Properties

Label 625.2.a.g.1.7
Level $625$
Weight $2$
Character 625.1
Self dual yes
Analytic conductor $4.991$
Analytic rank $0$
Dimension $8$
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] = [625,2,Mod(1,625)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(625, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("625.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 625 = 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 625.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: \(4.99065012633\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.6152203125.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} - 8x^{6} + 20x^{5} + 26x^{4} - 35x^{3} - 27x^{2} + 16x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-1.47435\) of defining polynomial
Character \(\chi\) \(=\) 625.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.47435 q^{2} +2.11675 q^{3} +4.12242 q^{4} +5.23759 q^{6} -0.973070 q^{7} +5.25163 q^{8} +1.48063 q^{9} +O(q^{10})\) \(q+2.47435 q^{2} +2.11675 q^{3} +4.12242 q^{4} +5.23759 q^{6} -0.973070 q^{7} +5.25163 q^{8} +1.48063 q^{9} -5.38225 q^{11} +8.72615 q^{12} -1.99670 q^{13} -2.40772 q^{14} +4.74954 q^{16} -2.04301 q^{17} +3.66361 q^{18} +6.20428 q^{19} -2.05975 q^{21} -13.3176 q^{22} +1.93813 q^{23} +11.1164 q^{24} -4.94054 q^{26} -3.21612 q^{27} -4.01141 q^{28} +4.81812 q^{29} -6.64014 q^{31} +1.24877 q^{32} -11.3929 q^{33} -5.05512 q^{34} +6.10380 q^{36} -0.978913 q^{37} +15.3516 q^{38} -4.22652 q^{39} +2.73319 q^{41} -5.09654 q^{42} +3.99413 q^{43} -22.1879 q^{44} +4.79562 q^{46} +7.21339 q^{47} +10.0536 q^{48} -6.05313 q^{49} -4.32454 q^{51} -8.23125 q^{52} +13.2746 q^{53} -7.95782 q^{54} -5.11020 q^{56} +13.1329 q^{57} +11.9217 q^{58} +6.54024 q^{59} +2.72672 q^{61} -16.4301 q^{62} -1.44076 q^{63} -6.40916 q^{64} -28.1900 q^{66} +9.56957 q^{67} -8.42215 q^{68} +4.10254 q^{69} -5.68520 q^{71} +7.77574 q^{72} -9.35281 q^{73} -2.42218 q^{74} +25.5767 q^{76} +5.23731 q^{77} -10.4579 q^{78} -3.18171 q^{79} -11.2496 q^{81} +6.76289 q^{82} +6.11387 q^{83} -8.49115 q^{84} +9.88290 q^{86} +10.1988 q^{87} -28.2656 q^{88} -3.00743 q^{89} +1.94293 q^{91} +7.98980 q^{92} -14.0555 q^{93} +17.8485 q^{94} +2.64334 q^{96} +5.95526 q^{97} -14.9776 q^{98} -7.96914 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 5 q^{2} + 5 q^{3} + 11 q^{4} - 4 q^{6} + 10 q^{7} + 15 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 5 q^{2} + 5 q^{3} + 11 q^{4} - 4 q^{6} + 10 q^{7} + 15 q^{8} + 9 q^{9} + q^{11} + 10 q^{12} + 10 q^{13} - 8 q^{14} + 13 q^{16} + 15 q^{17} - 5 q^{18} - 10 q^{19} - 14 q^{21} - 5 q^{22} + 30 q^{23} + 5 q^{24} + 11 q^{26} + 20 q^{27} - 5 q^{28} + 10 q^{29} - 9 q^{31} + 30 q^{32} + 5 q^{33} + 7 q^{34} + 3 q^{36} - 10 q^{37} + 20 q^{38} + 8 q^{39} - 4 q^{41} - 35 q^{42} - 18 q^{44} - 9 q^{46} + 30 q^{47} + 5 q^{48} - 4 q^{49} - 14 q^{51} + 5 q^{52} + 10 q^{53} - 20 q^{54} - 10 q^{57} - 30 q^{58} - 5 q^{59} + 6 q^{61} + 10 q^{62} - 9 q^{64} - 18 q^{66} + 10 q^{67} + 40 q^{68} + 3 q^{69} - 9 q^{71} - 15 q^{72} - 18 q^{74} - 10 q^{76} + 5 q^{77} - 30 q^{78} - 20 q^{79} + 8 q^{81} - 45 q^{82} + 40 q^{83} - 28 q^{84} - 24 q^{86} + 40 q^{87} - 40 q^{88} - 5 q^{89} + 6 q^{91} + 15 q^{92} - 40 q^{93} + 47 q^{94} + 71 q^{96} - 30 q^{98} - 22 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.47435 1.74963 0.874816 0.484455i \(-0.160982\pi\)
0.874816 + 0.484455i \(0.160982\pi\)
\(3\) 2.11675 1.22211 0.611053 0.791589i \(-0.290746\pi\)
0.611053 + 0.791589i \(0.290746\pi\)
\(4\) 4.12242 2.06121
\(5\) 0 0
\(6\) 5.23759 2.13824
\(7\) −0.973070 −0.367786 −0.183893 0.982946i \(-0.558870\pi\)
−0.183893 + 0.982946i \(0.558870\pi\)
\(8\) 5.25163 1.85673
\(9\) 1.48063 0.493544
\(10\) 0 0
\(11\) −5.38225 −1.62281 −0.811405 0.584484i \(-0.801297\pi\)
−0.811405 + 0.584484i \(0.801297\pi\)
\(12\) 8.72615 2.51902
\(13\) −1.99670 −0.553785 −0.276892 0.960901i \(-0.589305\pi\)
−0.276892 + 0.960901i \(0.589305\pi\)
\(14\) −2.40772 −0.643490
\(15\) 0 0
\(16\) 4.74954 1.18738
\(17\) −2.04301 −0.495502 −0.247751 0.968824i \(-0.579692\pi\)
−0.247751 + 0.968824i \(0.579692\pi\)
\(18\) 3.66361 0.863521
\(19\) 6.20428 1.42336 0.711680 0.702504i \(-0.247935\pi\)
0.711680 + 0.702504i \(0.247935\pi\)
\(20\) 0 0
\(21\) −2.05975 −0.449474
\(22\) −13.3176 −2.83932
\(23\) 1.93813 0.404128 0.202064 0.979372i \(-0.435235\pi\)
0.202064 + 0.979372i \(0.435235\pi\)
\(24\) 11.1164 2.26912
\(25\) 0 0
\(26\) −4.94054 −0.968920
\(27\) −3.21612 −0.618943
\(28\) −4.01141 −0.758085
\(29\) 4.81812 0.894702 0.447351 0.894359i \(-0.352368\pi\)
0.447351 + 0.894359i \(0.352368\pi\)
\(30\) 0 0
\(31\) −6.64014 −1.19260 −0.596302 0.802760i \(-0.703364\pi\)
−0.596302 + 0.802760i \(0.703364\pi\)
\(32\) 1.24877 0.220754
\(33\) −11.3929 −1.98325
\(34\) −5.05512 −0.866947
\(35\) 0 0
\(36\) 6.10380 1.01730
\(37\) −0.978913 −0.160932 −0.0804662 0.996757i \(-0.525641\pi\)
−0.0804662 + 0.996757i \(0.525641\pi\)
\(38\) 15.3516 2.49035
\(39\) −4.22652 −0.676784
\(40\) 0 0
\(41\) 2.73319 0.426853 0.213427 0.976959i \(-0.431538\pi\)
0.213427 + 0.976959i \(0.431538\pi\)
\(42\) −5.09654 −0.786413
\(43\) 3.99413 0.609100 0.304550 0.952496i \(-0.401494\pi\)
0.304550 + 0.952496i \(0.401494\pi\)
\(44\) −22.1879 −3.34496
\(45\) 0 0
\(46\) 4.79562 0.707076
\(47\) 7.21339 1.05218 0.526091 0.850428i \(-0.323657\pi\)
0.526091 + 0.850428i \(0.323657\pi\)
\(48\) 10.0536 1.45111
\(49\) −6.05313 −0.864734
\(50\) 0 0
\(51\) −4.32454 −0.605557
\(52\) −8.23125 −1.14147
\(53\) 13.2746 1.82340 0.911700 0.410857i \(-0.134770\pi\)
0.911700 + 0.410857i \(0.134770\pi\)
\(54\) −7.95782 −1.08292
\(55\) 0 0
\(56\) −5.11020 −0.682880
\(57\) 13.1329 1.73950
\(58\) 11.9217 1.56540
\(59\) 6.54024 0.851467 0.425733 0.904849i \(-0.360016\pi\)
0.425733 + 0.904849i \(0.360016\pi\)
\(60\) 0 0
\(61\) 2.72672 0.349121 0.174560 0.984646i \(-0.444150\pi\)
0.174560 + 0.984646i \(0.444150\pi\)
\(62\) −16.4301 −2.08662
\(63\) −1.44076 −0.181519
\(64\) −6.40916 −0.801146
\(65\) 0 0
\(66\) −28.1900 −3.46995
\(67\) 9.56957 1.16911 0.584555 0.811354i \(-0.301269\pi\)
0.584555 + 0.811354i \(0.301269\pi\)
\(68\) −8.42215 −1.02134
\(69\) 4.10254 0.493888
\(70\) 0 0
\(71\) −5.68520 −0.674710 −0.337355 0.941378i \(-0.609532\pi\)
−0.337355 + 0.941378i \(0.609532\pi\)
\(72\) 7.77574 0.916379
\(73\) −9.35281 −1.09466 −0.547332 0.836916i \(-0.684356\pi\)
−0.547332 + 0.836916i \(0.684356\pi\)
\(74\) −2.42218 −0.281572
\(75\) 0 0
\(76\) 25.5767 2.93385
\(77\) 5.23731 0.596847
\(78\) −10.4579 −1.18412
\(79\) −3.18171 −0.357970 −0.178985 0.983852i \(-0.557281\pi\)
−0.178985 + 0.983852i \(0.557281\pi\)
\(80\) 0 0
\(81\) −11.2496 −1.24996
\(82\) 6.76289 0.746836
\(83\) 6.11387 0.671085 0.335542 0.942025i \(-0.391080\pi\)
0.335542 + 0.942025i \(0.391080\pi\)
\(84\) −8.49115 −0.926460
\(85\) 0 0
\(86\) 9.88290 1.06570
\(87\) 10.1988 1.09342
\(88\) −28.2656 −3.01312
\(89\) −3.00743 −0.318787 −0.159393 0.987215i \(-0.550954\pi\)
−0.159393 + 0.987215i \(0.550954\pi\)
\(90\) 0 0
\(91\) 1.94293 0.203674
\(92\) 7.98980 0.832994
\(93\) −14.0555 −1.45749
\(94\) 17.8485 1.84093
\(95\) 0 0
\(96\) 2.64334 0.269785
\(97\) 5.95526 0.604666 0.302333 0.953202i \(-0.402235\pi\)
0.302333 + 0.953202i \(0.402235\pi\)
\(98\) −14.9776 −1.51297
\(99\) −7.96914 −0.800929
\(100\) 0 0
\(101\) −7.77373 −0.773515 −0.386758 0.922181i \(-0.626405\pi\)
−0.386758 + 0.922181i \(0.626405\pi\)
\(102\) −10.7004 −1.05950
\(103\) 4.95056 0.487793 0.243897 0.969801i \(-0.421574\pi\)
0.243897 + 0.969801i \(0.421574\pi\)
\(104\) −10.4859 −1.02823
\(105\) 0 0
\(106\) 32.8459 3.19028
\(107\) 1.16798 0.112913 0.0564567 0.998405i \(-0.482020\pi\)
0.0564567 + 0.998405i \(0.482020\pi\)
\(108\) −13.2582 −1.27577
\(109\) −17.6879 −1.69420 −0.847099 0.531435i \(-0.821653\pi\)
−0.847099 + 0.531435i \(0.821653\pi\)
\(110\) 0 0
\(111\) −2.07212 −0.196676
\(112\) −4.62163 −0.436703
\(113\) 5.03643 0.473787 0.236894 0.971536i \(-0.423871\pi\)
0.236894 + 0.971536i \(0.423871\pi\)
\(114\) 32.4955 3.04348
\(115\) 0 0
\(116\) 19.8623 1.84417
\(117\) −2.95638 −0.273317
\(118\) 16.1829 1.48975
\(119\) 1.98799 0.182239
\(120\) 0 0
\(121\) 17.9687 1.63351
\(122\) 6.74687 0.610833
\(123\) 5.78549 0.521660
\(124\) −27.3735 −2.45821
\(125\) 0 0
\(126\) −3.56495 −0.317591
\(127\) 7.79457 0.691656 0.345828 0.938298i \(-0.387598\pi\)
0.345828 + 0.938298i \(0.387598\pi\)
\(128\) −18.3561 −1.62246
\(129\) 8.45458 0.744385
\(130\) 0 0
\(131\) −9.79015 −0.855369 −0.427685 0.903928i \(-0.640671\pi\)
−0.427685 + 0.903928i \(0.640671\pi\)
\(132\) −46.9663 −4.08789
\(133\) −6.03720 −0.523491
\(134\) 23.6785 2.04551
\(135\) 0 0
\(136\) −10.7291 −0.920015
\(137\) −3.63326 −0.310411 −0.155205 0.987882i \(-0.549604\pi\)
−0.155205 + 0.987882i \(0.549604\pi\)
\(138\) 10.1511 0.864122
\(139\) −1.86079 −0.157830 −0.0789149 0.996881i \(-0.525146\pi\)
−0.0789149 + 0.996881i \(0.525146\pi\)
\(140\) 0 0
\(141\) 15.2690 1.28588
\(142\) −14.0672 −1.18049
\(143\) 10.7467 0.898688
\(144\) 7.03232 0.586027
\(145\) 0 0
\(146\) −23.1421 −1.91526
\(147\) −12.8130 −1.05680
\(148\) −4.03550 −0.331716
\(149\) 16.8530 1.38066 0.690328 0.723497i \(-0.257466\pi\)
0.690328 + 0.723497i \(0.257466\pi\)
\(150\) 0 0
\(151\) −14.3201 −1.16536 −0.582678 0.812703i \(-0.697995\pi\)
−0.582678 + 0.812703i \(0.697995\pi\)
\(152\) 32.5826 2.64280
\(153\) −3.02495 −0.244552
\(154\) 12.9590 1.04426
\(155\) 0 0
\(156\) −17.4235 −1.39500
\(157\) −21.6869 −1.73080 −0.865400 0.501081i \(-0.832936\pi\)
−0.865400 + 0.501081i \(0.832936\pi\)
\(158\) −7.87266 −0.626315
\(159\) 28.0989 2.22839
\(160\) 0 0
\(161\) −1.88594 −0.148633
\(162\) −27.8355 −2.18697
\(163\) −19.7210 −1.54467 −0.772334 0.635217i \(-0.780911\pi\)
−0.772334 + 0.635217i \(0.780911\pi\)
\(164\) 11.2674 0.879835
\(165\) 0 0
\(166\) 15.1279 1.17415
\(167\) 6.07947 0.470444 0.235222 0.971942i \(-0.424418\pi\)
0.235222 + 0.971942i \(0.424418\pi\)
\(168\) −10.8170 −0.834552
\(169\) −9.01319 −0.693322
\(170\) 0 0
\(171\) 9.18626 0.702491
\(172\) 16.4655 1.25548
\(173\) 15.0309 1.14278 0.571388 0.820680i \(-0.306405\pi\)
0.571388 + 0.820680i \(0.306405\pi\)
\(174\) 25.2353 1.91308
\(175\) 0 0
\(176\) −25.5632 −1.92690
\(177\) 13.8441 1.04058
\(178\) −7.44144 −0.557759
\(179\) −20.0167 −1.49612 −0.748060 0.663631i \(-0.769014\pi\)
−0.748060 + 0.663631i \(0.769014\pi\)
\(180\) 0 0
\(181\) 20.2993 1.50883 0.754417 0.656395i \(-0.227920\pi\)
0.754417 + 0.656395i \(0.227920\pi\)
\(182\) 4.80749 0.356355
\(183\) 5.77178 0.426663
\(184\) 10.1783 0.750357
\(185\) 0 0
\(186\) −34.7783 −2.55007
\(187\) 10.9960 0.804106
\(188\) 29.7367 2.16877
\(189\) 3.12951 0.227638
\(190\) 0 0
\(191\) −9.47681 −0.685718 −0.342859 0.939387i \(-0.611395\pi\)
−0.342859 + 0.939387i \(0.611395\pi\)
\(192\) −13.5666 −0.979085
\(193\) 2.72478 0.196134 0.0980671 0.995180i \(-0.468734\pi\)
0.0980671 + 0.995180i \(0.468734\pi\)
\(194\) 14.7354 1.05794
\(195\) 0 0
\(196\) −24.9536 −1.78240
\(197\) 0.780036 0.0555753 0.0277876 0.999614i \(-0.491154\pi\)
0.0277876 + 0.999614i \(0.491154\pi\)
\(198\) −19.7185 −1.40133
\(199\) −7.42526 −0.526363 −0.263181 0.964746i \(-0.584772\pi\)
−0.263181 + 0.964746i \(0.584772\pi\)
\(200\) 0 0
\(201\) 20.2564 1.42878
\(202\) −19.2350 −1.35337
\(203\) −4.68836 −0.329059
\(204\) −17.8276 −1.24818
\(205\) 0 0
\(206\) 12.2494 0.853459
\(207\) 2.86966 0.199455
\(208\) −9.48340 −0.657555
\(209\) −33.3930 −2.30984
\(210\) 0 0
\(211\) −2.98128 −0.205240 −0.102620 0.994721i \(-0.532723\pi\)
−0.102620 + 0.994721i \(0.532723\pi\)
\(212\) 54.7233 3.75841
\(213\) −12.0342 −0.824567
\(214\) 2.89001 0.197557
\(215\) 0 0
\(216\) −16.8899 −1.14921
\(217\) 6.46132 0.438623
\(218\) −43.7662 −2.96422
\(219\) −19.7976 −1.33779
\(220\) 0 0
\(221\) 4.07927 0.274402
\(222\) −5.12715 −0.344111
\(223\) 24.8720 1.66556 0.832778 0.553607i \(-0.186749\pi\)
0.832778 + 0.553607i \(0.186749\pi\)
\(224\) −1.21514 −0.0811903
\(225\) 0 0
\(226\) 12.4619 0.828953
\(227\) −17.5594 −1.16546 −0.582729 0.812667i \(-0.698015\pi\)
−0.582729 + 0.812667i \(0.698015\pi\)
\(228\) 54.1394 3.58547
\(229\) 7.01849 0.463795 0.231897 0.972740i \(-0.425507\pi\)
0.231897 + 0.972740i \(0.425507\pi\)
\(230\) 0 0
\(231\) 11.0861 0.729410
\(232\) 25.3030 1.66122
\(233\) −8.69126 −0.569383 −0.284692 0.958619i \(-0.591891\pi\)
−0.284692 + 0.958619i \(0.591891\pi\)
\(234\) −7.31513 −0.478205
\(235\) 0 0
\(236\) 26.9617 1.75505
\(237\) −6.73488 −0.437477
\(238\) 4.91899 0.318851
\(239\) 21.3430 1.38057 0.690283 0.723540i \(-0.257486\pi\)
0.690283 + 0.723540i \(0.257486\pi\)
\(240\) 0 0
\(241\) −21.3897 −1.37783 −0.688916 0.724841i \(-0.741913\pi\)
−0.688916 + 0.724841i \(0.741913\pi\)
\(242\) 44.4608 2.85805
\(243\) −14.1643 −0.908639
\(244\) 11.2407 0.719612
\(245\) 0 0
\(246\) 14.3153 0.912713
\(247\) −12.3881 −0.788235
\(248\) −34.8716 −2.21435
\(249\) 12.9415 0.820137
\(250\) 0 0
\(251\) 14.9016 0.940580 0.470290 0.882512i \(-0.344149\pi\)
0.470290 + 0.882512i \(0.344149\pi\)
\(252\) −5.93942 −0.374148
\(253\) −10.4315 −0.655823
\(254\) 19.2865 1.21014
\(255\) 0 0
\(256\) −32.6011 −2.03757
\(257\) 19.0689 1.18948 0.594741 0.803917i \(-0.297254\pi\)
0.594741 + 0.803917i \(0.297254\pi\)
\(258\) 20.9196 1.30240
\(259\) 0.952551 0.0591887
\(260\) 0 0
\(261\) 7.13386 0.441575
\(262\) −24.2243 −1.49658
\(263\) −6.78341 −0.418283 −0.209142 0.977885i \(-0.567067\pi\)
−0.209142 + 0.977885i \(0.567067\pi\)
\(264\) −59.8312 −3.68236
\(265\) 0 0
\(266\) −14.9382 −0.915917
\(267\) −6.36597 −0.389591
\(268\) 39.4498 2.40978
\(269\) 0.494663 0.0301601 0.0150801 0.999886i \(-0.495200\pi\)
0.0150801 + 0.999886i \(0.495200\pi\)
\(270\) 0 0
\(271\) 9.82610 0.596893 0.298446 0.954426i \(-0.403532\pi\)
0.298446 + 0.954426i \(0.403532\pi\)
\(272\) −9.70334 −0.588352
\(273\) 4.11270 0.248912
\(274\) −8.98998 −0.543104
\(275\) 0 0
\(276\) 16.9124 1.01801
\(277\) 25.1399 1.51051 0.755255 0.655431i \(-0.227513\pi\)
0.755255 + 0.655431i \(0.227513\pi\)
\(278\) −4.60424 −0.276144
\(279\) −9.83161 −0.588603
\(280\) 0 0
\(281\) −7.77050 −0.463549 −0.231775 0.972770i \(-0.574453\pi\)
−0.231775 + 0.972770i \(0.574453\pi\)
\(282\) 37.7808 2.24981
\(283\) 28.9617 1.72160 0.860798 0.508946i \(-0.169965\pi\)
0.860798 + 0.508946i \(0.169965\pi\)
\(284\) −23.4368 −1.39072
\(285\) 0 0
\(286\) 26.5912 1.57237
\(287\) −2.65959 −0.156991
\(288\) 1.84898 0.108952
\(289\) −12.8261 −0.754477
\(290\) 0 0
\(291\) 12.6058 0.738966
\(292\) −38.5562 −2.25633
\(293\) −0.154329 −0.00901602 −0.00450801 0.999990i \(-0.501435\pi\)
−0.00450801 + 0.999990i \(0.501435\pi\)
\(294\) −31.7038 −1.84901
\(295\) 0 0
\(296\) −5.14089 −0.298808
\(297\) 17.3100 1.00443
\(298\) 41.7004 2.41564
\(299\) −3.86987 −0.223800
\(300\) 0 0
\(301\) −3.88657 −0.224018
\(302\) −35.4330 −2.03894
\(303\) −16.4551 −0.945318
\(304\) 29.4675 1.69007
\(305\) 0 0
\(306\) −7.48478 −0.427877
\(307\) −14.4875 −0.826845 −0.413423 0.910539i \(-0.635667\pi\)
−0.413423 + 0.910539i \(0.635667\pi\)
\(308\) 21.5904 1.23023
\(309\) 10.4791 0.596135
\(310\) 0 0
\(311\) 18.5385 1.05122 0.525610 0.850726i \(-0.323837\pi\)
0.525610 + 0.850726i \(0.323837\pi\)
\(312\) −22.1961 −1.25661
\(313\) 6.85703 0.387582 0.193791 0.981043i \(-0.437922\pi\)
0.193791 + 0.981043i \(0.437922\pi\)
\(314\) −53.6610 −3.02826
\(315\) 0 0
\(316\) −13.1163 −0.737852
\(317\) −27.2810 −1.53225 −0.766127 0.642689i \(-0.777819\pi\)
−0.766127 + 0.642689i \(0.777819\pi\)
\(318\) 69.5266 3.89886
\(319\) −25.9323 −1.45193
\(320\) 0 0
\(321\) 2.47233 0.137992
\(322\) −4.66647 −0.260052
\(323\) −12.6754 −0.705278
\(324\) −46.3757 −2.57643
\(325\) 0 0
\(326\) −48.7967 −2.70260
\(327\) −37.4410 −2.07049
\(328\) 14.3537 0.792551
\(329\) −7.01914 −0.386978
\(330\) 0 0
\(331\) −13.0429 −0.716901 −0.358450 0.933549i \(-0.616695\pi\)
−0.358450 + 0.933549i \(0.616695\pi\)
\(332\) 25.2040 1.38325
\(333\) −1.44941 −0.0794273
\(334\) 15.0428 0.823103
\(335\) 0 0
\(336\) −9.78284 −0.533698
\(337\) −19.6507 −1.07044 −0.535222 0.844712i \(-0.679772\pi\)
−0.535222 + 0.844712i \(0.679772\pi\)
\(338\) −22.3018 −1.21306
\(339\) 10.6609 0.579018
\(340\) 0 0
\(341\) 35.7389 1.93537
\(342\) 22.7301 1.22910
\(343\) 12.7016 0.685823
\(344\) 20.9757 1.13093
\(345\) 0 0
\(346\) 37.1917 1.99944
\(347\) 4.38943 0.235637 0.117818 0.993035i \(-0.462410\pi\)
0.117818 + 0.993035i \(0.462410\pi\)
\(348\) 42.0436 2.25377
\(349\) 27.1955 1.45574 0.727872 0.685713i \(-0.240510\pi\)
0.727872 + 0.685713i \(0.240510\pi\)
\(350\) 0 0
\(351\) 6.42163 0.342761
\(352\) −6.72122 −0.358242
\(353\) −15.9317 −0.847960 −0.423980 0.905672i \(-0.639367\pi\)
−0.423980 + 0.905672i \(0.639367\pi\)
\(354\) 34.2551 1.82064
\(355\) 0 0
\(356\) −12.3979 −0.657087
\(357\) 4.20808 0.222715
\(358\) −49.5284 −2.61766
\(359\) −5.01508 −0.264686 −0.132343 0.991204i \(-0.542250\pi\)
−0.132343 + 0.991204i \(0.542250\pi\)
\(360\) 0 0
\(361\) 19.4931 1.02595
\(362\) 50.2276 2.63991
\(363\) 38.0352 1.99633
\(364\) 8.00958 0.419816
\(365\) 0 0
\(366\) 14.2814 0.746502
\(367\) 28.3669 1.48074 0.740370 0.672200i \(-0.234651\pi\)
0.740370 + 0.672200i \(0.234651\pi\)
\(368\) 9.20522 0.479855
\(369\) 4.04686 0.210671
\(370\) 0 0
\(371\) −12.9171 −0.670621
\(372\) −57.9428 −3.00420
\(373\) −32.7218 −1.69427 −0.847135 0.531377i \(-0.821675\pi\)
−0.847135 + 0.531377i \(0.821675\pi\)
\(374\) 27.2080 1.40689
\(375\) 0 0
\(376\) 37.8821 1.95362
\(377\) −9.62033 −0.495472
\(378\) 7.74352 0.398283
\(379\) 4.48483 0.230370 0.115185 0.993344i \(-0.463254\pi\)
0.115185 + 0.993344i \(0.463254\pi\)
\(380\) 0 0
\(381\) 16.4992 0.845278
\(382\) −23.4490 −1.19975
\(383\) 16.2839 0.832069 0.416034 0.909349i \(-0.363420\pi\)
0.416034 + 0.909349i \(0.363420\pi\)
\(384\) −38.8553 −1.98282
\(385\) 0 0
\(386\) 6.74208 0.343163
\(387\) 5.91385 0.300618
\(388\) 24.5501 1.24634
\(389\) 21.4899 1.08958 0.544791 0.838572i \(-0.316609\pi\)
0.544791 + 0.838572i \(0.316609\pi\)
\(390\) 0 0
\(391\) −3.95962 −0.200246
\(392\) −31.7888 −1.60558
\(393\) −20.7233 −1.04535
\(394\) 1.93008 0.0972363
\(395\) 0 0
\(396\) −32.8522 −1.65088
\(397\) −20.8870 −1.04829 −0.524145 0.851629i \(-0.675615\pi\)
−0.524145 + 0.851629i \(0.675615\pi\)
\(398\) −18.3727 −0.920941
\(399\) −12.7792 −0.639762
\(400\) 0 0
\(401\) −0.633423 −0.0316316 −0.0158158 0.999875i \(-0.505035\pi\)
−0.0158158 + 0.999875i \(0.505035\pi\)
\(402\) 50.1215 2.49983
\(403\) 13.2584 0.660447
\(404\) −32.0466 −1.59438
\(405\) 0 0
\(406\) −11.6007 −0.575732
\(407\) 5.26876 0.261163
\(408\) −22.7109 −1.12436
\(409\) −4.29822 −0.212534 −0.106267 0.994338i \(-0.533890\pi\)
−0.106267 + 0.994338i \(0.533890\pi\)
\(410\) 0 0
\(411\) −7.69071 −0.379355
\(412\) 20.4083 1.00545
\(413\) −6.36411 −0.313158
\(414\) 7.10055 0.348973
\(415\) 0 0
\(416\) −2.49343 −0.122250
\(417\) −3.93882 −0.192885
\(418\) −82.6261 −4.04137
\(419\) 33.6215 1.64252 0.821258 0.570556i \(-0.193272\pi\)
0.821258 + 0.570556i \(0.193272\pi\)
\(420\) 0 0
\(421\) −21.3768 −1.04184 −0.520920 0.853605i \(-0.674411\pi\)
−0.520920 + 0.853605i \(0.674411\pi\)
\(422\) −7.37674 −0.359094
\(423\) 10.6804 0.519298
\(424\) 69.7130 3.38556
\(425\) 0 0
\(426\) −29.7768 −1.44269
\(427\) −2.65329 −0.128402
\(428\) 4.81493 0.232738
\(429\) 22.7482 1.09829
\(430\) 0 0
\(431\) 26.0905 1.25673 0.628367 0.777917i \(-0.283723\pi\)
0.628367 + 0.777917i \(0.283723\pi\)
\(432\) −15.2751 −0.734923
\(433\) 11.7676 0.565513 0.282757 0.959192i \(-0.408751\pi\)
0.282757 + 0.959192i \(0.408751\pi\)
\(434\) 15.9876 0.767429
\(435\) 0 0
\(436\) −72.9172 −3.49210
\(437\) 12.0247 0.575220
\(438\) −48.9862 −2.34065
\(439\) 4.92572 0.235092 0.117546 0.993067i \(-0.462497\pi\)
0.117546 + 0.993067i \(0.462497\pi\)
\(440\) 0 0
\(441\) −8.96247 −0.426784
\(442\) 10.0936 0.480102
\(443\) 9.35961 0.444689 0.222344 0.974968i \(-0.428629\pi\)
0.222344 + 0.974968i \(0.428629\pi\)
\(444\) −8.54214 −0.405392
\(445\) 0 0
\(446\) 61.5422 2.91411
\(447\) 35.6737 1.68731
\(448\) 6.23657 0.294650
\(449\) 7.41602 0.349983 0.174992 0.984570i \(-0.444010\pi\)
0.174992 + 0.984570i \(0.444010\pi\)
\(450\) 0 0
\(451\) −14.7107 −0.692702
\(452\) 20.7623 0.976576
\(453\) −30.3121 −1.42419
\(454\) −43.4481 −2.03912
\(455\) 0 0
\(456\) 68.9692 3.22978
\(457\) 8.79714 0.411513 0.205756 0.978603i \(-0.434035\pi\)
0.205756 + 0.978603i \(0.434035\pi\)
\(458\) 17.3662 0.811471
\(459\) 6.57056 0.306688
\(460\) 0 0
\(461\) −42.1526 −1.96324 −0.981621 0.190838i \(-0.938879\pi\)
−0.981621 + 0.190838i \(0.938879\pi\)
\(462\) 27.4309 1.27620
\(463\) 23.0565 1.07153 0.535763 0.844368i \(-0.320024\pi\)
0.535763 + 0.844368i \(0.320024\pi\)
\(464\) 22.8838 1.06235
\(465\) 0 0
\(466\) −21.5053 −0.996212
\(467\) 4.37964 0.202666 0.101333 0.994853i \(-0.467689\pi\)
0.101333 + 0.994853i \(0.467689\pi\)
\(468\) −12.1875 −0.563365
\(469\) −9.31186 −0.429982
\(470\) 0 0
\(471\) −45.9057 −2.11522
\(472\) 34.3469 1.58095
\(473\) −21.4974 −0.988453
\(474\) −16.6645 −0.765424
\(475\) 0 0
\(476\) 8.19534 0.375633
\(477\) 19.6547 0.899929
\(478\) 52.8102 2.41548
\(479\) 14.1913 0.648417 0.324209 0.945986i \(-0.394902\pi\)
0.324209 + 0.945986i \(0.394902\pi\)
\(480\) 0 0
\(481\) 1.95460 0.0891219
\(482\) −52.9257 −2.41070
\(483\) −3.99206 −0.181645
\(484\) 74.0744 3.36702
\(485\) 0 0
\(486\) −35.0475 −1.58978
\(487\) 22.0694 1.00006 0.500029 0.866009i \(-0.333323\pi\)
0.500029 + 0.866009i \(0.333323\pi\)
\(488\) 14.3197 0.648223
\(489\) −41.7444 −1.88775
\(490\) 0 0
\(491\) −7.95704 −0.359096 −0.179548 0.983749i \(-0.557464\pi\)
−0.179548 + 0.983749i \(0.557464\pi\)
\(492\) 23.8502 1.07525
\(493\) −9.84345 −0.443327
\(494\) −30.6525 −1.37912
\(495\) 0 0
\(496\) −31.5376 −1.41608
\(497\) 5.53210 0.248149
\(498\) 32.0220 1.43494
\(499\) −4.09384 −0.183266 −0.0916328 0.995793i \(-0.529209\pi\)
−0.0916328 + 0.995793i \(0.529209\pi\)
\(500\) 0 0
\(501\) 12.8687 0.574932
\(502\) 36.8718 1.64567
\(503\) 26.9554 1.20188 0.600941 0.799293i \(-0.294792\pi\)
0.600941 + 0.799293i \(0.294792\pi\)
\(504\) −7.56633 −0.337031
\(505\) 0 0
\(506\) −25.8112 −1.14745
\(507\) −19.0787 −0.847314
\(508\) 32.1325 1.42565
\(509\) 33.8837 1.50187 0.750935 0.660376i \(-0.229603\pi\)
0.750935 + 0.660376i \(0.229603\pi\)
\(510\) 0 0
\(511\) 9.10093 0.402602
\(512\) −43.9545 −1.94253
\(513\) −19.9537 −0.880978
\(514\) 47.1831 2.08116
\(515\) 0 0
\(516\) 34.8534 1.53433
\(517\) −38.8243 −1.70749
\(518\) 2.35695 0.103558
\(519\) 31.8166 1.39659
\(520\) 0 0
\(521\) −26.6469 −1.16742 −0.583711 0.811962i \(-0.698400\pi\)
−0.583711 + 0.811962i \(0.698400\pi\)
\(522\) 17.6517 0.772594
\(523\) −1.80935 −0.0791172 −0.0395586 0.999217i \(-0.512595\pi\)
−0.0395586 + 0.999217i \(0.512595\pi\)
\(524\) −40.3592 −1.76310
\(525\) 0 0
\(526\) −16.7846 −0.731842
\(527\) 13.5659 0.590938
\(528\) −54.1109 −2.35488
\(529\) −19.2437 −0.836680
\(530\) 0 0
\(531\) 9.68370 0.420237
\(532\) −24.8879 −1.07903
\(533\) −5.45737 −0.236385
\(534\) −15.7517 −0.681641
\(535\) 0 0
\(536\) 50.2558 2.17072
\(537\) −42.3704 −1.82842
\(538\) 1.22397 0.0527691
\(539\) 32.5795 1.40330
\(540\) 0 0
\(541\) −33.5233 −1.44128 −0.720640 0.693309i \(-0.756152\pi\)
−0.720640 + 0.693309i \(0.756152\pi\)
\(542\) 24.3132 1.04434
\(543\) 42.9685 1.84396
\(544\) −2.55126 −0.109384
\(545\) 0 0
\(546\) 10.1763 0.435504
\(547\) −27.5146 −1.17644 −0.588219 0.808702i \(-0.700171\pi\)
−0.588219 + 0.808702i \(0.700171\pi\)
\(548\) −14.9779 −0.639822
\(549\) 4.03727 0.172306
\(550\) 0 0
\(551\) 29.8929 1.27348
\(552\) 21.5450 0.917017
\(553\) 3.09602 0.131656
\(554\) 62.2050 2.64284
\(555\) 0 0
\(556\) −7.67095 −0.325321
\(557\) 2.16761 0.0918448 0.0459224 0.998945i \(-0.485377\pi\)
0.0459224 + 0.998945i \(0.485377\pi\)
\(558\) −24.3269 −1.02984
\(559\) −7.97509 −0.337310
\(560\) 0 0
\(561\) 23.2758 0.982704
\(562\) −19.2270 −0.811040
\(563\) 33.4836 1.41116 0.705582 0.708628i \(-0.250686\pi\)
0.705582 + 0.708628i \(0.250686\pi\)
\(564\) 62.9451 2.65047
\(565\) 0 0
\(566\) 71.6616 3.01216
\(567\) 10.9467 0.459717
\(568\) −29.8566 −1.25275
\(569\) 17.2323 0.722415 0.361208 0.932485i \(-0.382365\pi\)
0.361208 + 0.932485i \(0.382365\pi\)
\(570\) 0 0
\(571\) 42.7928 1.79082 0.895411 0.445240i \(-0.146882\pi\)
0.895411 + 0.445240i \(0.146882\pi\)
\(572\) 44.3027 1.85239
\(573\) −20.0600 −0.838020
\(574\) −6.58076 −0.274676
\(575\) 0 0
\(576\) −9.48962 −0.395401
\(577\) −31.8385 −1.32546 −0.662728 0.748860i \(-0.730601\pi\)
−0.662728 + 0.748860i \(0.730601\pi\)
\(578\) −31.7363 −1.32006
\(579\) 5.76769 0.239697
\(580\) 0 0
\(581\) −5.94923 −0.246816
\(582\) 31.1912 1.29292
\(583\) −71.4470 −2.95903
\(584\) −49.1175 −2.03250
\(585\) 0 0
\(586\) −0.381866 −0.0157747
\(587\) −24.8027 −1.02372 −0.511858 0.859070i \(-0.671043\pi\)
−0.511858 + 0.859070i \(0.671043\pi\)
\(588\) −52.8205 −2.17828
\(589\) −41.1973 −1.69750
\(590\) 0 0
\(591\) 1.65114 0.0679189
\(592\) −4.64938 −0.191089
\(593\) 17.8431 0.732728 0.366364 0.930472i \(-0.380603\pi\)
0.366364 + 0.930472i \(0.380603\pi\)
\(594\) 42.8310 1.75738
\(595\) 0 0
\(596\) 69.4754 2.84582
\(597\) −15.7174 −0.643271
\(598\) −9.57541 −0.391568
\(599\) −28.1028 −1.14825 −0.574126 0.818767i \(-0.694658\pi\)
−0.574126 + 0.818767i \(0.694658\pi\)
\(600\) 0 0
\(601\) −26.6296 −1.08624 −0.543121 0.839654i \(-0.682758\pi\)
−0.543121 + 0.839654i \(0.682758\pi\)
\(602\) −9.61675 −0.391949
\(603\) 14.1690 0.577007
\(604\) −59.0336 −2.40204
\(605\) 0 0
\(606\) −40.7156 −1.65396
\(607\) −19.2757 −0.782375 −0.391187 0.920311i \(-0.627936\pi\)
−0.391187 + 0.920311i \(0.627936\pi\)
\(608\) 7.74774 0.314213
\(609\) −9.92410 −0.402145
\(610\) 0 0
\(611\) −14.4030 −0.582682
\(612\) −12.4701 −0.504074
\(613\) −0.465680 −0.0188086 −0.00940432 0.999956i \(-0.502994\pi\)
−0.00940432 + 0.999956i \(0.502994\pi\)
\(614\) −35.8472 −1.44668
\(615\) 0 0
\(616\) 27.5044 1.10818
\(617\) −44.2765 −1.78251 −0.891253 0.453507i \(-0.850173\pi\)
−0.891253 + 0.453507i \(0.850173\pi\)
\(618\) 25.9290 1.04302
\(619\) −36.1620 −1.45347 −0.726736 0.686917i \(-0.758964\pi\)
−0.726736 + 0.686917i \(0.758964\pi\)
\(620\) 0 0
\(621\) −6.23326 −0.250132
\(622\) 45.8707 1.83925
\(623\) 2.92644 0.117245
\(624\) −20.0740 −0.803603
\(625\) 0 0
\(626\) 16.9667 0.678126
\(627\) −70.6847 −2.82287
\(628\) −89.4025 −3.56755
\(629\) 1.99993 0.0797424
\(630\) 0 0
\(631\) −0.199431 −0.00793921 −0.00396961 0.999992i \(-0.501264\pi\)
−0.00396961 + 0.999992i \(0.501264\pi\)
\(632\) −16.7091 −0.664654
\(633\) −6.31062 −0.250825
\(634\) −67.5028 −2.68088
\(635\) 0 0
\(636\) 115.836 4.59318
\(637\) 12.0863 0.478876
\(638\) −64.1657 −2.54035
\(639\) −8.41770 −0.332999
\(640\) 0 0
\(641\) −30.9078 −1.22079 −0.610393 0.792099i \(-0.708988\pi\)
−0.610393 + 0.792099i \(0.708988\pi\)
\(642\) 6.11742 0.241436
\(643\) −22.2489 −0.877412 −0.438706 0.898631i \(-0.644563\pi\)
−0.438706 + 0.898631i \(0.644563\pi\)
\(644\) −7.77463 −0.306363
\(645\) 0 0
\(646\) −31.3634 −1.23398
\(647\) −35.0696 −1.37873 −0.689364 0.724415i \(-0.742110\pi\)
−0.689364 + 0.724415i \(0.742110\pi\)
\(648\) −59.0789 −2.32084
\(649\) −35.2012 −1.38177
\(650\) 0 0
\(651\) 13.6770 0.536044
\(652\) −81.2983 −3.18389
\(653\) −4.00012 −0.156537 −0.0782683 0.996932i \(-0.524939\pi\)
−0.0782683 + 0.996932i \(0.524939\pi\)
\(654\) −92.6422 −3.62260
\(655\) 0 0
\(656\) 12.9814 0.506839
\(657\) −13.8481 −0.540265
\(658\) −17.3678 −0.677068
\(659\) 10.0349 0.390904 0.195452 0.980713i \(-0.437383\pi\)
0.195452 + 0.980713i \(0.437383\pi\)
\(660\) 0 0
\(661\) −48.5672 −1.88905 −0.944523 0.328445i \(-0.893475\pi\)
−0.944523 + 0.328445i \(0.893475\pi\)
\(662\) −32.2727 −1.25431
\(663\) 8.63481 0.335348
\(664\) 32.1078 1.24602
\(665\) 0 0
\(666\) −3.58636 −0.138968
\(667\) 9.33814 0.361574
\(668\) 25.0622 0.969684
\(669\) 52.6479 2.03549
\(670\) 0 0
\(671\) −14.6759 −0.566557
\(672\) −2.57216 −0.0992232
\(673\) 5.06306 0.195167 0.0975834 0.995227i \(-0.468889\pi\)
0.0975834 + 0.995227i \(0.468889\pi\)
\(674\) −48.6229 −1.87288
\(675\) 0 0
\(676\) −37.1562 −1.42908
\(677\) 33.6669 1.29393 0.646963 0.762521i \(-0.276039\pi\)
0.646963 + 0.762521i \(0.276039\pi\)
\(678\) 26.3787 1.01307
\(679\) −5.79489 −0.222387
\(680\) 0 0
\(681\) −37.1689 −1.42431
\(682\) 88.4307 3.38619
\(683\) 32.4523 1.24175 0.620877 0.783908i \(-0.286777\pi\)
0.620877 + 0.783908i \(0.286777\pi\)
\(684\) 37.8697 1.44798
\(685\) 0 0
\(686\) 31.4283 1.19994
\(687\) 14.8564 0.566807
\(688\) 18.9703 0.723235
\(689\) −26.5053 −1.00977
\(690\) 0 0
\(691\) −30.7811 −1.17097 −0.585484 0.810684i \(-0.699095\pi\)
−0.585484 + 0.810684i \(0.699095\pi\)
\(692\) 61.9637 2.35550
\(693\) 7.75453 0.294570
\(694\) 10.8610 0.412278
\(695\) 0 0
\(696\) 53.5601 2.03019
\(697\) −5.58394 −0.211507
\(698\) 67.2914 2.54702
\(699\) −18.3972 −0.695847
\(700\) 0 0
\(701\) −13.1755 −0.497633 −0.248817 0.968551i \(-0.580042\pi\)
−0.248817 + 0.968551i \(0.580042\pi\)
\(702\) 15.8894 0.599706
\(703\) −6.07345 −0.229065
\(704\) 34.4958 1.30011
\(705\) 0 0
\(706\) −39.4207 −1.48362
\(707\) 7.56439 0.284488
\(708\) 57.0711 2.14486
\(709\) −10.6307 −0.399246 −0.199623 0.979873i \(-0.563972\pi\)
−0.199623 + 0.979873i \(0.563972\pi\)
\(710\) 0 0
\(711\) −4.71094 −0.176674
\(712\) −15.7939 −0.591901
\(713\) −12.8695 −0.481965
\(714\) 10.4123 0.389670
\(715\) 0 0
\(716\) −82.5174 −3.08382
\(717\) 45.1779 1.68720
\(718\) −12.4091 −0.463102
\(719\) −15.3464 −0.572324 −0.286162 0.958181i \(-0.592380\pi\)
−0.286162 + 0.958181i \(0.592380\pi\)
\(720\) 0 0
\(721\) −4.81724 −0.179404
\(722\) 48.2328 1.79504
\(723\) −45.2767 −1.68386
\(724\) 83.6823 3.11003
\(725\) 0 0
\(726\) 94.1124 3.49284
\(727\) 23.8156 0.883272 0.441636 0.897194i \(-0.354398\pi\)
0.441636 + 0.897194i \(0.354398\pi\)
\(728\) 10.2035 0.378168
\(729\) 3.76661 0.139504
\(730\) 0 0
\(731\) −8.16005 −0.301810
\(732\) 23.7937 0.879442
\(733\) −4.09180 −0.151134 −0.0755670 0.997141i \(-0.524077\pi\)
−0.0755670 + 0.997141i \(0.524077\pi\)
\(734\) 70.1897 2.59075
\(735\) 0 0
\(736\) 2.42029 0.0892130
\(737\) −51.5059 −1.89724
\(738\) 10.0134 0.368597
\(739\) 2.45909 0.0904590 0.0452295 0.998977i \(-0.485598\pi\)
0.0452295 + 0.998977i \(0.485598\pi\)
\(740\) 0 0
\(741\) −26.2225 −0.963307
\(742\) −31.9614 −1.17334
\(743\) −11.1921 −0.410598 −0.205299 0.978699i \(-0.565817\pi\)
−0.205299 + 0.978699i \(0.565817\pi\)
\(744\) −73.8144 −2.70617
\(745\) 0 0
\(746\) −80.9653 −2.96435
\(747\) 9.05240 0.331210
\(748\) 45.3301 1.65743
\(749\) −1.13653 −0.0415279
\(750\) 0 0
\(751\) −10.3616 −0.378101 −0.189051 0.981967i \(-0.560541\pi\)
−0.189051 + 0.981967i \(0.560541\pi\)
\(752\) 34.2603 1.24934
\(753\) 31.5430 1.14949
\(754\) −23.8041 −0.866894
\(755\) 0 0
\(756\) 12.9012 0.469211
\(757\) −5.84722 −0.212521 −0.106260 0.994338i \(-0.533888\pi\)
−0.106260 + 0.994338i \(0.533888\pi\)
\(758\) 11.0971 0.403063
\(759\) −22.0809 −0.801486
\(760\) 0 0
\(761\) 41.6931 1.51137 0.755687 0.654932i \(-0.227303\pi\)
0.755687 + 0.654932i \(0.227303\pi\)
\(762\) 40.8248 1.47893
\(763\) 17.2116 0.623102
\(764\) −39.0674 −1.41341
\(765\) 0 0
\(766\) 40.2921 1.45581
\(767\) −13.0589 −0.471530
\(768\) −69.0084 −2.49013
\(769\) −17.3076 −0.624129 −0.312065 0.950061i \(-0.601021\pi\)
−0.312065 + 0.950061i \(0.601021\pi\)
\(770\) 0 0
\(771\) 40.3640 1.45367
\(772\) 11.2327 0.404274
\(773\) 17.8695 0.642723 0.321361 0.946957i \(-0.395860\pi\)
0.321361 + 0.946957i \(0.395860\pi\)
\(774\) 14.6329 0.525970
\(775\) 0 0
\(776\) 31.2748 1.12270
\(777\) 2.01631 0.0723348
\(778\) 53.1736 1.90637
\(779\) 16.9575 0.607565
\(780\) 0 0
\(781\) 30.5992 1.09493
\(782\) −9.79749 −0.350358
\(783\) −15.4956 −0.553769
\(784\) −28.7496 −1.02677
\(785\) 0 0
\(786\) −51.2768 −1.82898
\(787\) 9.91104 0.353290 0.176645 0.984275i \(-0.443475\pi\)
0.176645 + 0.984275i \(0.443475\pi\)
\(788\) 3.21564 0.114552
\(789\) −14.3588 −0.511187
\(790\) 0 0
\(791\) −4.90080 −0.174252
\(792\) −41.8510 −1.48711
\(793\) −5.44444 −0.193338
\(794\) −51.6819 −1.83412
\(795\) 0 0
\(796\) −30.6101 −1.08495
\(797\) 39.7590 1.40834 0.704168 0.710034i \(-0.251320\pi\)
0.704168 + 0.710034i \(0.251320\pi\)
\(798\) −31.6204 −1.11935
\(799\) −14.7370 −0.521359
\(800\) 0 0
\(801\) −4.45290 −0.157335
\(802\) −1.56731 −0.0553437
\(803\) 50.3392 1.77643
\(804\) 83.5055 2.94501
\(805\) 0 0
\(806\) 32.8059 1.15554
\(807\) 1.04708 0.0368589
\(808\) −40.8248 −1.43621
\(809\) −5.05110 −0.177587 −0.0887935 0.996050i \(-0.528301\pi\)
−0.0887935 + 0.996050i \(0.528301\pi\)
\(810\) 0 0
\(811\) 11.7743 0.413452 0.206726 0.978399i \(-0.433719\pi\)
0.206726 + 0.978399i \(0.433719\pi\)
\(812\) −19.3274 −0.678260
\(813\) 20.7994 0.729467
\(814\) 13.0368 0.456939
\(815\) 0 0
\(816\) −20.5396 −0.719028
\(817\) 24.7807 0.866968
\(818\) −10.6353 −0.371855
\(819\) 2.87676 0.100522
\(820\) 0 0
\(821\) 5.25401 0.183366 0.0916832 0.995788i \(-0.470775\pi\)
0.0916832 + 0.995788i \(0.470775\pi\)
\(822\) −19.0295 −0.663732
\(823\) −37.0042 −1.28989 −0.644943 0.764230i \(-0.723119\pi\)
−0.644943 + 0.764230i \(0.723119\pi\)
\(824\) 25.9985 0.905701
\(825\) 0 0
\(826\) −15.7471 −0.547910
\(827\) 28.1147 0.977645 0.488822 0.872383i \(-0.337427\pi\)
0.488822 + 0.872383i \(0.337427\pi\)
\(828\) 11.8300 0.411119
\(829\) −14.4532 −0.501979 −0.250990 0.967990i \(-0.580756\pi\)
−0.250990 + 0.967990i \(0.580756\pi\)
\(830\) 0 0
\(831\) 53.2149 1.84600
\(832\) 12.7972 0.443662
\(833\) 12.3666 0.428477
\(834\) −9.74604 −0.337478
\(835\) 0 0
\(836\) −137.660 −4.76108
\(837\) 21.3555 0.738154
\(838\) 83.1914 2.87380
\(839\) 16.3694 0.565134 0.282567 0.959248i \(-0.408814\pi\)
0.282567 + 0.959248i \(0.408814\pi\)
\(840\) 0 0
\(841\) −5.78575 −0.199509
\(842\) −52.8937 −1.82284
\(843\) −16.4482 −0.566506
\(844\) −12.2901 −0.423043
\(845\) 0 0
\(846\) 26.4271 0.908581
\(847\) −17.4848 −0.600784
\(848\) 63.0480 2.16508
\(849\) 61.3048 2.10397
\(850\) 0 0
\(851\) −1.89726 −0.0650373
\(852\) −49.6099 −1.69961
\(853\) −46.8216 −1.60314 −0.801571 0.597900i \(-0.796002\pi\)
−0.801571 + 0.597900i \(0.796002\pi\)
\(854\) −6.56517 −0.224656
\(855\) 0 0
\(856\) 6.13382 0.209650
\(857\) 28.9569 0.989148 0.494574 0.869136i \(-0.335324\pi\)
0.494574 + 0.869136i \(0.335324\pi\)
\(858\) 56.2870 1.92161
\(859\) 24.4996 0.835917 0.417959 0.908466i \(-0.362746\pi\)
0.417959 + 0.908466i \(0.362746\pi\)
\(860\) 0 0
\(861\) −5.62968 −0.191859
\(862\) 64.5571 2.19882
\(863\) 10.4696 0.356391 0.178195 0.983995i \(-0.442974\pi\)
0.178195 + 0.983995i \(0.442974\pi\)
\(864\) −4.01621 −0.136634
\(865\) 0 0
\(866\) 29.1171 0.989440
\(867\) −27.1497 −0.922052
\(868\) 26.6363 0.904096
\(869\) 17.1247 0.580917
\(870\) 0 0
\(871\) −19.1076 −0.647435
\(872\) −92.8905 −3.14567
\(873\) 8.81756 0.298429
\(874\) 29.7534 1.00642
\(875\) 0 0
\(876\) −81.6139 −2.75748
\(877\) 39.9121 1.34774 0.673868 0.738852i \(-0.264632\pi\)
0.673868 + 0.738852i \(0.264632\pi\)
\(878\) 12.1880 0.411324
\(879\) −0.326677 −0.0110185
\(880\) 0 0
\(881\) −7.88965 −0.265809 −0.132904 0.991129i \(-0.542430\pi\)
−0.132904 + 0.991129i \(0.542430\pi\)
\(882\) −22.1763 −0.746716
\(883\) −15.1304 −0.509178 −0.254589 0.967049i \(-0.581940\pi\)
−0.254589 + 0.967049i \(0.581940\pi\)
\(884\) 16.8165 0.565600
\(885\) 0 0
\(886\) 23.1590 0.778042
\(887\) 47.6886 1.60123 0.800614 0.599180i \(-0.204507\pi\)
0.800614 + 0.599180i \(0.204507\pi\)
\(888\) −10.8820 −0.365175
\(889\) −7.58466 −0.254381
\(890\) 0 0
\(891\) 60.5483 2.02845
\(892\) 102.533 3.43306
\(893\) 44.7539 1.49763
\(894\) 88.2693 2.95217
\(895\) 0 0
\(896\) 17.8618 0.596719
\(897\) −8.19154 −0.273508
\(898\) 18.3498 0.612342
\(899\) −31.9930 −1.06703
\(900\) 0 0
\(901\) −27.1200 −0.903499
\(902\) −36.3996 −1.21197
\(903\) −8.22690 −0.273774
\(904\) 26.4494 0.879696
\(905\) 0 0
\(906\) −75.0029 −2.49180
\(907\) −38.9321 −1.29272 −0.646359 0.763033i \(-0.723709\pi\)
−0.646359 + 0.763033i \(0.723709\pi\)
\(908\) −72.3873 −2.40226
\(909\) −11.5100 −0.381764
\(910\) 0 0
\(911\) 37.6740 1.24819 0.624097 0.781347i \(-0.285467\pi\)
0.624097 + 0.781347i \(0.285467\pi\)
\(912\) 62.3752 2.06545
\(913\) −32.9064 −1.08904
\(914\) 21.7672 0.719996
\(915\) 0 0
\(916\) 28.9332 0.955980
\(917\) 9.52650 0.314593
\(918\) 16.2579 0.536590
\(919\) 14.4646 0.477145 0.238572 0.971125i \(-0.423321\pi\)
0.238572 + 0.971125i \(0.423321\pi\)
\(920\) 0 0
\(921\) −30.6664 −1.01049
\(922\) −104.300 −3.43495
\(923\) 11.3516 0.373644
\(924\) 45.7015 1.50347
\(925\) 0 0
\(926\) 57.0499 1.87478
\(927\) 7.32997 0.240748
\(928\) 6.01674 0.197509
\(929\) −40.8990 −1.34185 −0.670927 0.741524i \(-0.734103\pi\)
−0.670927 + 0.741524i \(0.734103\pi\)
\(930\) 0 0
\(931\) −37.5553 −1.23083
\(932\) −35.8291 −1.17362
\(933\) 39.2413 1.28470
\(934\) 10.8368 0.354590
\(935\) 0 0
\(936\) −15.5258 −0.507477
\(937\) 33.2563 1.08644 0.543218 0.839592i \(-0.317206\pi\)
0.543218 + 0.839592i \(0.317206\pi\)
\(938\) −23.0408 −0.752310
\(939\) 14.5146 0.473666
\(940\) 0 0
\(941\) 39.1577 1.27650 0.638252 0.769827i \(-0.279658\pi\)
0.638252 + 0.769827i \(0.279658\pi\)
\(942\) −113.587 −3.70086
\(943\) 5.29728 0.172503
\(944\) 31.0631 1.01102
\(945\) 0 0
\(946\) −53.1923 −1.72943
\(947\) −28.5840 −0.928854 −0.464427 0.885611i \(-0.653740\pi\)
−0.464427 + 0.885611i \(0.653740\pi\)
\(948\) −27.7640 −0.901734
\(949\) 18.6747 0.606208
\(950\) 0 0
\(951\) −57.7471 −1.87258
\(952\) 10.4402 0.338368
\(953\) −43.3568 −1.40446 −0.702232 0.711948i \(-0.747813\pi\)
−0.702232 + 0.711948i \(0.747813\pi\)
\(954\) 48.6328 1.57454
\(955\) 0 0
\(956\) 87.9850 2.84564
\(957\) −54.8923 −1.77442
\(958\) 35.1143 1.13449
\(959\) 3.53542 0.114165
\(960\) 0 0
\(961\) 13.0915 0.422306
\(962\) 4.83636 0.155931
\(963\) 1.72936 0.0557277
\(964\) −88.1775 −2.84001
\(965\) 0 0
\(966\) −9.87776 −0.317812
\(967\) 32.2551 1.03726 0.518628 0.855000i \(-0.326443\pi\)
0.518628 + 0.855000i \(0.326443\pi\)
\(968\) 94.3647 3.03300
\(969\) −26.8306 −0.861925
\(970\) 0 0
\(971\) 10.9579 0.351654 0.175827 0.984421i \(-0.443740\pi\)
0.175827 + 0.984421i \(0.443740\pi\)
\(972\) −58.3912 −1.87290
\(973\) 1.81068 0.0580476
\(974\) 54.6074 1.74973
\(975\) 0 0
\(976\) 12.9507 0.414540
\(977\) −20.1536 −0.644770 −0.322385 0.946609i \(-0.604484\pi\)
−0.322385 + 0.946609i \(0.604484\pi\)
\(978\) −103.290 −3.30286
\(979\) 16.1867 0.517330
\(980\) 0 0
\(981\) −26.1894 −0.836162
\(982\) −19.6885 −0.628286
\(983\) −16.0511 −0.511951 −0.255975 0.966683i \(-0.582397\pi\)
−0.255975 + 0.966683i \(0.582397\pi\)
\(984\) 30.3832 0.968582
\(985\) 0 0
\(986\) −24.3562 −0.775659
\(987\) −14.8578 −0.472928
\(988\) −51.0689 −1.62472
\(989\) 7.74115 0.246154
\(990\) 0 0
\(991\) 20.5569 0.653010 0.326505 0.945195i \(-0.394129\pi\)
0.326505 + 0.945195i \(0.394129\pi\)
\(992\) −8.29204 −0.263272
\(993\) −27.6085 −0.876129
\(994\) 13.6884 0.434169
\(995\) 0 0
\(996\) 53.3506 1.69048
\(997\) 21.7296 0.688183 0.344092 0.938936i \(-0.388187\pi\)
0.344092 + 0.938936i \(0.388187\pi\)
\(998\) −10.1296 −0.320647
\(999\) 3.14830 0.0996079
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 625.2.a.g.1.7 yes 8
3.2 odd 2 5625.2.a.s.1.2 8
4.3 odd 2 10000.2.a.be.1.3 8
5.2 odd 4 625.2.b.d.624.15 16
5.3 odd 4 625.2.b.d.624.2 16
5.4 even 2 625.2.a.e.1.2 8
15.14 odd 2 5625.2.a.be.1.7 8
20.19 odd 2 10000.2.a.bn.1.6 8
25.2 odd 20 625.2.e.k.124.8 32
25.3 odd 20 625.2.e.j.249.1 32
25.4 even 10 625.2.d.q.376.4 16
25.6 even 5 625.2.d.m.251.1 16
25.8 odd 20 625.2.e.j.374.8 32
25.9 even 10 625.2.d.p.126.1 16
25.11 even 5 625.2.d.n.501.4 16
25.12 odd 20 625.2.e.k.499.1 32
25.13 odd 20 625.2.e.k.499.8 32
25.14 even 10 625.2.d.p.501.1 16
25.16 even 5 625.2.d.n.126.4 16
25.17 odd 20 625.2.e.j.374.1 32
25.19 even 10 625.2.d.q.251.4 16
25.21 even 5 625.2.d.m.376.1 16
25.22 odd 20 625.2.e.j.249.8 32
25.23 odd 20 625.2.e.k.124.1 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
625.2.a.e.1.2 8 5.4 even 2
625.2.a.g.1.7 yes 8 1.1 even 1 trivial
625.2.b.d.624.2 16 5.3 odd 4
625.2.b.d.624.15 16 5.2 odd 4
625.2.d.m.251.1 16 25.6 even 5
625.2.d.m.376.1 16 25.21 even 5
625.2.d.n.126.4 16 25.16 even 5
625.2.d.n.501.4 16 25.11 even 5
625.2.d.p.126.1 16 25.9 even 10
625.2.d.p.501.1 16 25.14 even 10
625.2.d.q.251.4 16 25.19 even 10
625.2.d.q.376.4 16 25.4 even 10
625.2.e.j.249.1 32 25.3 odd 20
625.2.e.j.249.8 32 25.22 odd 20
625.2.e.j.374.1 32 25.17 odd 20
625.2.e.j.374.8 32 25.8 odd 20
625.2.e.k.124.1 32 25.23 odd 20
625.2.e.k.124.8 32 25.2 odd 20
625.2.e.k.499.1 32 25.12 odd 20
625.2.e.k.499.8 32 25.13 odd 20
5625.2.a.s.1.2 8 3.2 odd 2
5625.2.a.be.1.7 8 15.14 odd 2
10000.2.a.be.1.3 8 4.3 odd 2
10000.2.a.bn.1.6 8 20.19 odd 2