Properties

Label 1250.2.a.l.1.3
Level $1250$
Weight $2$
Character 1250.1
Self dual yes
Analytic conductor $9.981$
Analytic rank $0$
Dimension $4$
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] = [1250,2,Mod(1,1250)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1250, 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("1250.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1250 = 2 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1250.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: \(9.98130025266\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.7625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 9x^{2} + 4x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 50)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.71472\) of defining polynomial
Character \(\chi\) \(=\) 1250.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.71472 q^{3} +1.00000 q^{4} +1.71472 q^{6} -2.77447 q^{7} +1.00000 q^{8} -0.0597522 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.71472 q^{3} +1.00000 q^{4} +1.71472 q^{6} -2.77447 q^{7} +1.00000 q^{8} -0.0597522 q^{9} +2.77447 q^{11} +1.71472 q^{12} +5.67779 q^{13} -2.77447 q^{14} +1.00000 q^{16} +5.15643 q^{17} -0.0597522 q^{18} +1.41238 q^{19} -4.75742 q^{21} +2.77447 q^{22} -0.654963 q^{23} +1.71472 q^{24} +5.67779 q^{26} -5.24660 q^{27} -2.77447 q^{28} +4.09668 q^{29} -7.12710 q^{31} +1.00000 q^{32} +4.75742 q^{33} +5.15643 q^{34} -0.0597522 q^{36} -1.04746 q^{37} +1.41238 q^{38} +9.73579 q^{39} +9.10722 q^{41} -4.75742 q^{42} +9.24660 q^{43} +2.77447 q^{44} -0.654963 q^{46} -2.77447 q^{47} +1.71472 q^{48} +0.697669 q^{49} +8.84181 q^{51} +5.67779 q^{52} -0.526111 q^{53} -5.24660 q^{54} -2.77447 q^{56} +2.42184 q^{57} +4.09668 q^{58} -3.78206 q^{59} -10.8325 q^{61} -7.12710 q^{62} +0.165781 q^{63} +1.00000 q^{64} +4.75742 q^{66} -4.32340 q^{67} +5.15643 q^{68} -1.12307 q^{69} -13.2466 q^{71} -0.0597522 q^{72} +4.21324 q^{73} -1.04746 q^{74} +1.41238 q^{76} -7.69767 q^{77} +9.73579 q^{78} -9.90157 q^{79} -8.81717 q^{81} +9.10722 q^{82} +4.67603 q^{83} -4.75742 q^{84} +9.24660 q^{86} +7.02464 q^{87} +2.77447 q^{88} -9.18401 q^{89} -15.7528 q^{91} -0.654963 q^{92} -12.2209 q^{93} -2.77447 q^{94} +1.71472 q^{96} -0.0901699 q^{97} +0.697669 q^{98} -0.165781 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + q^{3} + 4 q^{4} + q^{6} + 2 q^{7} + 4 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + q^{3} + 4 q^{4} + q^{6} + 2 q^{7} + 4 q^{8} + 7 q^{9} - 2 q^{11} + q^{12} + 11 q^{13} + 2 q^{14} + 4 q^{16} + 12 q^{17} + 7 q^{18} - 5 q^{19} - 7 q^{21} - 2 q^{22} - 4 q^{23} + q^{24} + 11 q^{26} + 10 q^{27} + 2 q^{28} + 15 q^{29} - 12 q^{31} + 4 q^{32} + 7 q^{33} + 12 q^{34} + 7 q^{36} + 12 q^{37} - 5 q^{38} - 11 q^{39} + 13 q^{41} - 7 q^{42} + 6 q^{43} - 2 q^{44} - 4 q^{46} + 2 q^{47} + q^{48} - 2 q^{49} + 13 q^{51} + 11 q^{52} + 11 q^{53} + 10 q^{54} + 2 q^{56} + 15 q^{58} + 8 q^{61} - 12 q^{62} + 21 q^{63} + 4 q^{64} + 7 q^{66} + 22 q^{67} + 12 q^{68} - 31 q^{69} - 22 q^{71} + 7 q^{72} + 21 q^{73} + 12 q^{74} - 5 q^{76} - 26 q^{77} - 11 q^{78} - 10 q^{79} - 16 q^{81} + 13 q^{82} - 24 q^{83} - 7 q^{84} + 6 q^{86} + 25 q^{87} - 2 q^{88} - 5 q^{89} - 12 q^{91} - 4 q^{92} - 23 q^{93} + 2 q^{94} + q^{96} + 22 q^{97} - 2 q^{98} - 21 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.71472 0.989991 0.494996 0.868895i \(-0.335170\pi\)
0.494996 + 0.868895i \(0.335170\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.71472 0.700029
\(7\) −2.77447 −1.04865 −0.524325 0.851518i \(-0.675682\pi\)
−0.524325 + 0.851518i \(0.675682\pi\)
\(8\) 1.00000 0.353553
\(9\) −0.0597522 −0.0199174
\(10\) 0 0
\(11\) 2.77447 0.836533 0.418267 0.908324i \(-0.362638\pi\)
0.418267 + 0.908324i \(0.362638\pi\)
\(12\) 1.71472 0.494996
\(13\) 5.67779 1.57473 0.787367 0.616484i \(-0.211444\pi\)
0.787367 + 0.616484i \(0.211444\pi\)
\(14\) −2.77447 −0.741508
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 5.15643 1.25062 0.625309 0.780377i \(-0.284973\pi\)
0.625309 + 0.780377i \(0.284973\pi\)
\(18\) −0.0597522 −0.0140837
\(19\) 1.41238 0.324023 0.162012 0.986789i \(-0.448202\pi\)
0.162012 + 0.986789i \(0.448202\pi\)
\(20\) 0 0
\(21\) −4.75742 −1.03815
\(22\) 2.77447 0.591518
\(23\) −0.654963 −0.136569 −0.0682846 0.997666i \(-0.521753\pi\)
−0.0682846 + 0.997666i \(0.521753\pi\)
\(24\) 1.71472 0.350015
\(25\) 0 0
\(26\) 5.67779 1.11351
\(27\) −5.24660 −1.00971
\(28\) −2.77447 −0.524325
\(29\) 4.09668 0.760735 0.380367 0.924836i \(-0.375798\pi\)
0.380367 + 0.924836i \(0.375798\pi\)
\(30\) 0 0
\(31\) −7.12710 −1.28006 −0.640032 0.768348i \(-0.721079\pi\)
−0.640032 + 0.768348i \(0.721079\pi\)
\(32\) 1.00000 0.176777
\(33\) 4.75742 0.828161
\(34\) 5.15643 0.884321
\(35\) 0 0
\(36\) −0.0597522 −0.00995870
\(37\) −1.04746 −0.172202 −0.0861010 0.996286i \(-0.527441\pi\)
−0.0861010 + 0.996286i \(0.527441\pi\)
\(38\) 1.41238 0.229119
\(39\) 9.73579 1.55897
\(40\) 0 0
\(41\) 9.10722 1.42231 0.711154 0.703036i \(-0.248173\pi\)
0.711154 + 0.703036i \(0.248173\pi\)
\(42\) −4.75742 −0.734086
\(43\) 9.24660 1.41009 0.705047 0.709161i \(-0.250926\pi\)
0.705047 + 0.709161i \(0.250926\pi\)
\(44\) 2.77447 0.418267
\(45\) 0 0
\(46\) −0.654963 −0.0965690
\(47\) −2.77447 −0.404698 −0.202349 0.979314i \(-0.564857\pi\)
−0.202349 + 0.979314i \(0.564857\pi\)
\(48\) 1.71472 0.247498
\(49\) 0.697669 0.0996670
\(50\) 0 0
\(51\) 8.84181 1.23810
\(52\) 5.67779 0.787367
\(53\) −0.526111 −0.0722669 −0.0361335 0.999347i \(-0.511504\pi\)
−0.0361335 + 0.999347i \(0.511504\pi\)
\(54\) −5.24660 −0.713972
\(55\) 0 0
\(56\) −2.77447 −0.370754
\(57\) 2.42184 0.320780
\(58\) 4.09668 0.537921
\(59\) −3.78206 −0.492382 −0.246191 0.969221i \(-0.579179\pi\)
−0.246191 + 0.969221i \(0.579179\pi\)
\(60\) 0 0
\(61\) −10.8325 −1.38696 −0.693478 0.720478i \(-0.743922\pi\)
−0.693478 + 0.720478i \(0.743922\pi\)
\(62\) −7.12710 −0.905142
\(63\) 0.165781 0.0208864
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 4.75742 0.585598
\(67\) −4.32340 −0.528188 −0.264094 0.964497i \(-0.585073\pi\)
−0.264094 + 0.964497i \(0.585073\pi\)
\(68\) 5.15643 0.625309
\(69\) −1.12307 −0.135202
\(70\) 0 0
\(71\) −13.2466 −1.57208 −0.786041 0.618174i \(-0.787873\pi\)
−0.786041 + 0.618174i \(0.787873\pi\)
\(72\) −0.0597522 −0.00704186
\(73\) 4.21324 0.493123 0.246561 0.969127i \(-0.420699\pi\)
0.246561 + 0.969127i \(0.420699\pi\)
\(74\) −1.04746 −0.121765
\(75\) 0 0
\(76\) 1.41238 0.162012
\(77\) −7.69767 −0.877231
\(78\) 9.73579 1.10236
\(79\) −9.90157 −1.11401 −0.557007 0.830508i \(-0.688050\pi\)
−0.557007 + 0.830508i \(0.688050\pi\)
\(80\) 0 0
\(81\) −8.81717 −0.979686
\(82\) 9.10722 1.00572
\(83\) 4.67603 0.513261 0.256631 0.966510i \(-0.417388\pi\)
0.256631 + 0.966510i \(0.417388\pi\)
\(84\) −4.75742 −0.519077
\(85\) 0 0
\(86\) 9.24660 0.997087
\(87\) 7.02464 0.753121
\(88\) 2.77447 0.295759
\(89\) −9.18401 −0.973504 −0.486752 0.873540i \(-0.661818\pi\)
−0.486752 + 0.873540i \(0.661818\pi\)
\(90\) 0 0
\(91\) −15.7528 −1.65135
\(92\) −0.654963 −0.0682846
\(93\) −12.2209 −1.26725
\(94\) −2.77447 −0.286164
\(95\) 0 0
\(96\) 1.71472 0.175007
\(97\) −0.0901699 −0.00915537 −0.00457769 0.999990i \(-0.501457\pi\)
−0.00457769 + 0.999990i \(0.501457\pi\)
\(98\) 0.697669 0.0704752
\(99\) −0.165781 −0.0166616
\(100\) 0 0
\(101\) −7.90632 −0.786709 −0.393354 0.919387i \(-0.628685\pi\)
−0.393354 + 0.919387i \(0.628685\pi\)
\(102\) 8.84181 0.875470
\(103\) 7.83422 0.771929 0.385964 0.922514i \(-0.373869\pi\)
0.385964 + 0.922514i \(0.373869\pi\)
\(104\) 5.67779 0.556753
\(105\) 0 0
\(106\) −0.526111 −0.0511004
\(107\) −10.2220 −0.988194 −0.494097 0.869407i \(-0.664501\pi\)
−0.494097 + 0.869407i \(0.664501\pi\)
\(108\) −5.24660 −0.504855
\(109\) 3.91091 0.374598 0.187299 0.982303i \(-0.440027\pi\)
0.187299 + 0.982303i \(0.440027\pi\)
\(110\) 0 0
\(111\) −1.79610 −0.170479
\(112\) −2.77447 −0.262163
\(113\) −6.37545 −0.599752 −0.299876 0.953978i \(-0.596945\pi\)
−0.299876 + 0.953978i \(0.596945\pi\)
\(114\) 2.42184 0.226826
\(115\) 0 0
\(116\) 4.09668 0.380367
\(117\) −0.339260 −0.0313646
\(118\) −3.78206 −0.348167
\(119\) −14.3064 −1.31146
\(120\) 0 0
\(121\) −3.30233 −0.300212
\(122\) −10.8325 −0.980725
\(123\) 15.6163 1.40807
\(124\) −7.12710 −0.640032
\(125\) 0 0
\(126\) 0.165781 0.0147689
\(127\) 9.54893 0.847331 0.423665 0.905819i \(-0.360743\pi\)
0.423665 + 0.905819i \(0.360743\pi\)
\(128\) 1.00000 0.0883883
\(129\) 15.8553 1.39598
\(130\) 0 0
\(131\) 5.59923 0.489207 0.244604 0.969623i \(-0.421342\pi\)
0.244604 + 0.969623i \(0.421342\pi\)
\(132\) 4.75742 0.414080
\(133\) −3.91861 −0.339787
\(134\) −4.32340 −0.373485
\(135\) 0 0
\(136\) 5.15643 0.442161
\(137\) 7.98414 0.682131 0.341066 0.940039i \(-0.389212\pi\)
0.341066 + 0.940039i \(0.389212\pi\)
\(138\) −1.12307 −0.0956025
\(139\) −22.0884 −1.87352 −0.936758 0.349979i \(-0.886189\pi\)
−0.936758 + 0.349979i \(0.886189\pi\)
\(140\) 0 0
\(141\) −4.75742 −0.400647
\(142\) −13.2466 −1.11163
\(143\) 15.7528 1.31732
\(144\) −0.0597522 −0.00497935
\(145\) 0 0
\(146\) 4.21324 0.348691
\(147\) 1.19630 0.0986694
\(148\) −1.04746 −0.0861010
\(149\) 4.18401 0.342768 0.171384 0.985204i \(-0.445176\pi\)
0.171384 + 0.985204i \(0.445176\pi\)
\(150\) 0 0
\(151\) −0.331561 −0.0269821 −0.0134910 0.999909i \(-0.504294\pi\)
−0.0134910 + 0.999909i \(0.504294\pi\)
\(152\) 1.41238 0.114559
\(153\) −0.308108 −0.0249091
\(154\) −7.69767 −0.620296
\(155\) 0 0
\(156\) 9.73579 0.779487
\(157\) 3.60750 0.287910 0.143955 0.989584i \(-0.454018\pi\)
0.143955 + 0.989584i \(0.454018\pi\)
\(158\) −9.90157 −0.787726
\(159\) −0.902131 −0.0715436
\(160\) 0 0
\(161\) 1.81717 0.143213
\(162\) −8.81717 −0.692743
\(163\) 5.09385 0.398981 0.199490 0.979900i \(-0.436071\pi\)
0.199490 + 0.979900i \(0.436071\pi\)
\(164\) 9.10722 0.711154
\(165\) 0 0
\(166\) 4.67603 0.362931
\(167\) 20.8629 1.61442 0.807209 0.590265i \(-0.200977\pi\)
0.807209 + 0.590265i \(0.200977\pi\)
\(168\) −4.75742 −0.367043
\(169\) 19.2373 1.47979
\(170\) 0 0
\(171\) −0.0843930 −0.00645370
\(172\) 9.24660 0.705047
\(173\) −11.9877 −0.911409 −0.455704 0.890131i \(-0.650613\pi\)
−0.455704 + 0.890131i \(0.650613\pi\)
\(174\) 7.02464 0.532537
\(175\) 0 0
\(176\) 2.77447 0.209133
\(177\) −6.48516 −0.487454
\(178\) −9.18401 −0.688371
\(179\) −8.54134 −0.638410 −0.319205 0.947686i \(-0.603416\pi\)
−0.319205 + 0.947686i \(0.603416\pi\)
\(180\) 0 0
\(181\) 7.82844 0.581884 0.290942 0.956741i \(-0.406031\pi\)
0.290942 + 0.956741i \(0.406031\pi\)
\(182\) −15.7528 −1.16768
\(183\) −18.5746 −1.37307
\(184\) −0.654963 −0.0482845
\(185\) 0 0
\(186\) −12.2209 −0.896083
\(187\) 14.3064 1.04618
\(188\) −2.77447 −0.202349
\(189\) 14.5565 1.05883
\(190\) 0 0
\(191\) 3.36611 0.243563 0.121782 0.992557i \(-0.461139\pi\)
0.121782 + 0.992557i \(0.461139\pi\)
\(192\) 1.71472 0.123749
\(193\) −15.4211 −1.11004 −0.555018 0.831839i \(-0.687288\pi\)
−0.555018 + 0.831839i \(0.687288\pi\)
\(194\) −0.0901699 −0.00647382
\(195\) 0 0
\(196\) 0.697669 0.0498335
\(197\) 8.80489 0.627322 0.313661 0.949535i \(-0.398444\pi\)
0.313661 + 0.949535i \(0.398444\pi\)
\(198\) −0.165781 −0.0117815
\(199\) −17.6222 −1.24920 −0.624601 0.780944i \(-0.714738\pi\)
−0.624601 + 0.780944i \(0.714738\pi\)
\(200\) 0 0
\(201\) −7.41340 −0.522901
\(202\) −7.90632 −0.556287
\(203\) −11.3661 −0.797744
\(204\) 8.84181 0.619051
\(205\) 0 0
\(206\) 7.83422 0.545836
\(207\) 0.0391355 0.00272010
\(208\) 5.67779 0.393684
\(209\) 3.91861 0.271056
\(210\) 0 0
\(211\) 6.10144 0.420040 0.210020 0.977697i \(-0.432647\pi\)
0.210020 + 0.977697i \(0.432647\pi\)
\(212\) −0.526111 −0.0361335
\(213\) −22.7142 −1.55635
\(214\) −10.2220 −0.698759
\(215\) 0 0
\(216\) −5.24660 −0.356986
\(217\) 19.7739 1.34234
\(218\) 3.91091 0.264880
\(219\) 7.22451 0.488187
\(220\) 0 0
\(221\) 29.2771 1.96939
\(222\) −1.79610 −0.120547
\(223\) −5.02866 −0.336744 −0.168372 0.985723i \(-0.553851\pi\)
−0.168372 + 0.985723i \(0.553851\pi\)
\(224\) −2.77447 −0.185377
\(225\) 0 0
\(226\) −6.37545 −0.424089
\(227\) −17.8382 −1.18397 −0.591983 0.805951i \(-0.701655\pi\)
−0.591983 + 0.805951i \(0.701655\pi\)
\(228\) 2.42184 0.160390
\(229\) −21.5566 −1.42450 −0.712251 0.701925i \(-0.752324\pi\)
−0.712251 + 0.701925i \(0.752324\pi\)
\(230\) 0 0
\(231\) −13.1993 −0.868451
\(232\) 4.09668 0.268960
\(233\) 14.7065 0.963452 0.481726 0.876322i \(-0.340010\pi\)
0.481726 + 0.876322i \(0.340010\pi\)
\(234\) −0.339260 −0.0221781
\(235\) 0 0
\(236\) −3.78206 −0.246191
\(237\) −16.9784 −1.10286
\(238\) −14.3064 −0.927343
\(239\) 6.94025 0.448927 0.224464 0.974482i \(-0.427937\pi\)
0.224464 + 0.974482i \(0.427937\pi\)
\(240\) 0 0
\(241\) 3.31003 0.213218 0.106609 0.994301i \(-0.466001\pi\)
0.106609 + 0.994301i \(0.466001\pi\)
\(242\) −3.30233 −0.212282
\(243\) 0.620870 0.0398288
\(244\) −10.8325 −0.693478
\(245\) 0 0
\(246\) 15.6163 0.995658
\(247\) 8.01921 0.510250
\(248\) −7.12710 −0.452571
\(249\) 8.01806 0.508124
\(250\) 0 0
\(251\) −9.46454 −0.597397 −0.298698 0.954348i \(-0.596552\pi\)
−0.298698 + 0.954348i \(0.596552\pi\)
\(252\) 0.165781 0.0104432
\(253\) −1.81717 −0.114245
\(254\) 9.54893 0.599153
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −7.07963 −0.441615 −0.220808 0.975317i \(-0.570869\pi\)
−0.220808 + 0.975317i \(0.570869\pi\)
\(258\) 15.8553 0.987107
\(259\) 2.90615 0.180580
\(260\) 0 0
\(261\) −0.244786 −0.0151519
\(262\) 5.59923 0.345922
\(263\) 1.43345 0.0883906 0.0441953 0.999023i \(-0.485928\pi\)
0.0441953 + 0.999023i \(0.485928\pi\)
\(264\) 4.75742 0.292799
\(265\) 0 0
\(266\) −3.91861 −0.240266
\(267\) −15.7480 −0.963760
\(268\) −4.32340 −0.264094
\(269\) −14.2830 −0.870848 −0.435424 0.900226i \(-0.643402\pi\)
−0.435424 + 0.900226i \(0.643402\pi\)
\(270\) 0 0
\(271\) 30.7855 1.87009 0.935044 0.354533i \(-0.115360\pi\)
0.935044 + 0.354533i \(0.115360\pi\)
\(272\) 5.15643 0.312655
\(273\) −27.0116 −1.63482
\(274\) 7.98414 0.482340
\(275\) 0 0
\(276\) −1.12307 −0.0676012
\(277\) 3.18969 0.191650 0.0958249 0.995398i \(-0.469451\pi\)
0.0958249 + 0.995398i \(0.469451\pi\)
\(278\) −22.0884 −1.32478
\(279\) 0.425860 0.0254956
\(280\) 0 0
\(281\) 20.3889 1.21630 0.608151 0.793822i \(-0.291912\pi\)
0.608151 + 0.793822i \(0.291912\pi\)
\(282\) −4.75742 −0.283300
\(283\) −13.2969 −0.790419 −0.395209 0.918591i \(-0.629328\pi\)
−0.395209 + 0.918591i \(0.629328\pi\)
\(284\) −13.2466 −0.786041
\(285\) 0 0
\(286\) 15.7528 0.931484
\(287\) −25.2677 −1.49150
\(288\) −0.0597522 −0.00352093
\(289\) 9.58880 0.564047
\(290\) 0 0
\(291\) −0.154616 −0.00906374
\(292\) 4.21324 0.246561
\(293\) −26.0420 −1.52139 −0.760696 0.649108i \(-0.775142\pi\)
−0.760696 + 0.649108i \(0.775142\pi\)
\(294\) 1.19630 0.0697698
\(295\) 0 0
\(296\) −1.04746 −0.0608826
\(297\) −14.5565 −0.844655
\(298\) 4.18401 0.242373
\(299\) −3.71874 −0.215060
\(300\) 0 0
\(301\) −25.6544 −1.47869
\(302\) −0.331561 −0.0190792
\(303\) −13.5571 −0.778835
\(304\) 1.41238 0.0810058
\(305\) 0 0
\(306\) −0.308108 −0.0176134
\(307\) 25.1000 1.43253 0.716267 0.697826i \(-0.245849\pi\)
0.716267 + 0.697826i \(0.245849\pi\)
\(308\) −7.69767 −0.438615
\(309\) 13.4335 0.764203
\(310\) 0 0
\(311\) 0.723328 0.0410162 0.0205081 0.999790i \(-0.493472\pi\)
0.0205081 + 0.999790i \(0.493472\pi\)
\(312\) 9.73579 0.551180
\(313\) 21.6163 1.22182 0.610912 0.791698i \(-0.290803\pi\)
0.610912 + 0.791698i \(0.290803\pi\)
\(314\) 3.60750 0.203583
\(315\) 0 0
\(316\) −9.90157 −0.557007
\(317\) −18.8337 −1.05780 −0.528902 0.848683i \(-0.677396\pi\)
−0.528902 + 0.848683i \(0.677396\pi\)
\(318\) −0.902131 −0.0505890
\(319\) 11.3661 0.636380
\(320\) 0 0
\(321\) −17.5278 −0.978304
\(322\) 1.81717 0.101267
\(323\) 7.28286 0.405229
\(324\) −8.81717 −0.489843
\(325\) 0 0
\(326\) 5.09385 0.282122
\(327\) 6.70610 0.370848
\(328\) 9.10722 0.502862
\(329\) 7.69767 0.424386
\(330\) 0 0
\(331\) 16.8759 0.927584 0.463792 0.885944i \(-0.346489\pi\)
0.463792 + 0.885944i \(0.346489\pi\)
\(332\) 4.67603 0.256631
\(333\) 0.0625883 0.00342982
\(334\) 20.8629 1.14157
\(335\) 0 0
\(336\) −4.75742 −0.259539
\(337\) 19.3521 1.05417 0.527087 0.849811i \(-0.323284\pi\)
0.527087 + 0.849811i \(0.323284\pi\)
\(338\) 19.2373 1.04637
\(339\) −10.9321 −0.593750
\(340\) 0 0
\(341\) −19.7739 −1.07082
\(342\) −0.0843930 −0.00456345
\(343\) 17.4856 0.944134
\(344\) 9.24660 0.498543
\(345\) 0 0
\(346\) −11.9877 −0.644463
\(347\) −9.03868 −0.485222 −0.242611 0.970124i \(-0.578004\pi\)
−0.242611 + 0.970124i \(0.578004\pi\)
\(348\) 7.02464 0.376560
\(349\) 36.7305 1.96614 0.983068 0.183240i \(-0.0586584\pi\)
0.983068 + 0.183240i \(0.0586584\pi\)
\(350\) 0 0
\(351\) −29.7891 −1.59002
\(352\) 2.77447 0.147880
\(353\) 18.3737 0.977933 0.488967 0.872302i \(-0.337374\pi\)
0.488967 + 0.872302i \(0.337374\pi\)
\(354\) −6.48516 −0.344682
\(355\) 0 0
\(356\) −9.18401 −0.486752
\(357\) −24.5313 −1.29834
\(358\) −8.54134 −0.451424
\(359\) −24.4259 −1.28915 −0.644574 0.764542i \(-0.722965\pi\)
−0.644574 + 0.764542i \(0.722965\pi\)
\(360\) 0 0
\(361\) −17.0052 −0.895009
\(362\) 7.82844 0.411454
\(363\) −5.66256 −0.297207
\(364\) −15.7528 −0.825673
\(365\) 0 0
\(366\) −18.5746 −0.970910
\(367\) 14.5224 0.758065 0.379032 0.925383i \(-0.376257\pi\)
0.379032 + 0.925383i \(0.376257\pi\)
\(368\) −0.654963 −0.0341423
\(369\) −0.544176 −0.0283287
\(370\) 0 0
\(371\) 1.45968 0.0757827
\(372\) −12.2209 −0.633626
\(373\) −23.2887 −1.20585 −0.602923 0.797800i \(-0.705997\pi\)
−0.602923 + 0.797800i \(0.705997\pi\)
\(374\) 14.3064 0.739764
\(375\) 0 0
\(376\) −2.77447 −0.143082
\(377\) 23.2601 1.19796
\(378\) 14.5565 0.748707
\(379\) 28.6176 1.46999 0.734993 0.678075i \(-0.237185\pi\)
0.734993 + 0.678075i \(0.237185\pi\)
\(380\) 0 0
\(381\) 16.3737 0.838850
\(382\) 3.36611 0.172225
\(383\) −16.6157 −0.849023 −0.424512 0.905422i \(-0.639554\pi\)
−0.424512 + 0.905422i \(0.639554\pi\)
\(384\) 1.71472 0.0875037
\(385\) 0 0
\(386\) −15.4211 −0.784913
\(387\) −0.552505 −0.0280854
\(388\) −0.0901699 −0.00457769
\(389\) 26.3767 1.33735 0.668677 0.743553i \(-0.266861\pi\)
0.668677 + 0.743553i \(0.266861\pi\)
\(390\) 0 0
\(391\) −3.37727 −0.170796
\(392\) 0.697669 0.0352376
\(393\) 9.60109 0.484311
\(394\) 8.80489 0.443584
\(395\) 0 0
\(396\) −0.165781 −0.00833078
\(397\) −17.9086 −0.898807 −0.449403 0.893329i \(-0.648363\pi\)
−0.449403 + 0.893329i \(0.648363\pi\)
\(398\) −17.6222 −0.883319
\(399\) −6.71930 −0.336386
\(400\) 0 0
\(401\) 32.0164 1.59882 0.799411 0.600785i \(-0.205145\pi\)
0.799411 + 0.600785i \(0.205145\pi\)
\(402\) −7.41340 −0.369747
\(403\) −40.4661 −2.01576
\(404\) −7.90632 −0.393354
\(405\) 0 0
\(406\) −11.3661 −0.564090
\(407\) −2.90615 −0.144053
\(408\) 8.84181 0.437735
\(409\) 18.0344 0.891743 0.445871 0.895097i \(-0.352894\pi\)
0.445871 + 0.895097i \(0.352894\pi\)
\(410\) 0 0
\(411\) 13.6905 0.675304
\(412\) 7.83422 0.385964
\(413\) 10.4932 0.516337
\(414\) 0.0391355 0.00192340
\(415\) 0 0
\(416\) 5.67779 0.278376
\(417\) −37.8753 −1.85476
\(418\) 3.91861 0.191666
\(419\) −19.5300 −0.954104 −0.477052 0.878875i \(-0.658295\pi\)
−0.477052 + 0.878875i \(0.658295\pi\)
\(420\) 0 0
\(421\) −19.5537 −0.952989 −0.476494 0.879178i \(-0.658093\pi\)
−0.476494 + 0.879178i \(0.658093\pi\)
\(422\) 6.10144 0.297013
\(423\) 0.165781 0.00806052
\(424\) −0.526111 −0.0255502
\(425\) 0 0
\(426\) −22.7142 −1.10050
\(427\) 30.0543 1.45443
\(428\) −10.2220 −0.494097
\(429\) 27.0116 1.30413
\(430\) 0 0
\(431\) −21.1963 −1.02099 −0.510495 0.859881i \(-0.670538\pi\)
−0.510495 + 0.859881i \(0.670538\pi\)
\(432\) −5.24660 −0.252427
\(433\) 8.54967 0.410871 0.205435 0.978671i \(-0.434139\pi\)
0.205435 + 0.978671i \(0.434139\pi\)
\(434\) 19.7739 0.949178
\(435\) 0 0
\(436\) 3.91091 0.187299
\(437\) −0.925059 −0.0442516
\(438\) 7.22451 0.345201
\(439\) −28.0714 −1.33977 −0.669887 0.742463i \(-0.733657\pi\)
−0.669887 + 0.742463i \(0.733657\pi\)
\(440\) 0 0
\(441\) −0.0416872 −0.00198511
\(442\) 29.2771 1.39257
\(443\) −33.0546 −1.57047 −0.785236 0.619197i \(-0.787458\pi\)
−0.785236 + 0.619197i \(0.787458\pi\)
\(444\) −1.79610 −0.0852393
\(445\) 0 0
\(446\) −5.02866 −0.238114
\(447\) 7.17439 0.339337
\(448\) −2.77447 −0.131081
\(449\) −37.1628 −1.75382 −0.876911 0.480652i \(-0.840400\pi\)
−0.876911 + 0.480652i \(0.840400\pi\)
\(450\) 0 0
\(451\) 25.2677 1.18981
\(452\) −6.37545 −0.299876
\(453\) −0.568533 −0.0267120
\(454\) −17.8382 −0.837190
\(455\) 0 0
\(456\) 2.42184 0.113413
\(457\) 29.3139 1.37125 0.685624 0.727956i \(-0.259529\pi\)
0.685624 + 0.727956i \(0.259529\pi\)
\(458\) −21.5566 −1.00728
\(459\) −27.0538 −1.26276
\(460\) 0 0
\(461\) 8.90741 0.414859 0.207430 0.978250i \(-0.433490\pi\)
0.207430 + 0.978250i \(0.433490\pi\)
\(462\) −13.1993 −0.614087
\(463\) −18.2771 −0.849410 −0.424705 0.905332i \(-0.639622\pi\)
−0.424705 + 0.905332i \(0.639622\pi\)
\(464\) 4.09668 0.190184
\(465\) 0 0
\(466\) 14.7065 0.681263
\(467\) −36.6841 −1.69754 −0.848768 0.528765i \(-0.822655\pi\)
−0.848768 + 0.528765i \(0.822655\pi\)
\(468\) −0.339260 −0.0156823
\(469\) 11.9951 0.553884
\(470\) 0 0
\(471\) 6.18583 0.285028
\(472\) −3.78206 −0.174084
\(473\) 25.6544 1.17959
\(474\) −16.9784 −0.779842
\(475\) 0 0
\(476\) −14.3064 −0.655731
\(477\) 0.0314363 0.00143937
\(478\) 6.94025 0.317440
\(479\) −12.5898 −0.575242 −0.287621 0.957744i \(-0.592864\pi\)
−0.287621 + 0.957744i \(0.592864\pi\)
\(480\) 0 0
\(481\) −5.94728 −0.271173
\(482\) 3.31003 0.150768
\(483\) 3.11593 0.141780
\(484\) −3.30233 −0.150106
\(485\) 0 0
\(486\) 0.620870 0.0281632
\(487\) −18.3256 −0.830411 −0.415205 0.909728i \(-0.636290\pi\)
−0.415205 + 0.909728i \(0.636290\pi\)
\(488\) −10.8325 −0.490363
\(489\) 8.73449 0.394987
\(490\) 0 0
\(491\) 21.7047 0.979519 0.489760 0.871857i \(-0.337085\pi\)
0.489760 + 0.871857i \(0.337085\pi\)
\(492\) 15.6163 0.704036
\(493\) 21.1243 0.951389
\(494\) 8.01921 0.360801
\(495\) 0 0
\(496\) −7.12710 −0.320016
\(497\) 36.7523 1.64856
\(498\) 8.01806 0.359298
\(499\) 15.8391 0.709055 0.354527 0.935046i \(-0.384642\pi\)
0.354527 + 0.935046i \(0.384642\pi\)
\(500\) 0 0
\(501\) 35.7739 1.59826
\(502\) −9.46454 −0.422423
\(503\) 27.5530 1.22853 0.614263 0.789101i \(-0.289453\pi\)
0.614263 + 0.789101i \(0.289453\pi\)
\(504\) 0.165781 0.00738445
\(505\) 0 0
\(506\) −1.81717 −0.0807832
\(507\) 32.9864 1.46498
\(508\) 9.54893 0.423665
\(509\) −27.6120 −1.22388 −0.611941 0.790903i \(-0.709611\pi\)
−0.611941 + 0.790903i \(0.709611\pi\)
\(510\) 0 0
\(511\) −11.6895 −0.517113
\(512\) 1.00000 0.0441942
\(513\) −7.41022 −0.327169
\(514\) −7.07963 −0.312269
\(515\) 0 0
\(516\) 15.8553 0.697990
\(517\) −7.69767 −0.338543
\(518\) 2.90615 0.127689
\(519\) −20.5555 −0.902287
\(520\) 0 0
\(521\) 17.0892 0.748689 0.374345 0.927290i \(-0.377868\pi\)
0.374345 + 0.927290i \(0.377868\pi\)
\(522\) −0.244786 −0.0107140
\(523\) 36.3152 1.58795 0.793977 0.607947i \(-0.208007\pi\)
0.793977 + 0.607947i \(0.208007\pi\)
\(524\) 5.59923 0.244604
\(525\) 0 0
\(526\) 1.43345 0.0625016
\(527\) −36.7504 −1.60087
\(528\) 4.75742 0.207040
\(529\) −22.5710 −0.981349
\(530\) 0 0
\(531\) 0.225987 0.00980698
\(532\) −3.91861 −0.169893
\(533\) 51.7088 2.23976
\(534\) −15.7480 −0.681481
\(535\) 0 0
\(536\) −4.32340 −0.186743
\(537\) −14.6460 −0.632020
\(538\) −14.2830 −0.615782
\(539\) 1.93566 0.0833747
\(540\) 0 0
\(541\) 24.9366 1.07211 0.536055 0.844183i \(-0.319914\pi\)
0.536055 + 0.844183i \(0.319914\pi\)
\(542\) 30.7855 1.32235
\(543\) 13.4235 0.576060
\(544\) 5.15643 0.221080
\(545\) 0 0
\(546\) −27.0116 −1.15599
\(547\) 6.67201 0.285275 0.142637 0.989775i \(-0.454442\pi\)
0.142637 + 0.989775i \(0.454442\pi\)
\(548\) 7.98414 0.341066
\(549\) 0.647264 0.0276245
\(550\) 0 0
\(551\) 5.78609 0.246496
\(552\) −1.12307 −0.0478012
\(553\) 27.4716 1.16821
\(554\) 3.18969 0.135517
\(555\) 0 0
\(556\) −22.0884 −0.936758
\(557\) −20.4328 −0.865765 −0.432882 0.901450i \(-0.642503\pi\)
−0.432882 + 0.901450i \(0.642503\pi\)
\(558\) 0.425860 0.0180281
\(559\) 52.5002 2.22052
\(560\) 0 0
\(561\) 24.5313 1.03571
\(562\) 20.3889 0.860055
\(563\) −0.602805 −0.0254052 −0.0127026 0.999919i \(-0.504043\pi\)
−0.0127026 + 0.999919i \(0.504043\pi\)
\(564\) −4.75742 −0.200324
\(565\) 0 0
\(566\) −13.2969 −0.558911
\(567\) 24.4630 1.02735
\(568\) −13.2466 −0.555815
\(569\) 33.8776 1.42022 0.710112 0.704089i \(-0.248644\pi\)
0.710112 + 0.704089i \(0.248644\pi\)
\(570\) 0 0
\(571\) −6.23012 −0.260722 −0.130361 0.991467i \(-0.541614\pi\)
−0.130361 + 0.991467i \(0.541614\pi\)
\(572\) 15.7528 0.658659
\(573\) 5.77192 0.241125
\(574\) −25.2677 −1.05465
\(575\) 0 0
\(576\) −0.0597522 −0.00248967
\(577\) 3.78104 0.157407 0.0787034 0.996898i \(-0.474922\pi\)
0.0787034 + 0.996898i \(0.474922\pi\)
\(578\) 9.58880 0.398842
\(579\) −26.4428 −1.09893
\(580\) 0 0
\(581\) −12.9735 −0.538232
\(582\) −0.154616 −0.00640903
\(583\) −1.45968 −0.0604537
\(584\) 4.21324 0.174345
\(585\) 0 0
\(586\) −26.0420 −1.07579
\(587\) 16.3526 0.674945 0.337473 0.941335i \(-0.390428\pi\)
0.337473 + 0.941335i \(0.390428\pi\)
\(588\) 1.19630 0.0493347
\(589\) −10.0662 −0.414770
\(590\) 0 0
\(591\) 15.0979 0.621043
\(592\) −1.04746 −0.0430505
\(593\) 9.53314 0.391479 0.195740 0.980656i \(-0.437289\pi\)
0.195740 + 0.980656i \(0.437289\pi\)
\(594\) −14.5565 −0.597262
\(595\) 0 0
\(596\) 4.18401 0.171384
\(597\) −30.2170 −1.23670
\(598\) −3.71874 −0.152071
\(599\) −16.0682 −0.656528 −0.328264 0.944586i \(-0.606464\pi\)
−0.328264 + 0.944586i \(0.606464\pi\)
\(600\) 0 0
\(601\) 15.0380 0.613413 0.306707 0.951804i \(-0.400773\pi\)
0.306707 + 0.951804i \(0.400773\pi\)
\(602\) −25.6544 −1.04560
\(603\) 0.258333 0.0105201
\(604\) −0.331561 −0.0134910
\(605\) 0 0
\(606\) −13.5571 −0.550719
\(607\) −44.8348 −1.81979 −0.909894 0.414840i \(-0.863837\pi\)
−0.909894 + 0.414840i \(0.863837\pi\)
\(608\) 1.41238 0.0572797
\(609\) −19.4896 −0.789760
\(610\) 0 0
\(611\) −15.7528 −0.637291
\(612\) −0.308108 −0.0124545
\(613\) −15.5488 −0.628011 −0.314006 0.949421i \(-0.601671\pi\)
−0.314006 + 0.949421i \(0.601671\pi\)
\(614\) 25.1000 1.01296
\(615\) 0 0
\(616\) −7.69767 −0.310148
\(617\) 17.5994 0.708525 0.354263 0.935146i \(-0.384732\pi\)
0.354263 + 0.935146i \(0.384732\pi\)
\(618\) 13.4335 0.540373
\(619\) −37.2887 −1.49876 −0.749381 0.662140i \(-0.769649\pi\)
−0.749381 + 0.662140i \(0.769649\pi\)
\(620\) 0 0
\(621\) 3.43633 0.137895
\(622\) 0.723328 0.0290028
\(623\) 25.4807 1.02086
\(624\) 9.73579 0.389743
\(625\) 0 0
\(626\) 21.6163 0.863960
\(627\) 6.71930 0.268343
\(628\) 3.60750 0.143955
\(629\) −5.40118 −0.215359
\(630\) 0 0
\(631\) 8.55551 0.340589 0.170295 0.985393i \(-0.445528\pi\)
0.170295 + 0.985393i \(0.445528\pi\)
\(632\) −9.90157 −0.393863
\(633\) 10.4622 0.415836
\(634\) −18.8337 −0.747980
\(635\) 0 0
\(636\) −0.902131 −0.0357718
\(637\) 3.96121 0.156949
\(638\) 11.3661 0.449989
\(639\) 0.791514 0.0313118
\(640\) 0 0
\(641\) −22.8156 −0.901162 −0.450581 0.892736i \(-0.648783\pi\)
−0.450581 + 0.892736i \(0.648783\pi\)
\(642\) −17.5278 −0.691765
\(643\) 34.7745 1.37137 0.685686 0.727898i \(-0.259503\pi\)
0.685686 + 0.727898i \(0.259503\pi\)
\(644\) 1.81717 0.0716067
\(645\) 0 0
\(646\) 7.28286 0.286540
\(647\) −16.4581 −0.647034 −0.323517 0.946222i \(-0.604865\pi\)
−0.323517 + 0.946222i \(0.604865\pi\)
\(648\) −8.81717 −0.346371
\(649\) −10.4932 −0.411894
\(650\) 0 0
\(651\) 33.9066 1.32890
\(652\) 5.09385 0.199490
\(653\) 30.2092 1.18218 0.591088 0.806607i \(-0.298699\pi\)
0.591088 + 0.806607i \(0.298699\pi\)
\(654\) 6.70610 0.262229
\(655\) 0 0
\(656\) 9.10722 0.355577
\(657\) −0.251751 −0.00982173
\(658\) 7.69767 0.300086
\(659\) −28.4942 −1.10998 −0.554989 0.831858i \(-0.687277\pi\)
−0.554989 + 0.831858i \(0.687277\pi\)
\(660\) 0 0
\(661\) 50.7342 1.97333 0.986666 0.162759i \(-0.0520393\pi\)
0.986666 + 0.162759i \(0.0520393\pi\)
\(662\) 16.8759 0.655901
\(663\) 50.2019 1.94968
\(664\) 4.67603 0.181465
\(665\) 0 0
\(666\) 0.0625883 0.00242525
\(667\) −2.68317 −0.103893
\(668\) 20.8629 0.807209
\(669\) −8.62273 −0.333374
\(670\) 0 0
\(671\) −30.0543 −1.16023
\(672\) −4.75742 −0.183521
\(673\) −20.4447 −0.788084 −0.394042 0.919093i \(-0.628923\pi\)
−0.394042 + 0.919093i \(0.628923\pi\)
\(674\) 19.3521 0.745414
\(675\) 0 0
\(676\) 19.2373 0.739894
\(677\) 8.53360 0.327973 0.163986 0.986463i \(-0.447565\pi\)
0.163986 + 0.986463i \(0.447565\pi\)
\(678\) −10.9321 −0.419844
\(679\) 0.250174 0.00960078
\(680\) 0 0
\(681\) −30.5875 −1.17212
\(682\) −19.7739 −0.757182
\(683\) −48.5684 −1.85842 −0.929210 0.369553i \(-0.879511\pi\)
−0.929210 + 0.369553i \(0.879511\pi\)
\(684\) −0.0843930 −0.00322685
\(685\) 0 0
\(686\) 17.4856 0.667604
\(687\) −36.9635 −1.41024
\(688\) 9.24660 0.352523
\(689\) −2.98715 −0.113801
\(690\) 0 0
\(691\) 44.8076 1.70456 0.852281 0.523084i \(-0.175219\pi\)
0.852281 + 0.523084i \(0.175219\pi\)
\(692\) −11.9877 −0.455704
\(693\) 0.459953 0.0174722
\(694\) −9.03868 −0.343104
\(695\) 0 0
\(696\) 7.02464 0.266268
\(697\) 46.9608 1.77877
\(698\) 36.7305 1.39027
\(699\) 25.2174 0.953809
\(700\) 0 0
\(701\) 2.21913 0.0838152 0.0419076 0.999121i \(-0.486656\pi\)
0.0419076 + 0.999121i \(0.486656\pi\)
\(702\) −29.7891 −1.12432
\(703\) −1.47942 −0.0557974
\(704\) 2.77447 0.104567
\(705\) 0 0
\(706\) 18.3737 0.691503
\(707\) 21.9358 0.824982
\(708\) −6.48516 −0.243727
\(709\) 34.3587 1.29037 0.645184 0.764027i \(-0.276781\pi\)
0.645184 + 0.764027i \(0.276781\pi\)
\(710\) 0 0
\(711\) 0.591640 0.0221882
\(712\) −9.18401 −0.344186
\(713\) 4.66799 0.174817
\(714\) −24.5313 −0.918062
\(715\) 0 0
\(716\) −8.54134 −0.319205
\(717\) 11.9005 0.444434
\(718\) −24.4259 −0.911565
\(719\) 11.9900 0.447151 0.223575 0.974687i \(-0.428227\pi\)
0.223575 + 0.974687i \(0.428227\pi\)
\(720\) 0 0
\(721\) −21.7358 −0.809483
\(722\) −17.0052 −0.632867
\(723\) 5.67576 0.211084
\(724\) 7.82844 0.290942
\(725\) 0 0
\(726\) −5.66256 −0.210157
\(727\) 42.1827 1.56447 0.782235 0.622983i \(-0.214080\pi\)
0.782235 + 0.622983i \(0.214080\pi\)
\(728\) −15.7528 −0.583839
\(729\) 27.5161 1.01912
\(730\) 0 0
\(731\) 47.6795 1.76349
\(732\) −18.5746 −0.686537
\(733\) −0.322383 −0.0119075 −0.00595375 0.999982i \(-0.501895\pi\)
−0.00595375 + 0.999982i \(0.501895\pi\)
\(734\) 14.5224 0.536033
\(735\) 0 0
\(736\) −0.654963 −0.0241423
\(737\) −11.9951 −0.441847
\(738\) −0.544176 −0.0200314
\(739\) −5.49978 −0.202313 −0.101156 0.994871i \(-0.532254\pi\)
−0.101156 + 0.994871i \(0.532254\pi\)
\(740\) 0 0
\(741\) 13.7507 0.505143
\(742\) 1.45968 0.0535865
\(743\) −6.03812 −0.221517 −0.110759 0.993847i \(-0.535328\pi\)
−0.110759 + 0.993847i \(0.535328\pi\)
\(744\) −12.2209 −0.448042
\(745\) 0 0
\(746\) −23.2887 −0.852662
\(747\) −0.279403 −0.0102228
\(748\) 14.3064 0.523092
\(749\) 28.3605 1.03627
\(750\) 0 0
\(751\) −13.5719 −0.495244 −0.247622 0.968857i \(-0.579649\pi\)
−0.247622 + 0.968857i \(0.579649\pi\)
\(752\) −2.77447 −0.101174
\(753\) −16.2290 −0.591417
\(754\) 23.2601 0.847082
\(755\) 0 0
\(756\) 14.5565 0.529416
\(757\) 33.0661 1.20181 0.600904 0.799321i \(-0.294807\pi\)
0.600904 + 0.799321i \(0.294807\pi\)
\(758\) 28.6176 1.03944
\(759\) −3.11593 −0.113101
\(760\) 0 0
\(761\) 5.18860 0.188087 0.0940434 0.995568i \(-0.470021\pi\)
0.0940434 + 0.995568i \(0.470021\pi\)
\(762\) 16.3737 0.593157
\(763\) −10.8507 −0.392822
\(764\) 3.36611 0.121782
\(765\) 0 0
\(766\) −16.6157 −0.600350
\(767\) −21.4737 −0.775372
\(768\) 1.71472 0.0618745
\(769\) 33.0698 1.19253 0.596264 0.802789i \(-0.296651\pi\)
0.596264 + 0.802789i \(0.296651\pi\)
\(770\) 0 0
\(771\) −12.1396 −0.437195
\(772\) −15.4211 −0.555018
\(773\) −1.13095 −0.0406776 −0.0203388 0.999793i \(-0.506474\pi\)
−0.0203388 + 0.999793i \(0.506474\pi\)
\(774\) −0.552505 −0.0198594
\(775\) 0 0
\(776\) −0.0901699 −0.00323691
\(777\) 4.98323 0.178772
\(778\) 26.3767 0.945652
\(779\) 12.8629 0.460861
\(780\) 0 0
\(781\) −36.7523 −1.31510
\(782\) −3.37727 −0.120771
\(783\) −21.4937 −0.768121
\(784\) 0.697669 0.0249167
\(785\) 0 0
\(786\) 9.60109 0.342460
\(787\) 15.7610 0.561819 0.280909 0.959734i \(-0.409364\pi\)
0.280909 + 0.959734i \(0.409364\pi\)
\(788\) 8.80489 0.313661
\(789\) 2.45797 0.0875059
\(790\) 0 0
\(791\) 17.6885 0.628930
\(792\) −0.165781 −0.00589075
\(793\) −61.5044 −2.18409
\(794\) −17.9086 −0.635552
\(795\) 0 0
\(796\) −17.6222 −0.624601
\(797\) −23.1397 −0.819649 −0.409824 0.912164i \(-0.634410\pi\)
−0.409824 + 0.912164i \(0.634410\pi\)
\(798\) −6.71930 −0.237861
\(799\) −14.3064 −0.506122
\(800\) 0 0
\(801\) 0.548765 0.0193897
\(802\) 32.0164 1.13054
\(803\) 11.6895 0.412514
\(804\) −7.41340 −0.261451
\(805\) 0 0
\(806\) −40.4661 −1.42536
\(807\) −24.4912 −0.862132
\(808\) −7.90632 −0.278144
\(809\) 33.5400 1.17921 0.589603 0.807694i \(-0.299284\pi\)
0.589603 + 0.807694i \(0.299284\pi\)
\(810\) 0 0
\(811\) −11.9626 −0.420064 −0.210032 0.977695i \(-0.567357\pi\)
−0.210032 + 0.977695i \(0.567357\pi\)
\(812\) −11.3661 −0.398872
\(813\) 52.7884 1.85137
\(814\) −2.90615 −0.101861
\(815\) 0 0
\(816\) 8.84181 0.309525
\(817\) 13.0598 0.456903
\(818\) 18.0344 0.630557
\(819\) 0.941266 0.0328905
\(820\) 0 0
\(821\) −10.6941 −0.373227 −0.186613 0.982433i \(-0.559751\pi\)
−0.186613 + 0.982433i \(0.559751\pi\)
\(822\) 13.6905 0.477512
\(823\) −12.0211 −0.419028 −0.209514 0.977806i \(-0.567188\pi\)
−0.209514 + 0.977806i \(0.567188\pi\)
\(824\) 7.83422 0.272918
\(825\) 0 0
\(826\) 10.4932 0.365105
\(827\) −4.13974 −0.143953 −0.0719764 0.997406i \(-0.522931\pi\)
−0.0719764 + 0.997406i \(0.522931\pi\)
\(828\) 0.0391355 0.00136005
\(829\) −20.3503 −0.706796 −0.353398 0.935473i \(-0.614974\pi\)
−0.353398 + 0.935473i \(0.614974\pi\)
\(830\) 0 0
\(831\) 5.46940 0.189732
\(832\) 5.67779 0.196842
\(833\) 3.59748 0.124645
\(834\) −37.8753 −1.31152
\(835\) 0 0
\(836\) 3.91861 0.135528
\(837\) 37.3931 1.29249
\(838\) −19.5300 −0.674654
\(839\) 3.13515 0.108237 0.0541186 0.998535i \(-0.482765\pi\)
0.0541186 + 0.998535i \(0.482765\pi\)
\(840\) 0 0
\(841\) −12.2172 −0.421283
\(842\) −19.5537 −0.673865
\(843\) 34.9612 1.20413
\(844\) 6.10144 0.210020
\(845\) 0 0
\(846\) 0.165781 0.00569965
\(847\) 9.16221 0.314817
\(848\) −0.526111 −0.0180667
\(849\) −22.8004 −0.782508
\(850\) 0 0
\(851\) 0.686050 0.0235175
\(852\) −22.7142 −0.778174
\(853\) 23.3471 0.799388 0.399694 0.916649i \(-0.369116\pi\)
0.399694 + 0.916649i \(0.369116\pi\)
\(854\) 30.0543 1.02844
\(855\) 0 0
\(856\) −10.2220 −0.349379
\(857\) 15.3036 0.522762 0.261381 0.965236i \(-0.415822\pi\)
0.261381 + 0.965236i \(0.415822\pi\)
\(858\) 27.0116 0.922161
\(859\) −2.96132 −0.101039 −0.0505194 0.998723i \(-0.516088\pi\)
−0.0505194 + 0.998723i \(0.516088\pi\)
\(860\) 0 0
\(861\) −43.3269 −1.47658
\(862\) −21.1963 −0.721949
\(863\) −37.8220 −1.28748 −0.643739 0.765246i \(-0.722618\pi\)
−0.643739 + 0.765246i \(0.722618\pi\)
\(864\) −5.24660 −0.178493
\(865\) 0 0
\(866\) 8.54967 0.290530
\(867\) 16.4421 0.558402
\(868\) 19.7739 0.671170
\(869\) −27.4716 −0.931909
\(870\) 0 0
\(871\) −24.5474 −0.831755
\(872\) 3.91091 0.132440
\(873\) 0.00538785 0.000182351 0
\(874\) −0.925059 −0.0312906
\(875\) 0 0
\(876\) 7.22451 0.244094
\(877\) −15.6011 −0.526810 −0.263405 0.964685i \(-0.584846\pi\)
−0.263405 + 0.964685i \(0.584846\pi\)
\(878\) −28.0714 −0.947363
\(879\) −44.6547 −1.50616
\(880\) 0 0
\(881\) 1.16438 0.0392289 0.0196144 0.999808i \(-0.493756\pi\)
0.0196144 + 0.999808i \(0.493756\pi\)
\(882\) −0.0416872 −0.00140368
\(883\) 48.8851 1.64511 0.822557 0.568682i \(-0.192547\pi\)
0.822557 + 0.568682i \(0.192547\pi\)
\(884\) 29.2771 0.984696
\(885\) 0 0
\(886\) −33.0546 −1.11049
\(887\) 8.62892 0.289731 0.144865 0.989451i \(-0.453725\pi\)
0.144865 + 0.989451i \(0.453725\pi\)
\(888\) −1.79610 −0.0602733
\(889\) −26.4932 −0.888554
\(890\) 0 0
\(891\) −24.4630 −0.819540
\(892\) −5.02866 −0.168372
\(893\) −3.91861 −0.131131
\(894\) 7.17439 0.239948
\(895\) 0 0
\(896\) −2.77447 −0.0926884
\(897\) −6.37658 −0.212908
\(898\) −37.1628 −1.24014
\(899\) −29.1975 −0.973790
\(900\) 0 0
\(901\) −2.71286 −0.0903784
\(902\) 25.2677 0.841322
\(903\) −43.9900 −1.46389
\(904\) −6.37545 −0.212044
\(905\) 0 0
\(906\) −0.568533 −0.0188882
\(907\) 29.5085 0.979815 0.489907 0.871774i \(-0.337031\pi\)
0.489907 + 0.871774i \(0.337031\pi\)
\(908\) −17.8382 −0.591983
\(909\) 0.472420 0.0156692
\(910\) 0 0
\(911\) −38.4088 −1.27254 −0.636270 0.771466i \(-0.719524\pi\)
−0.636270 + 0.771466i \(0.719524\pi\)
\(912\) 2.42184 0.0801950
\(913\) 12.9735 0.429360
\(914\) 29.3139 0.969619
\(915\) 0 0
\(916\) −21.5566 −0.712251
\(917\) −15.5349 −0.513007
\(918\) −27.0538 −0.892907
\(919\) 41.6905 1.37524 0.687622 0.726069i \(-0.258655\pi\)
0.687622 + 0.726069i \(0.258655\pi\)
\(920\) 0 0
\(921\) 43.0394 1.41820
\(922\) 8.90741 0.293350
\(923\) −75.2114 −2.47561
\(924\) −13.1993 −0.434225
\(925\) 0 0
\(926\) −18.2771 −0.600624
\(927\) −0.468112 −0.0153748
\(928\) 4.09668 0.134480
\(929\) 28.1047 0.922086 0.461043 0.887378i \(-0.347475\pi\)
0.461043 + 0.887378i \(0.347475\pi\)
\(930\) 0 0
\(931\) 0.985376 0.0322944
\(932\) 14.7065 0.481726
\(933\) 1.24030 0.0406056
\(934\) −36.6841 −1.20034
\(935\) 0 0
\(936\) −0.339260 −0.0110891
\(937\) −12.8754 −0.420620 −0.210310 0.977635i \(-0.567447\pi\)
−0.210310 + 0.977635i \(0.567447\pi\)
\(938\) 11.9951 0.391655
\(939\) 37.0658 1.20960
\(940\) 0 0
\(941\) 10.1510 0.330913 0.165457 0.986217i \(-0.447090\pi\)
0.165457 + 0.986217i \(0.447090\pi\)
\(942\) 6.18583 0.201545
\(943\) −5.96489 −0.194243
\(944\) −3.78206 −0.123096
\(945\) 0 0
\(946\) 25.6544 0.834096
\(947\) −57.3750 −1.86444 −0.932218 0.361896i \(-0.882129\pi\)
−0.932218 + 0.361896i \(0.882129\pi\)
\(948\) −16.9784 −0.551432
\(949\) 23.9219 0.776538
\(950\) 0 0
\(951\) −32.2944 −1.04722
\(952\) −14.3064 −0.463672
\(953\) 5.17264 0.167558 0.0837791 0.996484i \(-0.473301\pi\)
0.0837791 + 0.996484i \(0.473301\pi\)
\(954\) 0.0314363 0.00101779
\(955\) 0 0
\(956\) 6.94025 0.224464
\(957\) 19.4896 0.630010
\(958\) −12.5898 −0.406757
\(959\) −22.1517 −0.715317
\(960\) 0 0
\(961\) 19.7955 0.638566
\(962\) −5.94728 −0.191748
\(963\) 0.610785 0.0196823
\(964\) 3.31003 0.106609
\(965\) 0 0
\(966\) 3.11593 0.100254
\(967\) −41.3979 −1.33127 −0.665634 0.746279i \(-0.731839\pi\)
−0.665634 + 0.746279i \(0.731839\pi\)
\(968\) −3.30233 −0.106141
\(969\) 12.4880 0.401173
\(970\) 0 0
\(971\) 43.0848 1.38266 0.691329 0.722540i \(-0.257025\pi\)
0.691329 + 0.722540i \(0.257025\pi\)
\(972\) 0.620870 0.0199144
\(973\) 61.2836 1.96466
\(974\) −18.3256 −0.587189
\(975\) 0 0
\(976\) −10.8325 −0.346739
\(977\) 40.4614 1.29448 0.647238 0.762288i \(-0.275924\pi\)
0.647238 + 0.762288i \(0.275924\pi\)
\(978\) 8.73449 0.279298
\(979\) −25.4807 −0.814368
\(980\) 0 0
\(981\) −0.233686 −0.00746101
\(982\) 21.7047 0.692625
\(983\) 40.3786 1.28788 0.643938 0.765078i \(-0.277299\pi\)
0.643938 + 0.765078i \(0.277299\pi\)
\(984\) 15.6163 0.497829
\(985\) 0 0
\(986\) 21.1243 0.672734
\(987\) 13.1993 0.420139
\(988\) 8.01921 0.255125
\(989\) −6.05618 −0.192575
\(990\) 0 0
\(991\) −58.1435 −1.84699 −0.923494 0.383613i \(-0.874680\pi\)
−0.923494 + 0.383613i \(0.874680\pi\)
\(992\) −7.12710 −0.226286
\(993\) 28.9374 0.918300
\(994\) 36.7523 1.16571
\(995\) 0 0
\(996\) 8.01806 0.254062
\(997\) 39.2968 1.24454 0.622271 0.782802i \(-0.286210\pi\)
0.622271 + 0.782802i \(0.286210\pi\)
\(998\) 15.8391 0.501377
\(999\) 5.49563 0.173874
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 1250.2.a.l.1.3 4
4.3 odd 2 10000.2.a.t.1.2 4
5.2 odd 4 1250.2.b.e.1249.6 8
5.3 odd 4 1250.2.b.e.1249.3 8
5.4 even 2 1250.2.a.f.1.2 4
20.19 odd 2 10000.2.a.x.1.3 4
25.3 odd 20 250.2.e.c.49.1 16
25.4 even 10 250.2.d.d.201.1 8
25.6 even 5 50.2.d.b.11.2 8
25.8 odd 20 250.2.e.c.199.4 16
25.17 odd 20 250.2.e.c.199.1 16
25.19 even 10 250.2.d.d.51.1 8
25.21 even 5 50.2.d.b.41.2 yes 8
25.22 odd 20 250.2.e.c.49.4 16
75.56 odd 10 450.2.h.e.361.1 8
75.71 odd 10 450.2.h.e.91.1 8
100.31 odd 10 400.2.u.d.161.1 8
100.71 odd 10 400.2.u.d.241.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.2.d.b.11.2 8 25.6 even 5
50.2.d.b.41.2 yes 8 25.21 even 5
250.2.d.d.51.1 8 25.19 even 10
250.2.d.d.201.1 8 25.4 even 10
250.2.e.c.49.1 16 25.3 odd 20
250.2.e.c.49.4 16 25.22 odd 20
250.2.e.c.199.1 16 25.17 odd 20
250.2.e.c.199.4 16 25.8 odd 20
400.2.u.d.161.1 8 100.31 odd 10
400.2.u.d.241.1 8 100.71 odd 10
450.2.h.e.91.1 8 75.71 odd 10
450.2.h.e.361.1 8 75.56 odd 10
1250.2.a.f.1.2 4 5.4 even 2
1250.2.a.l.1.3 4 1.1 even 1 trivial
1250.2.b.e.1249.3 8 5.3 odd 4
1250.2.b.e.1249.6 8 5.2 odd 4
10000.2.a.t.1.2 4 4.3 odd 2
10000.2.a.x.1.3 4 20.19 odd 2