -3.28083 q^{76} -3.22521 q^{77} +0.807315 q^{79} +3.62565 q^{81} +1.84117 q^{82} +16.3327 q^{83} +3.65279 q^{84} -3.97285 q^{86} +6.33513 q^{87} +2.45473 q^{88} -14.7289 q^{89} -7.78986 q^{92} -4.73125 q^{93} +4.26875 q^{94} -4.28621 q^{96} -3.13169 q^{97} +0.137063 q^{98} +2.82371 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} + 2 q^{3} - q^{6} + 3 q^{7} + 3 q^{8} - 3 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.554958 0.392415 0.196207 0.980562i \(-0.437137\pi\)
0.196207 + 0.980562i \(0.437137\pi\)
\(3\) −0.801938 −0.462999 −0.231499 0.972835i \(-0.574363\pi\)
−0.231499 + 0.972835i \(0.574363\pi\)
\(4\) −1.69202 −0.846011
\(5\) 0 0
\(6\) −0.445042 −0.181688
\(7\) 2.69202 1.01749 0.508744 0.860918i \(-0.330110\pi\)
0.508744 + 0.860918i \(0.330110\pi\)
\(8\) −2.04892 −0.724402
\(9\) −2.35690 −0.785632
\(10\) 0 0
\(11\) −1.19806 −0.361229 −0.180615 0.983554i \(-0.557809\pi\)
−0.180615 + 0.983554i \(0.557809\pi\)
\(12\) 1.35690 0.391702
\(13\) 0 0
\(14\) 1.49396 0.399277
\(15\) 0 0
\(16\) 2.24698 0.561745
\(17\) −1.13706 −0.275778 −0.137889 0.990448i \(-0.544032\pi\)
−0.137889 + 0.990448i \(0.544032\pi\)
\(18\) −1.30798 −0.308293
\(19\) 1.93900 0.444837 0.222419 0.974951i \(-0.428605\pi\)
0.222419 + 0.974951i \(0.428605\pi\)
\(20\) 0 0
\(21\) −2.15883 −0.471096
\(22\) −0.664874 −0.141752
\(23\) 4.60388 0.959974 0.479987 0.877275i \(-0.340641\pi\)
0.479987 + 0.877275i \(0.340641\pi\)
\(24\) 1.64310 0.335397
\(25\) 0 0
\(26\) 0 0
\(27\) 4.29590 0.826746
\(28\) −4.55496 −0.860806
\(29\) −7.89977 −1.46695 −0.733475 0.679716i \(-0.762103\pi\)
−0.733475 + 0.679716i \(0.762103\pi\)
\(30\) 0 0
\(31\) 5.89977 1.05963 0.529815 0.848113i \(-0.322261\pi\)
0.529815 + 0.848113i \(0.322261\pi\)
\(32\) 5.34481 0.944839
\(33\) 0.960771 0.167249
\(34\) −0.631023 −0.108219
\(35\) 0 0
\(36\) 3.98792 0.664653
\(37\) −0.951083 −0.156357 −0.0781785 0.996939i \(-0.524910\pi\)
−0.0781785 + 0.996939i \(0.524910\pi\)
\(38\) 1.07606 0.174561
\(39\) 0 0
\(40\) 0 0
\(41\) 3.31767 0.518133 0.259066 0.965860i \(-0.416585\pi\)
0.259066 + 0.965860i \(0.416585\pi\)
\(42\) −1.19806 −0.184865
\(43\) −7.15883 −1.09171 −0.545856 0.837879i \(-0.683795\pi\)
−0.545856 + 0.837879i \(0.683795\pi\)
\(44\) 2.02715 0.305604
\(45\) 0 0
\(46\) 2.55496 0.376708
\(47\) 7.69202 1.12200 0.560998 0.827817i \(-0.310417\pi\)
0.560998 + 0.827817i \(0.310417\pi\)
\(48\) −1.80194 −0.260087
\(49\) 0.246980 0.0352828
\(50\) 0 0
\(51\) 0.911854 0.127685
\(52\) 0 0
\(53\) −5.87263 −0.806667 −0.403334 0.915053i \(-0.632149\pi\)
−0.403334 + 0.915053i \(0.632149\pi\)
\(54\) 2.38404 0.324427
\(55\) 0 0
\(56\) −5.51573 −0.737070
\(57\) −1.55496 −0.205959
\(58\) −4.38404 −0.575653
\(59\) −0.0120816 −0.00157289 −0.000786444 1.00000i \(-0.500250\pi\)
−0.000786444 1.00000i \(0.500250\pi\)
\(60\) 0 0
\(61\) −8.03684 −1.02901 −0.514506 0.857487i \(-0.672025\pi\)
−0.514506 + 0.857487i \(0.672025\pi\)
\(62\) 3.27413 0.415815
\(63\) −6.34481 −0.799371
\(64\) −1.52781 −0.190976
\(65\) 0 0
\(66\) 0.533188 0.0656309
\(67\) 9.25667 1.13088 0.565441 0.824789i \(-0.308706\pi\)
0.565441 + 0.824789i \(0.308706\pi\)
\(68\) 1.92394 0.233311
\(69\) −3.69202 −0.444467
\(70\) 0 0
\(71\) −13.7409 −1.63075 −0.815375 0.578934i \(-0.803469\pi\)
−0.815375 + 0.578934i \(0.803469\pi\)
\(72\) 4.82908 0.569113
\(73\) −12.8170 −1.50012 −0.750058 0.661372i \(-0.769975\pi\)
−0.750058 + 0.661372i \(0.769975\pi\)
\(74\) −0.527811 −0.0613568
\(75\) 0 0
\(76\) −3.28083 −0.376337
\(77\) −3.22521 −0.367547
\(78\) 0 0
\(79\) 0.807315 0.0908300 0.0454150 0.998968i \(-0.485539\pi\)
0.0454150 + 0.998968i \(0.485539\pi\)
\(80\) 0 0
\(81\) 3.62565 0.402850
\(82\) 1.84117 0.203323
\(83\) 16.3327 1.79275 0.896375 0.443296i \(-0.146191\pi\)
0.896375 + 0.443296i \(0.146191\pi\)
\(84\) 3.65279 0.398552
\(85\) 0 0
\(86\) −3.97285 −0.428404
\(87\) 6.33513 0.679197
\(88\) 2.45473 0.261675
\(89\) −14.7289 −1.56126 −0.780628 0.624996i \(-0.785101\pi\)
−0.780628 + 0.624996i \(0.785101\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −7.78986 −0.812149
\(93\) −4.73125 −0.490608
\(94\) 4.26875 0.440288
\(95\) 0 0
\(96\) −4.28621 −0.437459
\(97\) −3.13169 −0.317975 −0.158987 0.987281i \(-0.550823\pi\)
−0.158987 + 0.987281i \(0.550823\pi\)
\(98\) 0.137063 0.0138455
\(99\) 2.82371 0.283793
\(100\) 0 0
\(101\) 5.29052 0.526426 0.263213 0.964738i \(-0.415218\pi\)
0.263213 + 0.964738i \(0.415218\pi\)
\(102\) 0.506041 0.0501055
\(103\) 13.5308 1.33323 0.666614 0.745403i \(-0.267743\pi\)
0.666614 + 0.745403i \(0.267743\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −3.25906 −0.316548
\(107\) −5.63102 −0.544371 −0.272186 0.962245i \(-0.587747\pi\)
−0.272186 + 0.962245i \(0.587747\pi\)
\(108\) −7.26875 −0.699436
\(109\) 4.17629 0.400016 0.200008 0.979794i \(-0.435903\pi\)
0.200008 + 0.979794i \(0.435903\pi\)
\(110\) 0 0
\(111\) 0.762709 0.0723931
\(112\) 6.04892 0.571569
\(113\) −7.64310 −0.719003 −0.359501 0.933145i \(-0.617053\pi\)
−0.359501 + 0.933145i \(0.617053\pi\)
\(114\) −0.862937 −0.0808214
\(115\) 0 0
\(116\) 13.3666 1.24106
\(117\) 0 0
\(118\) −0.00670477 −0.000617224 0
\(119\) −3.06100 −0.280601
\(120\) 0 0
\(121\) −9.56465 −0.869513
\(122\) −4.46011 −0.403799
\(123\) −2.66056 −0.239895
\(124\) −9.98254 −0.896459
\(125\) 0 0
\(126\) −3.52111 −0.313685
\(127\) −6.77777 −0.601430 −0.300715 0.953714i \(-0.597225\pi\)
−0.300715 + 0.953714i \(0.597225\pi\)
\(128\) −11.5375 −1.01978
\(129\) 5.74094 0.505461
\(130\) 0 0
\(131\) −13.6799 −1.19522 −0.597611 0.801786i \(-0.703883\pi\)
−0.597611 + 0.801786i \(0.703883\pi\)
\(132\) −1.62565 −0.141494
\(133\) 5.21983 0.452617
\(134\) 5.13706 0.443775
\(135\) 0 0
\(136\) 2.32975 0.199774
\(137\) −12.9879 −1.10963 −0.554816 0.831973i \(-0.687211\pi\)
−0.554816 + 0.831973i \(0.687211\pi\)
\(138\) −2.04892 −0.174415
\(139\) 12.0465 1.02177 0.510886 0.859648i \(-0.329317\pi\)
0.510886 + 0.859648i \(0.329317\pi\)
\(140\) 0 0
\(141\) −6.16852 −0.519483
\(142\) −7.62565 −0.639930
\(143\) 0 0
\(144\) −5.29590 −0.441325
\(145\) 0 0
\(146\) −7.11290 −0.588668
\(147\) −0.198062 −0.0163359
\(148\) 1.60925 0.132280
\(149\) −0.740939 −0.0607001 −0.0303500 0.999539i \(-0.509662\pi\)
−0.0303500 + 0.999539i \(0.509662\pi\)
\(150\) 0 0
\(151\) −19.0737 −1.55219 −0.776097 0.630614i \(-0.782803\pi\)
−0.776097 + 0.630614i \(0.782803\pi\)
\(152\) −3.97285 −0.322241
\(153\) 2.67994 0.216660
\(154\) −1.78986 −0.144231
\(155\) 0 0
\(156\) 0 0
\(157\) 4.02177 0.320972 0.160486 0.987038i \(-0.448694\pi\)
0.160486 + 0.987038i \(0.448694\pi\)
\(158\) 0.448026 0.0356430
\(159\) 4.70948 0.373486
\(160\) 0 0
\(161\) 12.3937 0.976763
\(162\) 2.01208 0.158084
\(163\) −15.1371 −1.18563 −0.592813 0.805340i \(-0.701983\pi\)
−0.592813 + 0.805340i \(0.701983\pi\)
\(164\) −5.61356 −0.438346
\(165\) 0 0
\(166\) 9.06398 0.703502
\(167\) 6.26337 0.484674 0.242337 0.970192i \(-0.422086\pi\)
0.242337 + 0.970192i \(0.422086\pi\)
\(168\) 4.42327 0.341263
\(169\) 0 0
\(170\) 0 0
\(171\) −4.57002 −0.349478
\(172\) 12.1129 0.923600
\(173\) −16.3913 −1.24621 −0.623105 0.782138i \(-0.714129\pi\)
−0.623105 + 0.782138i \(0.714129\pi\)
\(174\) 3.51573 0.266527
\(175\) 0 0
\(176\) −2.69202 −0.202919
\(177\) 0.00968868 0.000728246 0
\(178\) −8.17390 −0.612660
\(179\) −2.45473 −0.183475 −0.0917376 0.995783i \(-0.529242\pi\)
−0.0917376 + 0.995783i \(0.529242\pi\)
\(180\) 0 0
\(181\) 11.8073 0.877631 0.438815 0.898577i \(-0.355398\pi\)
0.438815 + 0.898577i \(0.355398\pi\)
\(182\) 0 0
\(183\) 6.44504 0.476431
\(184\) −9.43296 −0.695407
\(185\) 0 0
\(186\) −2.62565 −0.192522
\(187\) 1.36227 0.0996192
\(188\) −13.0151 −0.949221
\(189\) 11.5646 0.841204
\(190\) 0 0
\(191\) −8.99330 −0.650732 −0.325366 0.945588i \(-0.605488\pi\)
−0.325366 + 0.945588i \(0.605488\pi\)
\(192\) 1.22521 0.0884219
\(193\) −13.5254 −0.973581 −0.486790 0.873519i \(-0.661832\pi\)
−0.486790 + 0.873519i \(0.661832\pi\)
\(194\) −1.73795 −0.124778
\(195\) 0 0
\(196\) −0.417895 −0.0298496
\(197\) −12.9758 −0.924490 −0.462245 0.886752i \(-0.652956\pi\)
−0.462245 + 0.886752i \(0.652956\pi\)
\(198\) 1.56704 0.111365
\(199\) −13.5864 −0.963116 −0.481558 0.876414i \(-0.659929\pi\)
−0.481558 + 0.876414i \(0.659929\pi\)
\(200\) 0 0
\(201\) −7.42327 −0.523597
\(202\) 2.93602 0.206577
\(203\) −21.2664 −1.49261
\(204\) −1.54288 −0.108023
\(205\) 0 0
\(206\) 7.50902 0.523179
\(207\) −10.8509 −0.754187
\(208\) 0 0
\(209\) −2.32304 −0.160688
\(210\) 0 0
\(211\) 10.4601 0.720103 0.360052 0.932932i \(-0.382759\pi\)
0.360052 + 0.932932i \(0.382759\pi\)
\(212\) 9.93661 0.682449
\(213\) 11.0194 0.755035
\(214\) −3.12498 −0.213619
\(215\) 0 0
\(216\) −8.80194 −0.598896
\(217\) 15.8823 1.07816
\(218\) 2.31767 0.156972
\(219\) 10.2784 0.694553
\(220\) 0 0
\(221\) 0 0
\(222\) 0.423272 0.0284081
\(223\) 11.4058 0.763790 0.381895 0.924206i \(-0.375272\pi\)
0.381895 + 0.924206i \(0.375272\pi\)
\(224\) 14.3884 0.961362
\(225\) 0 0
\(226\) −4.24160 −0.282147
\(227\) −10.6407 −0.706249 −0.353124 0.935576i \(-0.614881\pi\)
−0.353124 + 0.935576i \(0.614881\pi\)
\(228\) 2.63102 0.174244
\(229\) −1.13946 −0.0752974 −0.0376487 0.999291i \(-0.511987\pi\)
−0.0376487 + 0.999291i \(0.511987\pi\)
\(230\) 0 0
\(231\) 2.58642 0.170174
\(232\) 16.1860 1.06266
\(233\) 10.8509 0.710863 0.355432 0.934702i \(-0.384334\pi\)
0.355432 + 0.934702i \(0.384334\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0.0204423 0.00133068
\(237\) −0.647416 −0.0420542
\(238\) −1.69873 −0.110112
\(239\) −11.9293 −0.771643 −0.385822 0.922573i \(-0.626082\pi\)
−0.385822 + 0.922573i \(0.626082\pi\)
\(240\) 0 0
\(241\) −3.64848 −0.235019 −0.117510 0.993072i \(-0.537491\pi\)
−0.117510 + 0.993072i \(0.537491\pi\)
\(242\) −5.30798 −0.341210
\(243\) −15.7952 −1.01326
\(244\) 13.5985 0.870555
\(245\) 0 0
\(246\) −1.47650 −0.0941383
\(247\) 0 0
\(248\) −12.0881 −0.767598
\(249\) −13.0978 −0.830042
\(250\) 0 0
\(251\) 1.37329 0.0866813 0.0433406 0.999060i \(-0.486200\pi\)
0.0433406 + 0.999060i \(0.486200\pi\)
\(252\) 10.7356 0.676277
\(253\) −5.51573 −0.346771
\(254\) −3.76138 −0.236010
\(255\) 0 0
\(256\) −3.34721 −0.209200
\(257\) −29.4359 −1.83616 −0.918082 0.396391i \(-0.870263\pi\)
−0.918082 + 0.396391i \(0.870263\pi\)
\(258\) 3.18598 0.198350
\(259\) −2.56033 −0.159091
\(260\) 0 0
\(261\) 18.6189 1.15248
\(262\) −7.59179 −0.469023
\(263\) −10.6963 −0.659564 −0.329782 0.944057i \(-0.606975\pi\)
−0.329782 + 0.944057i \(0.606975\pi\)
\(264\) −1.96854 −0.121155
\(265\) 0 0
\(266\) 2.89679 0.177613
\(267\) 11.8116 0.722860
\(268\) −15.6625 −0.956738
\(269\) −10.1860 −0.621050 −0.310525 0.950565i \(-0.600505\pi\)
−0.310525 + 0.950565i \(0.600505\pi\)
\(270\) 0 0
\(271\) −29.4523 −1.78910 −0.894551 0.446966i \(-0.852505\pi\)
−0.894551 + 0.446966i \(0.852505\pi\)
\(272\) −2.55496 −0.154917
\(273\) 0 0
\(274\) −7.20775 −0.435436
\(275\) 0 0
\(276\) 6.24698 0.376024
\(277\) 10.2446 0.615538 0.307769 0.951461i \(-0.400418\pi\)
0.307769 + 0.951461i \(0.400418\pi\)
\(278\) 6.68532 0.400959
\(279\) −13.9051 −0.832480
\(280\) 0 0
\(281\) −11.5646 −0.689889 −0.344944 0.938623i \(-0.612102\pi\)
−0.344944 + 0.938623i \(0.612102\pi\)
\(282\) −3.42327 −0.203853
\(283\) 30.7090 1.82546 0.912730 0.408562i \(-0.133970\pi\)
0.912730 + 0.408562i \(0.133970\pi\)
\(284\) 23.2500 1.37963
\(285\) 0 0
\(286\) 0 0
\(287\) 8.93123 0.527194
\(288\) −12.5972 −0.742295
\(289\) −15.7071 −0.923946
\(290\) 0 0
\(291\) 2.51142 0.147222
\(292\) 21.6866 1.26911
\(293\) 18.6082 1.08710 0.543551 0.839376i \(-0.317079\pi\)
0.543551 + 0.839376i \(0.317079\pi\)
\(294\) −0.109916 −0.00641045
\(295\) 0 0
\(296\) 1.94869 0.113265
\(297\) −5.14675 −0.298645
\(298\) −0.411190 −0.0238196
\(299\) 0 0
\(300\) 0 0
\(301\) −19.2717 −1.11080
\(302\) −10.5851 −0.609103
\(303\) −4.24267 −0.243735
\(304\) 4.35690 0.249885
\(305\) 0 0
\(306\) 1.48725 0.0850207
\(307\) −8.94438 −0.510483 −0.255241 0.966877i \(-0.582155\pi\)
−0.255241 + 0.966877i \(0.582155\pi\)
\(308\) 5.45712 0.310948
\(309\) −10.8509 −0.617284
\(310\) 0 0
\(311\) 21.0398 1.19306 0.596529 0.802591i \(-0.296546\pi\)
0.596529 + 0.802591i \(0.296546\pi\)
\(312\) 0 0
\(313\) 7.12737 0.402863 0.201432 0.979503i \(-0.435441\pi\)
0.201432 + 0.979503i \(0.435441\pi\)
\(314\) 2.23191 0.125954
\(315\) 0 0
\(316\) −1.36599 −0.0768431
\(317\) −23.9651 −1.34601 −0.673007 0.739636i \(-0.734997\pi\)
−0.673007 + 0.739636i \(0.734997\pi\)
\(318\) 2.61356 0.146561
\(319\) 9.46442 0.529906
\(320\) 0 0
\(321\) 4.51573 0.252043
\(322\) 6.87800 0.383296
\(323\) −2.20477 −0.122677
\(324\) −6.13467 −0.340815
\(325\) 0 0
\(326\) −8.40044 −0.465257
\(327\) −3.34913 −0.185207
\(328\) −6.79763 −0.375336
\(329\) 20.7071 1.14162
\(330\) 0 0
\(331\) 2.89546 0.159149 0.0795745 0.996829i \(-0.474644\pi\)
0.0795745 + 0.996829i \(0.474644\pi\)
\(332\) −27.6353 −1.51669
\(333\) 2.24160 0.122839
\(334\) 3.47591 0.190193
\(335\) 0 0
\(336\) −4.85086 −0.264636
\(337\) 3.10560 0.169173 0.0845865 0.996416i \(-0.473043\pi\)
0.0845865 + 0.996416i \(0.473043\pi\)
\(338\) 0 0
\(339\) 6.12929 0.332898
\(340\) 0 0
\(341\) −7.06829 −0.382770
\(342\) −2.53617 −0.137140
\(343\) −18.1793 −0.981589
\(344\) 14.6679 0.790838
\(345\) 0 0
\(346\) −9.09651 −0.489031
\(347\) 11.3787 0.610839 0.305419 0.952218i \(-0.401203\pi\)
0.305419 + 0.952218i \(0.401203\pi\)
\(348\) −10.7192 −0.574608
\(349\) −3.34721 −0.179172 −0.0895859 0.995979i \(-0.528554\pi\)
−0.0895859 + 0.995979i \(0.528554\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −6.40342 −0.341303
\(353\) −0.637727 −0.0339428 −0.0169714 0.999856i \(-0.505402\pi\)
−0.0169714 + 0.999856i \(0.505402\pi\)
\(354\) 0.00537681 0.000285774 0
\(355\) 0 0
\(356\) 24.9215 1.32084
\(357\) 2.45473 0.129918
\(358\) −1.36227 −0.0719983
\(359\) −21.4590 −1.13256 −0.566282 0.824211i \(-0.691619\pi\)
−0.566282 + 0.824211i \(0.691619\pi\)
\(360\) 0 0
\(361\) −15.2403 −0.802120
\(362\) 6.55257 0.344395
\(363\) 7.67025 0.402584
\(364\) 0 0
\(365\) 0 0
\(366\) 3.57673 0.186959
\(367\) 9.38703 0.489999 0.244999 0.969523i \(-0.421212\pi\)
0.244999 + 0.969523i \(0.421212\pi\)
\(368\) 10.3448 0.539261
\(369\) −7.81940 −0.407062
\(370\) 0 0
\(371\) −15.8092 −0.820775
\(372\) 8.00538 0.415059
\(373\) −27.7265 −1.43562 −0.717811 0.696238i \(-0.754856\pi\)
−0.717811 + 0.696238i \(0.754856\pi\)
\(374\) 0.756004 0.0390921
\(375\) 0 0
\(376\) −15.7603 −0.812776
\(377\) 0 0
\(378\) 6.41789 0.330101
\(379\) 35.8702 1.84253 0.921265 0.388935i \(-0.127157\pi\)
0.921265 + 0.388935i \(0.127157\pi\)
\(380\) 0 0
\(381\) 5.43535 0.278462
\(382\) −4.99090 −0.255357
\(383\) −4.85517 −0.248087 −0.124044 0.992277i \(-0.539586\pi\)
−0.124044 + 0.992277i \(0.539586\pi\)
\(384\) 9.25236 0.472157
\(385\) 0 0
\(386\) −7.50604 −0.382047
\(387\) 16.8726 0.857684
\(388\) 5.29888 0.269010
\(389\) 2.38537 0.120943 0.0604716 0.998170i \(-0.480740\pi\)
0.0604716 + 0.998170i \(0.480740\pi\)
\(390\) 0 0
\(391\) −5.23490 −0.264740
\(392\) −0.506041 −0.0255589
\(393\) 10.9705 0.553387
\(394\) −7.20105 −0.362783
\(395\) 0 0
\(396\) −4.77777 −0.240092
\(397\) 15.2664 0.766196 0.383098 0.923708i \(-0.374857\pi\)
0.383098 + 0.923708i \(0.374857\pi\)
\(398\) −7.53989 −0.377941
\(399\) −4.18598 −0.209561
\(400\) 0 0
\(401\) 12.7584 0.637124 0.318562 0.947902i \(-0.396800\pi\)
0.318562 + 0.947902i \(0.396800\pi\)
\(402\) −4.11960 −0.205467
\(403\) 0 0
\(404\) −8.95167 −0.445362
\(405\) 0 0
\(406\) −11.8019 −0.585720
\(407\) 1.13946 0.0564807
\(408\) −1.86831 −0.0924953
\(409\) −25.3588 −1.25391 −0.626956 0.779054i \(-0.715700\pi\)
−0.626956 + 0.779054i \(0.715700\pi\)
\(410\) 0 0
\(411\) 10.4155 0.513759
\(412\) −22.8944 −1.12793
\(413\) −0.0325239 −0.00160040
\(414\) −6.02177 −0.295954
\(415\) 0 0
\(416\) 0 0
\(417\) −9.66056 −0.473080
\(418\) −1.28919 −0.0630565
\(419\) −11.6673 −0.569983 −0.284992 0.958530i \(-0.591991\pi\)
−0.284992 + 0.958530i \(0.591991\pi\)
\(420\) 0 0
\(421\) −8.29291 −0.404172 −0.202086 0.979368i \(-0.564772\pi\)
−0.202086 + 0.979368i \(0.564772\pi\)
\(422\) 5.80492 0.282579
\(423\) −18.1293 −0.881476
\(424\) 12.0325 0.584351
\(425\) 0 0
\(426\) 6.11529 0.296287
\(427\) −21.6353 −1.04701
\(428\) 9.52781 0.460544
\(429\) 0 0
\(430\) 0 0
\(431\) 0.932296 0.0449071 0.0224536 0.999748i \(-0.492852\pi\)
0.0224536 + 0.999748i \(0.492852\pi\)
\(432\) 9.65279 0.464420
\(433\) 13.3502 0.641569 0.320785 0.947152i \(-0.396053\pi\)
0.320785 + 0.947152i \(0.396053\pi\)
\(434\) 8.81402 0.423086
\(435\) 0 0
\(436\) −7.06638 −0.338418
\(437\) 8.92692 0.427032
\(438\) 5.70410 0.272553
\(439\) 13.9922 0.667813 0.333906 0.942606i \(-0.391633\pi\)
0.333906 + 0.942606i \(0.391633\pi\)
\(440\) 0 0
\(441\) −0.582105 −0.0277193
\(442\) 0 0
\(443\) −23.7017 −1.12610 −0.563051 0.826422i \(-0.690373\pi\)
−0.563051 + 0.826422i \(0.690373\pi\)
\(444\) −1.29052 −0.0612454
\(445\) 0 0
\(446\) 6.32975 0.299722
\(447\) 0.594187 0.0281041
\(448\) −4.11290 −0.194316
\(449\) −12.5864 −0.593990 −0.296995 0.954879i \(-0.595984\pi\)
−0.296995 + 0.954879i \(0.595984\pi\)
\(450\) 0 0
\(451\) −3.97477 −0.187165
\(452\) 12.9323 0.608284
\(453\) 15.2959 0.718664
\(454\) −5.90515 −0.277142
\(455\) 0 0
\(456\) 3.18598 0.149197
\(457\) 33.6383 1.57353 0.786767 0.617250i \(-0.211753\pi\)
0.786767 + 0.617250i \(0.211753\pi\)
\(458\) −0.632351 −0.0295478
\(459\) −4.88471 −0.227999
\(460\) 0 0
\(461\) −1.40283 −0.0653363 −0.0326681 0.999466i \(-0.510400\pi\)
−0.0326681 + 0.999466i \(0.510400\pi\)
\(462\) 1.43535 0.0667787
\(463\) 15.2010 0.706453 0.353226 0.935538i \(-0.385085\pi\)
0.353226 + 0.935538i \(0.385085\pi\)
\(464\) −17.7506 −0.824052
\(465\) 0 0
\(466\) 6.02177 0.278953
\(467\) 39.3414 1.82050 0.910250 0.414058i \(-0.135889\pi\)
0.910250 + 0.414058i \(0.135889\pi\)
\(468\) 0 0
\(469\) 24.9191 1.15066
\(470\) 0 0
\(471\) −3.22521 −0.148610
\(472\) 0.0247542 0.00113940
\(473\) 8.57673 0.394358
\(474\) −0.359289 −0.0165027
\(475\) 0 0
\(476\) 5.17928 0.237392
\(477\) 13.8412 0.633743
\(478\) −6.62027 −0.302804
\(479\) −22.3690 −1.02206 −0.511032 0.859561i \(-0.670737\pi\)
−0.511032 + 0.859561i \(0.670737\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −2.02475 −0.0922250
\(483\) −9.93900 −0.452240
\(484\) 16.1836 0.735618
\(485\) 0 0
\(486\) −8.76569 −0.397620
\(487\) −22.9205 −1.03863 −0.519313 0.854584i \(-0.673812\pi\)
−0.519313 + 0.854584i \(0.673812\pi\)
\(488\) 16.4668 0.745418
\(489\) 12.1390 0.548944
\(490\) 0 0
\(491\) 1.84356 0.0831987 0.0415993 0.999134i \(-0.486755\pi\)
0.0415993 + 0.999134i \(0.486755\pi\)
\(492\) 4.50173 0.202954
\(493\) 8.98254 0.404553
\(494\) 0 0
\(495\) 0 0
\(496\) 13.2567 0.595242
\(497\) −36.9909 −1.65927
\(498\) −7.26875 −0.325720
\(499\) −12.0344 −0.538736 −0.269368 0.963037i \(-0.586815\pi\)
−0.269368 + 0.963037i \(0.586815\pi\)
\(500\) 0 0
\(501\) −5.02284 −0.224404
\(502\) 0.762118 0.0340150
\(503\) 30.5056 1.36018 0.680088 0.733130i \(-0.261942\pi\)
0.680088 + 0.733130i \(0.261942\pi\)
\(504\) 13.0000 0.579066
\(505\) 0 0
\(506\) −3.06100 −0.136078
\(507\) 0 0
\(508\) 11.4681 0.508816
\(509\) −1.51142 −0.0669924 −0.0334962 0.999439i \(-0.510664\pi\)
−0.0334962 + 0.999439i \(0.510664\pi\)
\(510\) 0 0
\(511\) −34.5036 −1.52635
\(512\) 21.2174 0.937687
\(513\) 8.32975 0.367767
\(514\) −16.3357 −0.720538
\(515\) 0 0
\(516\) −9.71379 −0.427626
\(517\) −9.21552 −0.405298
\(518\) −1.42088 −0.0624298
\(519\) 13.1448 0.576994
\(520\) 0 0
\(521\) −5.64012 −0.247098 −0.123549 0.992338i \(-0.539428\pi\)
−0.123549 + 0.992338i \(0.539428\pi\)
\(522\) 10.3327 0.452251
\(523\) 31.7506 1.38836 0.694179 0.719802i \(-0.255768\pi\)
0.694179 + 0.719802i \(0.255768\pi\)
\(524\) 23.1468 1.01117
\(525\) 0 0
\(526\) −5.93602 −0.258823
\(527\) −6.70841 −0.292223
\(528\) 2.15883 0.0939512
\(529\) −1.80433 −0.0784492
\(530\) 0 0
\(531\) 0.0284750 0.00123571
\(532\) −8.83207 −0.382919
\(533\) 0 0
\(534\) 6.55496 0.283661
\(535\) 0 0
\(536\) −18.9661 −0.819213
\(537\) 1.96854 0.0849488
\(538\) −5.65279 −0.243709
\(539\) −0.295897 −0.0127452
\(540\) 0 0
\(541\) 24.3297 1.04602 0.523009 0.852327i \(-0.324809\pi\)
0.523009 + 0.852327i \(0.324809\pi\)
\(542\) −16.3448 −0.702070
\(543\) −9.46873 −0.406342
\(544\) −6.07739 −0.260566
\(545\) 0 0
\(546\) 0 0
\(547\) 8.18896 0.350135 0.175067 0.984556i \(-0.443986\pi\)
0.175067 + 0.984556i \(0.443986\pi\)
\(548\) 21.9758 0.938761
\(549\) 18.9420 0.808424
\(550\) 0 0
\(551\) −15.3177 −0.652555
\(552\) 7.56465 0.321973
\(553\) 2.17331 0.0924185
\(554\) 5.68532 0.241546
\(555\) 0 0
\(556\) −20.3830 −0.864431
\(557\) −25.3327 −1.07338 −0.536691 0.843779i \(-0.680326\pi\)
−0.536691 + 0.843779i \(0.680326\pi\)
\(558\) −7.71678 −0.326677
\(559\) 0 0
\(560\) 0 0
\(561\) −1.09246 −0.0461236
\(562\) −6.41789 −0.270723
\(563\) 25.3937 1.07022 0.535109 0.844783i \(-0.320270\pi\)
0.535109 + 0.844783i \(0.320270\pi\)
\(564\) 10.4373 0.439488
\(565\) 0 0
\(566\) 17.0422 0.716338
\(567\) 9.76032 0.409895
\(568\) 28.1540 1.18132
\(569\) −31.1347 −1.30523 −0.652617 0.757688i \(-0.726329\pi\)
−0.652617 + 0.757688i \(0.726329\pi\)
\(570\) 0 0
\(571\) −20.5090 −0.858276 −0.429138 0.903239i \(-0.641183\pi\)
−0.429138 + 0.903239i \(0.641183\pi\)
\(572\) 0 0
\(573\) 7.21206 0.301288
\(574\) 4.95646 0.206879
\(575\) 0 0
\(576\) 3.60089 0.150037
\(577\) −15.6890 −0.653143 −0.326572 0.945172i \(-0.605893\pi\)
−0.326572 + 0.945172i \(0.605893\pi\)
\(578\) −8.71678 −0.362570
\(579\) 10.8465 0.450767
\(580\) 0 0
\(581\) 43.9681 1.82410
\(582\) 1.39373 0.0577720
\(583\) 7.03577 0.291392
\(584\) 26.2610 1.08669
\(585\) 0 0
\(586\) 10.3268 0.426595
\(587\) −30.5687 −1.26171 −0.630853 0.775903i \(-0.717295\pi\)
−0.630853 + 0.775903i \(0.717295\pi\)
\(588\) 0.335126 0.0138203
\(589\) 11.4397 0.471363
\(590\) 0 0
\(591\) 10.4058 0.428038
\(592\) −2.13706 −0.0878328
\(593\) 29.6883 1.21915 0.609576 0.792727i \(-0.291340\pi\)
0.609576 + 0.792727i \(0.291340\pi\)
\(594\) −2.85623 −0.117193
\(595\) 0 0
\(596\) 1.25368 0.0513529
\(597\) 10.8955 0.445922
\(598\) 0 0
\(599\) 24.2325 0.990113 0.495057 0.868861i \(-0.335147\pi\)
0.495057 + 0.868861i \(0.335147\pi\)
\(600\) 0 0
\(601\) 16.4819 0.672310 0.336155 0.941807i \(-0.390873\pi\)
0.336155 + 0.941807i \(0.390873\pi\)
\(602\) −10.6950 −0.435896
\(603\) −21.8170 −0.888457
\(604\) 32.2731 1.31317
\(605\) 0 0
\(606\) −2.35450 −0.0956451
\(607\) −1.43190 −0.0581188 −0.0290594 0.999578i \(-0.509251\pi\)
−0.0290594 + 0.999578i \(0.509251\pi\)
\(608\) 10.3636 0.420300
\(609\) 17.0543 0.691075
\(610\) 0 0
\(611\) 0 0
\(612\) −4.53452 −0.183297
\(613\) −3.84846 −0.155438 −0.0777190 0.996975i \(-0.524764\pi\)
−0.0777190 + 0.996975i \(0.524764\pi\)
\(614\) −4.96376 −0.200321
\(615\) 0 0
\(616\) 6.60819 0.266251
\(617\) 15.0388 0.605437 0.302719 0.953080i \(-0.402106\pi\)
0.302719 + 0.953080i \(0.402106\pi\)
\(618\) −6.02177 −0.242231
\(619\) 12.8170 0.515159 0.257579 0.966257i \(-0.417075\pi\)
0.257579 + 0.966257i \(0.417075\pi\)
\(620\) 0 0
\(621\) 19.7778 0.793655
\(622\) 11.6762 0.468174
\(623\) −39.6504 −1.58856
\(624\) 0 0
\(625\) 0 0
\(626\) 3.95539 0.158089
\(627\) 1.86294 0.0743985
\(628\) −6.80492 −0.271546
\(629\) 1.08144 0.0431199
\(630\) 0 0
\(631\) 25.7517 1.02516 0.512579 0.858640i \(-0.328690\pi\)
0.512579 + 0.858640i \(0.328690\pi\)
\(632\) −1.65412 −0.0657974
\(633\) −8.38835 −0.333407
\(634\) −13.2996 −0.528195
\(635\) 0 0
\(636\) −7.96854 −0.315973
\(637\) 0 0
\(638\) 5.25236 0.207943
\(639\) 32.3860 1.28117
\(640\) 0 0
\(641\) 24.4571 0.965998 0.482999 0.875621i \(-0.339547\pi\)
0.482999 + 0.875621i \(0.339547\pi\)
\(642\) 2.50604 0.0989055
\(643\) 9.97344 0.393314 0.196657 0.980472i \(-0.436991\pi\)
0.196657 + 0.980472i \(0.436991\pi\)
\(644\) −20.9705 −0.826352
\(645\) 0 0
\(646\) −1.22355 −0.0481401
\(647\) −11.8431 −0.465600 −0.232800 0.972525i \(-0.574789\pi\)
−0.232800 + 0.972525i \(0.574789\pi\)
\(648\) −7.42865 −0.291825
\(649\) 0.0144745 0.000568173 0
\(650\) 0 0
\(651\) −12.7366 −0.499188
\(652\) 25.6122 1.00305
\(653\) 7.47411 0.292484 0.146242 0.989249i \(-0.453282\pi\)
0.146242 + 0.989249i \(0.453282\pi\)
\(654\) −1.85862 −0.0726780
\(655\) 0 0
\(656\) 7.45473 0.291058
\(657\) 30.2083 1.17854
\(658\) 11.4916 0.447988
\(659\) 34.1739 1.33123 0.665613 0.746297i \(-0.268170\pi\)
0.665613 + 0.746297i \(0.268170\pi\)
\(660\) 0 0
\(661\) 33.6088 1.30723 0.653615 0.756827i \(-0.273252\pi\)
0.653615 + 0.756827i \(0.273252\pi\)
\(662\) 1.60686 0.0624524
\(663\) 0 0
\(664\) −33.4644 −1.29867
\(665\) 0 0
\(666\) 1.24400 0.0482039
\(667\) −36.3696 −1.40824
\(668\) −10.5978 −0.410040
\(669\) −9.14675 −0.353634
\(670\) 0 0
\(671\) 9.62863 0.371709
\(672\) −11.5386 −0.445110
\(673\) −48.0320 −1.85150 −0.925750 0.378137i \(-0.876565\pi\)
−0.925750 + 0.378137i \(0.876565\pi\)
\(674\) 1.72348 0.0663860
\(675\) 0 0
\(676\) 0 0
\(677\) 33.6582 1.29359 0.646794 0.762665i \(-0.276109\pi\)
0.646794 + 0.762665i \(0.276109\pi\)
\(678\) 3.40150 0.130634
\(679\) −8.43057 −0.323535
\(680\) 0 0
\(681\) 8.53319 0.326992
\(682\) −3.92261 −0.150204
\(683\) −15.9041 −0.608553 −0.304276 0.952584i \(-0.598415\pi\)
−0.304276 + 0.952584i \(0.598415\pi\)
\(684\) 7.73258 0.295663
\(685\) 0 0
\(686\) −10.0887 −0.385190
\(687\) 0.913773 0.0348626
\(688\) −16.0858 −0.613264
\(689\) 0 0
\(690\) 0 0
\(691\) 33.1903 1.26262 0.631309 0.775531i \(-0.282518\pi\)
0.631309 + 0.775531i \(0.282518\pi\)
\(692\) 27.7345 1.05431
\(693\) 7.60148 0.288756
\(694\) 6.31468 0.239702
\(695\) 0 0
\(696\) −12.9801 −0.492011
\(697\) −3.77240 −0.142890
\(698\) −1.85756 −0.0703097
\(699\) −8.70171 −0.329129
\(700\) 0 0
\(701\) −14.9129 −0.563253 −0.281627 0.959524i \(-0.590874\pi\)
−0.281627 + 0.959524i \(0.590874\pi\)
\(702\) 0 0
\(703\) −1.84415 −0.0695534
\(704\) 1.83041 0.0689863
\(705\) 0 0
\(706\) −0.353912 −0.0133197
\(707\) 14.2422 0.535633
\(708\) −0.0163935 −0.000616104 0
\(709\) 38.4312 1.44331 0.721656 0.692252i \(-0.243381\pi\)
0.721656 + 0.692252i \(0.243381\pi\)
\(710\) 0 0
\(711\) −1.90276 −0.0713589
\(712\) 30.1782 1.13098
\(713\) 27.1618 1.01722
\(714\) 1.36227 0.0509818
\(715\) 0 0
\(716\) 4.15346 0.155222
\(717\) 9.56657 0.357270
\(718\) −11.9089 −0.444435
\(719\) −11.4373 −0.426538 −0.213269 0.976993i \(-0.568411\pi\)
−0.213269 + 0.976993i \(0.568411\pi\)
\(720\) 0 0
\(721\) 36.4252 1.35654
\(722\) −8.45771 −0.314764
\(723\) 2.92585 0.108814
\(724\) −19.9782 −0.742485
\(725\) 0 0
\(726\) 4.25667 0.157980
\(727\) −3.63640 −0.134867 −0.0674333 0.997724i \(-0.521481\pi\)
−0.0674333 + 0.997724i \(0.521481\pi\)
\(728\) 0 0
\(729\) 1.78986 0.0662910
\(730\) 0 0
\(731\) 8.14005 0.301071
\(732\) −10.9051 −0.403066
\(733\) 3.52217 0.130094 0.0650472 0.997882i \(-0.479280\pi\)
0.0650472 + 0.997882i \(0.479280\pi\)
\(734\) 5.20941 0.192283
\(735\) 0 0
\(736\) 24.6069 0.907021
\(737\) −11.0901 −0.408508
\(738\) −4.33944 −0.159737
\(739\) 0.420288 0.0154605 0.00773027 0.999970i \(-0.497539\pi\)
0.00773027 + 0.999970i \(0.497539\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −8.77346 −0.322084
\(743\) 25.3623 0.930452 0.465226 0.885192i \(-0.345973\pi\)
0.465226 + 0.885192i \(0.345973\pi\)
\(744\) 9.69394 0.355397
\(745\) 0 0
\(746\) −15.3870 −0.563359
\(747\) −38.4946 −1.40844
\(748\) −2.30499 −0.0842790
\(749\) −15.1588 −0.553892
\(750\) 0 0
\(751\) 0.650874 0.0237507 0.0118754 0.999929i \(-0.496220\pi\)
0.0118754 + 0.999929i \(0.496220\pi\)
\(752\) 17.2838 0.630276
\(753\) −1.10129 −0.0401333
\(754\) 0 0
\(755\) 0 0
\(756\) −19.5676 −0.711668
\(757\) 16.7909 0.610276 0.305138 0.952308i \(-0.401297\pi\)
0.305138 + 0.952308i \(0.401297\pi\)
\(758\) 19.9065 0.723036
\(759\) 4.42327 0.160555
\(760\) 0 0
\(761\) 30.9221 1.12093 0.560463 0.828179i \(-0.310623\pi\)
0.560463 + 0.828179i \(0.310623\pi\)
\(762\) 3.01639 0.109272
\(763\) 11.2427 0.407012
\(764\) 15.2168 0.550526
\(765\) 0 0
\(766\) −2.69441 −0.0973531
\(767\) 0 0
\(768\) 2.68425 0.0968596
\(769\) −43.7689 −1.57835 −0.789174 0.614169i \(-0.789491\pi\)
−0.789174 + 0.614169i \(0.789491\pi\)
\(770\) 0 0
\(771\) 23.6058 0.850142
\(772\) 22.8853 0.823660
\(773\) 42.4209 1.52577 0.762886 0.646532i \(-0.223782\pi\)
0.762886 + 0.646532i \(0.223782\pi\)
\(774\) 9.36360 0.336568
\(775\) 0 0
\(776\) 6.41657 0.230341
\(777\) 2.05323 0.0736592
\(778\) 1.32378 0.0474598
\(779\) 6.43296 0.230485
\(780\) 0 0
\(781\) 16.4625 0.589075
\(782\) −2.90515 −0.103888
\(783\) −33.9366 −1.21280
\(784\) 0.554958 0.0198199
\(785\) 0 0
\(786\) 6.08815 0.217157
\(787\) 36.0116 1.28368 0.641838 0.766841i \(-0.278172\pi\)
0.641838 + 0.766841i \(0.278172\pi\)
\(788\) 21.9554 0.782129
\(789\) 8.57779 0.305378
\(790\) 0 0
\(791\) −20.5754 −0.731577
\(792\) −5.78554 −0.205580
\(793\) 0 0
\(794\) 8.47219 0.300667
\(795\) 0 0
\(796\) 22.9885 0.814806
\(797\) 31.7101 1.12323 0.561614 0.827399i \(-0.310181\pi\)
0.561614 + 0.827399i \(0.310181\pi\)
\(798\) −2.32304 −0.0822349
\(799\) −8.74632 −0.309422
\(800\) 0 0
\(801\) 34.7144 1.22657
\(802\) 7.08038 0.250017
\(803\) 15.3556 0.541886
\(804\) 12.5603 0.442969
\(805\) 0 0
\(806\) 0 0
\(807\) 8.16852 0.287546
\(808\) −10.8398 −0.381344
\(809\) −45.2814 −1.59201 −0.796005 0.605290i \(-0.793057\pi\)
−0.796005 + 0.605290i \(0.793057\pi\)
\(810\) 0 0
\(811\) −42.8635 −1.50514 −0.752571 0.658511i \(-0.771187\pi\)
−0.752571 + 0.658511i \(0.771187\pi\)
\(812\) 35.9831 1.26276
\(813\) 23.6189 0.828352
\(814\) 0.632351 0.0221639
\(815\) 0 0
\(816\) 2.04892 0.0717265
\(817\) −13.8810 −0.485634
\(818\) −14.0731 −0.492054
\(819\) 0 0
\(820\) 0 0
\(821\) −7.82776 −0.273191 −0.136595 0.990627i \(-0.543616\pi\)
−0.136595 + 0.990627i \(0.543616\pi\)
\(822\) 5.78017 0.201606
\(823\) 36.7754 1.28191 0.640955 0.767579i \(-0.278539\pi\)
0.640955 + 0.767579i \(0.278539\pi\)
\(824\) −27.7235 −0.965793
\(825\) 0 0
\(826\) −0.0180494 −0.000628019 0
\(827\) −47.3293 −1.64580 −0.822900 0.568186i \(-0.807645\pi\)
−0.822900 + 0.568186i \(0.807645\pi\)
\(828\) 18.3599 0.638050
\(829\) 25.2687 0.877620 0.438810 0.898580i \(-0.355400\pi\)
0.438810 + 0.898580i \(0.355400\pi\)
\(830\) 0 0
\(831\) −8.21552 −0.284993
\(832\) 0 0
\(833\) −0.280831 −0.00973023
\(834\) −5.36121 −0.185643
\(835\) 0 0
\(836\) 3.93064 0.135944
\(837\) 25.3448 0.876045
\(838\) −6.47484 −0.223670
\(839\) −37.6883 −1.30114 −0.650572 0.759444i \(-0.725471\pi\)
−0.650572 + 0.759444i \(0.725471\pi\)
\(840\) 0 0
\(841\) 33.4064 1.15194
\(842\) −4.60222 −0.158603
\(843\) 9.27413 0.319418
\(844\) −17.6987 −0.609215
\(845\) 0 0
\(846\) −10.0610 −0.345904
\(847\) −25.7482 −0.884720
\(848\) −13.1957 −0.453141
\(849\) −24.6267 −0.845187
\(850\) 0 0
\(851\) −4.37867 −0.150099
\(852\) −18.6450 −0.638768
\(853\) 31.0121 1.06183 0.530917 0.847424i \(-0.321848\pi\)
0.530917 + 0.847424i \(0.321848\pi\)
\(854\) −12.0067 −0.410861
\(855\) 0 0
\(856\) 11.5375 0.394344
\(857\) −12.4692 −0.425940 −0.212970 0.977059i \(-0.568314\pi\)
−0.212970 + 0.977059i \(0.568314\pi\)
\(858\) 0 0
\(859\) −17.3163 −0.590826 −0.295413 0.955370i \(-0.595457\pi\)
−0.295413 + 0.955370i \(0.595457\pi\)
\(860\) 0 0
\(861\) −7.16229 −0.244090
\(862\) 0.517385 0.0176222
\(863\) 3.46383 0.117910 0.0589550 0.998261i \(-0.481223\pi\)
0.0589550 + 0.998261i \(0.481223\pi\)
\(864\) 22.9608 0.781141
\(865\) 0 0
\(866\) 7.40880 0.251761
\(867\) 12.5961 0.427786
\(868\) −26.8732 −0.912136
\(869\) −0.967213 −0.0328105
\(870\) 0 0
\(871\) 0 0
\(872\) −8.55688 −0.289772
\(873\) 7.38106 0.249811
\(874\) 4.95407 0.167574
\(875\) 0 0
\(876\) −17.3913 −0.587599
\(877\) −57.2549 −1.93336 −0.966680 0.255989i \(-0.917599\pi\)
−0.966680 + 0.255989i \(0.917599\pi\)
\(878\) 7.76510 0.262059
\(879\) −14.9226 −0.503327
\(880\) 0 0
\(881\) −43.1782 −1.45471 −0.727355 0.686261i \(-0.759251\pi\)
−0.727355 + 0.686261i \(0.759251\pi\)
\(882\) −0.323044 −0.0108775
\(883\) −49.9560 −1.68115 −0.840576 0.541693i \(-0.817784\pi\)
−0.840576 + 0.541693i \(0.817784\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −13.1535 −0.441899
\(887\) −17.6746 −0.593454 −0.296727 0.954962i \(-0.595895\pi\)
−0.296727 + 0.954962i \(0.595895\pi\)
\(888\) −1.56273 −0.0524417
\(889\) −18.2459 −0.611948
\(890\) 0 0
\(891\) −4.34375 −0.145521
\(892\) −19.2989 −0.646174
\(893\) 14.9148 0.499106
\(894\) 0.329749 0.0110284
\(895\) 0 0
\(896\) −31.0592 −1.03761
\(897\) 0 0
\(898\) −6.98493 −0.233090
\(899\) −46.6069 −1.55443
\(900\) 0 0
\(901\) 6.67755 0.222461
\(902\) −2.20583 −0.0734462
\(903\) 15.4547 0.514301
\(904\) 15.6601 0.520847
\(905\) 0 0
\(906\) 8.48858 0.282014
\(907\) −7.73423 −0.256811 −0.128406 0.991722i \(-0.540986\pi\)
−0.128406 + 0.991722i \(0.540986\pi\)
\(908\) 18.0043 0.597494
\(909\) −12.4692 −0.413577
\(910\) 0 0
\(911\) 39.6179 1.31260 0.656299 0.754501i \(-0.272121\pi\)
0.656299 + 0.754501i \(0.272121\pi\)
\(912\) −3.49396 −0.115697
\(913\) −19.5676 −0.647594
\(914\) 18.6679 0.617478
\(915\) 0 0
\(916\) 1.92798 0.0637024
\(917\) −36.8267 −1.21612
\(918\) −2.71081 −0.0894700
\(919\) 14.6213 0.482313 0.241157 0.970486i \(-0.422473\pi\)
0.241157 + 0.970486i \(0.422473\pi\)
\(920\) 0 0
\(921\) 7.17283 0.236353
\(922\) −0.778512 −0.0256389
\(923\) 0 0
\(924\) −4.37627 −0.143969
\(925\) 0 0
\(926\) 8.43594 0.277222
\(927\) −31.8907 −1.04743
\(928\) −42.2228 −1.38603
\(929\) 3.55735 0.116713 0.0583565 0.998296i \(-0.481414\pi\)
0.0583565 + 0.998296i \(0.481414\pi\)
\(930\) 0 0
\(931\) 0.478894 0.0156951
\(932\) −18.3599 −0.601398
\(933\) −16.8726 −0.552385
\(934\) 21.8328 0.714391
\(935\) 0 0
\(936\) 0 0
\(937\) −34.5526 −1.12878 −0.564392 0.825507i \(-0.690889\pi\)
−0.564392 + 0.825507i \(0.690889\pi\)
\(938\) 13.8291 0.451536
\(939\) −5.71571 −0.186525
\(940\) 0 0
\(941\) 20.6233 0.672299 0.336149 0.941809i \(-0.390875\pi\)
0.336149 + 0.941809i \(0.390875\pi\)
\(942\) −1.78986 −0.0583167
\(943\) 15.2741 0.497394
\(944\) −0.0271471 −0.000883562 0
\(945\) 0 0
\(946\) 4.75973 0.154752
\(947\) −29.4999 −0.958619 −0.479309 0.877646i \(-0.659113\pi\)
−0.479309 + 0.877646i \(0.659113\pi\)
\(948\) 1.09544 0.0355783
\(949\) 0 0
\(950\) 0 0
\(951\) 19.2185 0.623203
\(952\) 6.27173 0.203268
\(953\) −26.2389 −0.849963 −0.424981 0.905202i \(-0.639719\pi\)
−0.424981 + 0.905202i \(0.639719\pi\)
\(954\) 7.68127 0.248690
\(955\) 0 0
\(956\) 20.1847 0.652818
\(957\) −7.58987 −0.245346
\(958\) −12.4138 −0.401073
\(959\) −34.9638 −1.12904
\(960\) 0 0
\(961\) 3.80731 0.122817
\(962\) 0 0
\(963\) 13.2717 0.427676
\(964\) 6.17331 0.198829
\(965\) 0 0
\(966\) −5.51573 −0.177466
\(967\) −17.5176 −0.563330 −0.281665 0.959513i \(-0.590887\pi\)
−0.281665 + 0.959513i \(0.590887\pi\)
\(968\) 19.5972 0.629877
\(969\) 1.76809 0.0567991
\(970\) 0 0
\(971\) 20.5120 0.658262 0.329131 0.944284i \(-0.393244\pi\)
0.329131 + 0.944284i \(0.393244\pi\)
\(972\) 26.7259 0.857233
\(973\) 32.4295 1.03964
\(974\) −12.7199 −0.407572
\(975\) 0 0
\(976\) −18.0586 −0.578042
\(977\) −25.4450 −0.814059 −0.407030 0.913415i \(-0.633435\pi\)
−0.407030 + 0.913415i \(0.633435\pi\)
\(978\) 6.73663 0.215414
\(979\) 17.6461 0.563971
\(980\) 0 0
\(981\) −9.84309 −0.314266
\(982\) 1.02310 0.0326484
\(983\) 39.5244 1.26063 0.630316 0.776339i \(-0.282926\pi\)
0.630316 + 0.776339i \(0.282926\pi\)
\(984\) 5.45127 0.173780
\(985\) 0 0
\(986\) 4.98493 0.158753
\(987\) −16.6058 −0.528568
\(988\) 0 0
\(989\) −32.9584 −1.04802
\(990\) 0 0
\(991\) −29.8377 −0.947826 −0.473913 0.880572i \(-0.657159\pi\)
−0.473913 + 0.880572i \(0.657159\pi\)
\(992\) 31.5332 1.00118
\(993\) −2.32198 −0.0736858
\(994\) −20.5284 −0.651121
\(995\) 0 0
\(996\) 22.1618 0.702224
\(997\) 4.93123 0.156174 0.0780868 0.996947i \(-0.475119\pi\)
0.0780868 + 0.996947i \(0.475119\pi\)
\(998\) −6.67861 −0.211408
\(999\) −4.08575 −0.129268
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.1.2 3
5.4 even 2 169.2.a.b.1.2 3
13.12 even 2 4225.2.a.bb.1.2 3
15.14 odd 2 1521.2.a.r.1.2 3
20.19 odd 2 2704.2.a.z.1.1 3
35.34 odd 2 8281.2.a.bf.1.2 3
65.4 even 6 169.2.c.b.146.2 6
65.9 even 6 169.2.c.c.146.2 6
65.19 odd 12 169.2.e.b.23.4 12
65.24 odd 12 169.2.e.b.147.4 12
65.29 even 6 169.2.c.c.22.2 6
65.34 odd 4 169.2.b.b.168.3 6
65.44 odd 4 169.2.b.b.168.4 6
65.49 even 6 169.2.c.b.22.2 6
65.54 odd 12 169.2.e.b.147.3 12
65.59 odd 12 169.2.e.b.23.3 12
65.64 even 2 169.2.a.c.1.2 yes 3
195.44 even 4 1521.2.b.l.1351.3 6
195.164 even 4 1521.2.b.l.1351.4 6
195.194 odd 2 1521.2.a.o.1.2 3
260.99 even 4 2704.2.f.o.337.1 6
260.239 even 4 2704.2.f.o.337.2 6
260.259 odd 2 2704.2.a.ba.1.1 3
455.454 odd 2 8281.2.a.bj.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.2 3 5.4 even 2
169.2.a.c.1.2 yes 3 65.64 even 2
169.2.b.b.168.3 6 65.34 odd 4
169.2.b.b.168.4 6 65.44 odd 4
169.2.c.b.22.2 6 65.49 even 6
169.2.c.b.146.2 6 65.4 even 6
169.2.c.c.22.2 6 65.29 even 6
169.2.c.c.146.2 6 65.9 even 6
169.2.e.b.23.3 12 65.59 odd 12
169.2.e.b.23.4 12 65.19 odd 12
169.2.e.b.147.3 12 65.54 odd 12
169.2.e.b.147.4 12 65.24 odd 12
1521.2.a.o.1.2 3 195.194 odd 2
1521.2.a.r.1.2 3 15.14 odd 2
1521.2.b.l.1351.3 6 195.44 even 4
1521.2.b.l.1351.4 6 195.164 even 4
2704.2.a.z.1.1 3 20.19 odd 2
2704.2.a.ba.1.1 3 260.259 odd 2
2704.2.f.o.337.1 6 260.99 even 4
2704.2.f.o.337.2 6 260.239 even 4
4225.2.a.bb.1.2 3 13.12 even 2
4225.2.a.bg.1.2 3 1.1 even 1 trivial
8281.2.a.bf.1.2 3 35.34 odd 2
8281.2.a.bj.1.2 3 455.454 odd 2