Properties

Label 4334.2.a.d.1.15
Level $4334$
Weight $2$
Character 4334.1
Self dual yes
Analytic conductor $34.607$
Analytic rank $1$
Dimension $17$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4334,2,Mod(1,4334)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4334, 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("4334.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4334 = 2 \cdot 11 \cdot 197 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4334.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: \(34.6071642360\)
Analytic rank: \(1\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{17} - 7 x^{16} - 7 x^{15} + 137 x^{14} - 98 x^{13} - 1048 x^{12} + 1313 x^{11} + 4085 x^{10} - 6021 x^{9} - 8879 x^{8} + 13530 x^{7} + 11150 x^{6} - 15676 x^{5} - 8037 x^{4} + \cdots - 400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.15
Root \(-1.87628\) of defining polynomial
Character \(\chi\) \(=\) 4334.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.87628 q^{3} +1.00000 q^{4} -2.40821 q^{5} +1.87628 q^{6} +1.70446 q^{7} +1.00000 q^{8} +0.520431 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.87628 q^{3} +1.00000 q^{4} -2.40821 q^{5} +1.87628 q^{6} +1.70446 q^{7} +1.00000 q^{8} +0.520431 q^{9} -2.40821 q^{10} -1.00000 q^{11} +1.87628 q^{12} +0.0419654 q^{13} +1.70446 q^{14} -4.51847 q^{15} +1.00000 q^{16} -4.96548 q^{17} +0.520431 q^{18} -8.54692 q^{19} -2.40821 q^{20} +3.19804 q^{21} -1.00000 q^{22} -2.98767 q^{23} +1.87628 q^{24} +0.799457 q^{25} +0.0419654 q^{26} -4.65237 q^{27} +1.70446 q^{28} +1.32735 q^{29} -4.51847 q^{30} -8.45192 q^{31} +1.00000 q^{32} -1.87628 q^{33} -4.96548 q^{34} -4.10468 q^{35} +0.520431 q^{36} +4.04689 q^{37} -8.54692 q^{38} +0.0787390 q^{39} -2.40821 q^{40} +10.0542 q^{41} +3.19804 q^{42} -8.36404 q^{43} -1.00000 q^{44} -1.25330 q^{45} -2.98767 q^{46} +0.374163 q^{47} +1.87628 q^{48} -4.09483 q^{49} +0.799457 q^{50} -9.31664 q^{51} +0.0419654 q^{52} +7.53465 q^{53} -4.65237 q^{54} +2.40821 q^{55} +1.70446 q^{56} -16.0364 q^{57} +1.32735 q^{58} -2.04260 q^{59} -4.51847 q^{60} +2.53017 q^{61} -8.45192 q^{62} +0.887052 q^{63} +1.00000 q^{64} -0.101061 q^{65} -1.87628 q^{66} +2.97228 q^{67} -4.96548 q^{68} -5.60571 q^{69} -4.10468 q^{70} -9.85440 q^{71} +0.520431 q^{72} +5.29069 q^{73} +4.04689 q^{74} +1.50001 q^{75} -8.54692 q^{76} -1.70446 q^{77} +0.0787390 q^{78} +0.855219 q^{79} -2.40821 q^{80} -10.2904 q^{81} +10.0542 q^{82} -2.00289 q^{83} +3.19804 q^{84} +11.9579 q^{85} -8.36404 q^{86} +2.49049 q^{87} -1.00000 q^{88} +3.10139 q^{89} -1.25330 q^{90} +0.0715283 q^{91} -2.98767 q^{92} -15.8582 q^{93} +0.374163 q^{94} +20.5827 q^{95} +1.87628 q^{96} +11.8510 q^{97} -4.09483 q^{98} -0.520431 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q + 17 q^{2} - 7 q^{3} + 17 q^{4} - 4 q^{5} - 7 q^{6} - 5 q^{7} + 17 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 17 q + 17 q^{2} - 7 q^{3} + 17 q^{4} - 4 q^{5} - 7 q^{6} - 5 q^{7} + 17 q^{8} + 12 q^{9} - 4 q^{10} - 17 q^{11} - 7 q^{12} - 18 q^{13} - 5 q^{14} - 16 q^{15} + 17 q^{16} - 10 q^{17} + 12 q^{18} - 31 q^{19} - 4 q^{20} - 13 q^{21} - 17 q^{22} - 6 q^{23} - 7 q^{24} + 3 q^{25} - 18 q^{26} - 37 q^{27} - 5 q^{28} - 16 q^{29} - 16 q^{30} - 30 q^{31} + 17 q^{32} + 7 q^{33} - 10 q^{34} - 36 q^{35} + 12 q^{36} - 23 q^{37} - 31 q^{38} - 15 q^{39} - 4 q^{40} - 7 q^{41} - 13 q^{42} - 23 q^{43} - 17 q^{44} - 19 q^{45} - 6 q^{46} - 19 q^{47} - 7 q^{48} - 8 q^{49} + 3 q^{50} - 18 q^{51} - 18 q^{52} - 30 q^{53} - 37 q^{54} + 4 q^{55} - 5 q^{56} + 10 q^{57} - 16 q^{58} - 28 q^{59} - 16 q^{60} - 19 q^{61} - 30 q^{62} + 2 q^{63} + 17 q^{64} + 23 q^{65} + 7 q^{66} - 35 q^{67} - 10 q^{68} + q^{69} - 36 q^{70} + q^{71} + 12 q^{72} - 10 q^{73} - 23 q^{74} - 33 q^{75} - 31 q^{76} + 5 q^{77} - 15 q^{78} - 27 q^{79} - 4 q^{80} + 13 q^{81} - 7 q^{82} - 40 q^{83} - 13 q^{84} - 11 q^{85} - 23 q^{86} - 6 q^{87} - 17 q^{88} - 17 q^{89} - 19 q^{90} - 19 q^{91} - 6 q^{92} + 10 q^{93} - 19 q^{94} - 27 q^{95} - 7 q^{96} - 34 q^{97} - 8 q^{98} - 12 q^{99}+O(q^{100}) \) 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
\(3\) 1.87628 1.08327 0.541636 0.840613i \(-0.317805\pi\)
0.541636 + 0.840613i \(0.317805\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.40821 −1.07698 −0.538491 0.842631i \(-0.681006\pi\)
−0.538491 + 0.842631i \(0.681006\pi\)
\(6\) 1.87628 0.765989
\(7\) 1.70446 0.644224 0.322112 0.946702i \(-0.395607\pi\)
0.322112 + 0.946702i \(0.395607\pi\)
\(8\) 1.00000 0.353553
\(9\) 0.520431 0.173477
\(10\) −2.40821 −0.761542
\(11\) −1.00000 −0.301511
\(12\) 1.87628 0.541636
\(13\) 0.0419654 0.0116391 0.00581956 0.999983i \(-0.498148\pi\)
0.00581956 + 0.999983i \(0.498148\pi\)
\(14\) 1.70446 0.455535
\(15\) −4.51847 −1.16666
\(16\) 1.00000 0.250000
\(17\) −4.96548 −1.20431 −0.602153 0.798380i \(-0.705690\pi\)
−0.602153 + 0.798380i \(0.705690\pi\)
\(18\) 0.520431 0.122667
\(19\) −8.54692 −1.96080 −0.980399 0.197022i \(-0.936873\pi\)
−0.980399 + 0.197022i \(0.936873\pi\)
\(20\) −2.40821 −0.538491
\(21\) 3.19804 0.697869
\(22\) −1.00000 −0.213201
\(23\) −2.98767 −0.622972 −0.311486 0.950251i \(-0.600827\pi\)
−0.311486 + 0.950251i \(0.600827\pi\)
\(24\) 1.87628 0.382994
\(25\) 0.799457 0.159891
\(26\) 0.0419654 0.00823010
\(27\) −4.65237 −0.895349
\(28\) 1.70446 0.322112
\(29\) 1.32735 0.246484 0.123242 0.992377i \(-0.460671\pi\)
0.123242 + 0.992377i \(0.460671\pi\)
\(30\) −4.51847 −0.824956
\(31\) −8.45192 −1.51801 −0.759005 0.651085i \(-0.774314\pi\)
−0.759005 + 0.651085i \(0.774314\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.87628 −0.326619
\(34\) −4.96548 −0.851573
\(35\) −4.10468 −0.693818
\(36\) 0.520431 0.0867385
\(37\) 4.04689 0.665305 0.332652 0.943050i \(-0.392056\pi\)
0.332652 + 0.943050i \(0.392056\pi\)
\(38\) −8.54692 −1.38649
\(39\) 0.0787390 0.0126083
\(40\) −2.40821 −0.380771
\(41\) 10.0542 1.57020 0.785098 0.619372i \(-0.212613\pi\)
0.785098 + 0.619372i \(0.212613\pi\)
\(42\) 3.19804 0.493468
\(43\) −8.36404 −1.27550 −0.637752 0.770242i \(-0.720136\pi\)
−0.637752 + 0.770242i \(0.720136\pi\)
\(44\) −1.00000 −0.150756
\(45\) −1.25330 −0.186832
\(46\) −2.98767 −0.440508
\(47\) 0.374163 0.0545773 0.0272886 0.999628i \(-0.491313\pi\)
0.0272886 + 0.999628i \(0.491313\pi\)
\(48\) 1.87628 0.270818
\(49\) −4.09483 −0.584975
\(50\) 0.799457 0.113060
\(51\) −9.31664 −1.30459
\(52\) 0.0419654 0.00581956
\(53\) 7.53465 1.03496 0.517482 0.855694i \(-0.326869\pi\)
0.517482 + 0.855694i \(0.326869\pi\)
\(54\) −4.65237 −0.633107
\(55\) 2.40821 0.324722
\(56\) 1.70446 0.227768
\(57\) −16.0364 −2.12408
\(58\) 1.32735 0.174290
\(59\) −2.04260 −0.265924 −0.132962 0.991121i \(-0.542449\pi\)
−0.132962 + 0.991121i \(0.542449\pi\)
\(60\) −4.51847 −0.583332
\(61\) 2.53017 0.323955 0.161978 0.986794i \(-0.448213\pi\)
0.161978 + 0.986794i \(0.448213\pi\)
\(62\) −8.45192 −1.07339
\(63\) 0.887052 0.111758
\(64\) 1.00000 0.125000
\(65\) −0.101061 −0.0125351
\(66\) −1.87628 −0.230954
\(67\) 2.97228 0.363121 0.181561 0.983380i \(-0.441885\pi\)
0.181561 + 0.983380i \(0.441885\pi\)
\(68\) −4.96548 −0.602153
\(69\) −5.60571 −0.674848
\(70\) −4.10468 −0.490603
\(71\) −9.85440 −1.16950 −0.584751 0.811213i \(-0.698808\pi\)
−0.584751 + 0.811213i \(0.698808\pi\)
\(72\) 0.520431 0.0613334
\(73\) 5.29069 0.619229 0.309614 0.950862i \(-0.399800\pi\)
0.309614 + 0.950862i \(0.399800\pi\)
\(74\) 4.04689 0.470441
\(75\) 1.50001 0.173206
\(76\) −8.54692 −0.980399
\(77\) −1.70446 −0.194241
\(78\) 0.0787390 0.00891543
\(79\) 0.855219 0.0962196 0.0481098 0.998842i \(-0.484680\pi\)
0.0481098 + 0.998842i \(0.484680\pi\)
\(80\) −2.40821 −0.269246
\(81\) −10.2904 −1.14338
\(82\) 10.0542 1.11030
\(83\) −2.00289 −0.219846 −0.109923 0.993940i \(-0.535060\pi\)
−0.109923 + 0.993940i \(0.535060\pi\)
\(84\) 3.19804 0.348935
\(85\) 11.9579 1.29702
\(86\) −8.36404 −0.901918
\(87\) 2.49049 0.267009
\(88\) −1.00000 −0.106600
\(89\) 3.10139 0.328747 0.164373 0.986398i \(-0.447440\pi\)
0.164373 + 0.986398i \(0.447440\pi\)
\(90\) −1.25330 −0.132110
\(91\) 0.0715283 0.00749820
\(92\) −2.98767 −0.311486
\(93\) −15.8582 −1.64442
\(94\) 0.374163 0.0385920
\(95\) 20.5827 2.11175
\(96\) 1.87628 0.191497
\(97\) 11.8510 1.20329 0.601643 0.798765i \(-0.294513\pi\)
0.601643 + 0.798765i \(0.294513\pi\)
\(98\) −4.09483 −0.413640
\(99\) −0.520431 −0.0523053
\(100\) 0.799457 0.0799457
\(101\) 1.68016 0.167182 0.0835910 0.996500i \(-0.473361\pi\)
0.0835910 + 0.996500i \(0.473361\pi\)
\(102\) −9.31664 −0.922485
\(103\) −2.42599 −0.239039 −0.119520 0.992832i \(-0.538135\pi\)
−0.119520 + 0.992832i \(0.538135\pi\)
\(104\) 0.0419654 0.00411505
\(105\) −7.70154 −0.751593
\(106\) 7.53465 0.731830
\(107\) −19.9178 −1.92552 −0.962762 0.270350i \(-0.912861\pi\)
−0.962762 + 0.270350i \(0.912861\pi\)
\(108\) −4.65237 −0.447674
\(109\) 12.6390 1.21060 0.605299 0.795998i \(-0.293053\pi\)
0.605299 + 0.795998i \(0.293053\pi\)
\(110\) 2.40821 0.229613
\(111\) 7.59310 0.720705
\(112\) 1.70446 0.161056
\(113\) 14.0958 1.32602 0.663011 0.748610i \(-0.269278\pi\)
0.663011 + 0.748610i \(0.269278\pi\)
\(114\) −16.0364 −1.50195
\(115\) 7.19493 0.670930
\(116\) 1.32735 0.123242
\(117\) 0.0218401 0.00201912
\(118\) −2.04260 −0.188037
\(119\) −8.46345 −0.775843
\(120\) −4.51847 −0.412478
\(121\) 1.00000 0.0909091
\(122\) 2.53017 0.229071
\(123\) 18.8644 1.70095
\(124\) −8.45192 −0.759005
\(125\) 10.1158 0.904782
\(126\) 0.887052 0.0790248
\(127\) −11.0676 −0.982093 −0.491047 0.871133i \(-0.663386\pi\)
−0.491047 + 0.871133i \(0.663386\pi\)
\(128\) 1.00000 0.0883883
\(129\) −15.6933 −1.38172
\(130\) −0.101061 −0.00886367
\(131\) −15.6723 −1.36930 −0.684648 0.728874i \(-0.740044\pi\)
−0.684648 + 0.728874i \(0.740044\pi\)
\(132\) −1.87628 −0.163309
\(133\) −14.5679 −1.26319
\(134\) 2.97228 0.256765
\(135\) 11.2039 0.964275
\(136\) −4.96548 −0.425787
\(137\) 3.35052 0.286254 0.143127 0.989704i \(-0.454284\pi\)
0.143127 + 0.989704i \(0.454284\pi\)
\(138\) −5.60571 −0.477190
\(139\) 11.4461 0.970847 0.485424 0.874279i \(-0.338665\pi\)
0.485424 + 0.874279i \(0.338665\pi\)
\(140\) −4.10468 −0.346909
\(141\) 0.702035 0.0591220
\(142\) −9.85440 −0.826963
\(143\) −0.0419654 −0.00350933
\(144\) 0.520431 0.0433692
\(145\) −3.19654 −0.265459
\(146\) 5.29069 0.437861
\(147\) −7.68305 −0.633687
\(148\) 4.04689 0.332652
\(149\) −2.80403 −0.229715 −0.114858 0.993382i \(-0.536641\pi\)
−0.114858 + 0.993382i \(0.536641\pi\)
\(150\) 1.50001 0.122475
\(151\) 5.19141 0.422471 0.211235 0.977435i \(-0.432251\pi\)
0.211235 + 0.977435i \(0.432251\pi\)
\(152\) −8.54692 −0.693247
\(153\) −2.58419 −0.208919
\(154\) −1.70446 −0.137349
\(155\) 20.3540 1.63487
\(156\) 0.0787390 0.00630416
\(157\) 1.64185 0.131034 0.0655170 0.997851i \(-0.479130\pi\)
0.0655170 + 0.997851i \(0.479130\pi\)
\(158\) 0.855219 0.0680376
\(159\) 14.1371 1.12115
\(160\) −2.40821 −0.190385
\(161\) −5.09235 −0.401334
\(162\) −10.2904 −0.808494
\(163\) −16.1985 −1.26876 −0.634382 0.773019i \(-0.718746\pi\)
−0.634382 + 0.773019i \(0.718746\pi\)
\(164\) 10.0542 0.785098
\(165\) 4.51847 0.351763
\(166\) −2.00289 −0.155455
\(167\) −6.30982 −0.488268 −0.244134 0.969741i \(-0.578504\pi\)
−0.244134 + 0.969741i \(0.578504\pi\)
\(168\) 3.19804 0.246734
\(169\) −12.9982 −0.999865
\(170\) 11.9579 0.917130
\(171\) −4.44808 −0.340153
\(172\) −8.36404 −0.637752
\(173\) −4.95476 −0.376704 −0.188352 0.982102i \(-0.560315\pi\)
−0.188352 + 0.982102i \(0.560315\pi\)
\(174\) 2.49049 0.188804
\(175\) 1.36264 0.103006
\(176\) −1.00000 −0.0753778
\(177\) −3.83249 −0.288068
\(178\) 3.10139 0.232459
\(179\) 18.5569 1.38701 0.693505 0.720452i \(-0.256065\pi\)
0.693505 + 0.720452i \(0.256065\pi\)
\(180\) −1.25330 −0.0934158
\(181\) 8.53533 0.634426 0.317213 0.948354i \(-0.397253\pi\)
0.317213 + 0.948354i \(0.397253\pi\)
\(182\) 0.0715283 0.00530203
\(183\) 4.74731 0.350931
\(184\) −2.98767 −0.220254
\(185\) −9.74574 −0.716521
\(186\) −15.8582 −1.16278
\(187\) 4.96548 0.363112
\(188\) 0.374163 0.0272886
\(189\) −7.92976 −0.576805
\(190\) 20.5827 1.49323
\(191\) −5.88789 −0.426033 −0.213016 0.977049i \(-0.568329\pi\)
−0.213016 + 0.977049i \(0.568329\pi\)
\(192\) 1.87628 0.135409
\(193\) 7.07445 0.509230 0.254615 0.967043i \(-0.418051\pi\)
0.254615 + 0.967043i \(0.418051\pi\)
\(194\) 11.8510 0.850851
\(195\) −0.189620 −0.0135789
\(196\) −4.09483 −0.292488
\(197\) 1.00000 0.0712470
\(198\) −0.520431 −0.0369854
\(199\) −6.66336 −0.472353 −0.236177 0.971710i \(-0.575894\pi\)
−0.236177 + 0.971710i \(0.575894\pi\)
\(200\) 0.799457 0.0565302
\(201\) 5.57682 0.393359
\(202\) 1.68016 0.118215
\(203\) 2.26242 0.158791
\(204\) −9.31664 −0.652296
\(205\) −24.2125 −1.69107
\(206\) −2.42599 −0.169026
\(207\) −1.55488 −0.108071
\(208\) 0.0419654 0.00290978
\(209\) 8.54692 0.591203
\(210\) −7.70154 −0.531457
\(211\) −1.97095 −0.135686 −0.0678430 0.997696i \(-0.521612\pi\)
−0.0678430 + 0.997696i \(0.521612\pi\)
\(212\) 7.53465 0.517482
\(213\) −18.4896 −1.26689
\(214\) −19.9178 −1.36155
\(215\) 20.1423 1.37370
\(216\) −4.65237 −0.316554
\(217\) −14.4059 −0.977938
\(218\) 12.6390 0.856023
\(219\) 9.92682 0.670793
\(220\) 2.40821 0.162361
\(221\) −0.208379 −0.0140171
\(222\) 7.59310 0.509616
\(223\) −11.0359 −0.739015 −0.369508 0.929228i \(-0.620474\pi\)
−0.369508 + 0.929228i \(0.620474\pi\)
\(224\) 1.70446 0.113884
\(225\) 0.416062 0.0277375
\(226\) 14.0958 0.937639
\(227\) 9.96423 0.661349 0.330675 0.943745i \(-0.392724\pi\)
0.330675 + 0.943745i \(0.392724\pi\)
\(228\) −16.0364 −1.06204
\(229\) −14.4346 −0.953866 −0.476933 0.878940i \(-0.658252\pi\)
−0.476933 + 0.878940i \(0.658252\pi\)
\(230\) 7.19493 0.474419
\(231\) −3.19804 −0.210416
\(232\) 1.32735 0.0871451
\(233\) 16.7948 1.10027 0.550133 0.835077i \(-0.314577\pi\)
0.550133 + 0.835077i \(0.314577\pi\)
\(234\) 0.0218401 0.00142773
\(235\) −0.901062 −0.0587788
\(236\) −2.04260 −0.132962
\(237\) 1.60463 0.104232
\(238\) −8.46345 −0.548604
\(239\) 24.8879 1.60987 0.804934 0.593365i \(-0.202201\pi\)
0.804934 + 0.593365i \(0.202201\pi\)
\(240\) −4.51847 −0.291666
\(241\) −17.5898 −1.13306 −0.566528 0.824043i \(-0.691714\pi\)
−0.566528 + 0.824043i \(0.691714\pi\)
\(242\) 1.00000 0.0642824
\(243\) −5.35066 −0.343245
\(244\) 2.53017 0.161978
\(245\) 9.86119 0.630008
\(246\) 18.8644 1.20275
\(247\) −0.358675 −0.0228220
\(248\) −8.45192 −0.536697
\(249\) −3.75799 −0.238153
\(250\) 10.1158 0.639778
\(251\) −27.1453 −1.71340 −0.856698 0.515818i \(-0.827488\pi\)
−0.856698 + 0.515818i \(0.827488\pi\)
\(252\) 0.887052 0.0558790
\(253\) 2.98767 0.187833
\(254\) −11.0676 −0.694445
\(255\) 22.4364 1.40502
\(256\) 1.00000 0.0625000
\(257\) −24.2591 −1.51324 −0.756620 0.653855i \(-0.773150\pi\)
−0.756620 + 0.653855i \(0.773150\pi\)
\(258\) −15.6933 −0.977021
\(259\) 6.89775 0.428605
\(260\) −0.101061 −0.00626756
\(261\) 0.690796 0.0427592
\(262\) −15.6723 −0.968238
\(263\) −8.84355 −0.545316 −0.272658 0.962111i \(-0.587903\pi\)
−0.272658 + 0.962111i \(0.587903\pi\)
\(264\) −1.87628 −0.115477
\(265\) −18.1450 −1.11464
\(266\) −14.5679 −0.893212
\(267\) 5.81908 0.356122
\(268\) 2.97228 0.181561
\(269\) −21.9464 −1.33810 −0.669048 0.743219i \(-0.733298\pi\)
−0.669048 + 0.743219i \(0.733298\pi\)
\(270\) 11.2039 0.681845
\(271\) 9.20821 0.559359 0.279680 0.960093i \(-0.409772\pi\)
0.279680 + 0.960093i \(0.409772\pi\)
\(272\) −4.96548 −0.301077
\(273\) 0.134207 0.00812259
\(274\) 3.35052 0.202412
\(275\) −0.799457 −0.0482091
\(276\) −5.60571 −0.337424
\(277\) −25.4499 −1.52914 −0.764568 0.644543i \(-0.777048\pi\)
−0.764568 + 0.644543i \(0.777048\pi\)
\(278\) 11.4461 0.686493
\(279\) −4.39864 −0.263340
\(280\) −4.10468 −0.245302
\(281\) −12.9781 −0.774211 −0.387106 0.922035i \(-0.626525\pi\)
−0.387106 + 0.922035i \(0.626525\pi\)
\(282\) 0.702035 0.0418056
\(283\) −4.06933 −0.241896 −0.120948 0.992659i \(-0.538593\pi\)
−0.120948 + 0.992659i \(0.538593\pi\)
\(284\) −9.85440 −0.584751
\(285\) 38.6190 2.28759
\(286\) −0.0419654 −0.00248147
\(287\) 17.1369 1.01156
\(288\) 0.520431 0.0306667
\(289\) 7.65603 0.450355
\(290\) −3.19654 −0.187708
\(291\) 22.2358 1.30348
\(292\) 5.29069 0.309614
\(293\) 34.2111 1.99863 0.999316 0.0369875i \(-0.0117762\pi\)
0.999316 + 0.0369875i \(0.0117762\pi\)
\(294\) −7.68305 −0.448084
\(295\) 4.91900 0.286395
\(296\) 4.04689 0.235221
\(297\) 4.65237 0.269958
\(298\) −2.80403 −0.162433
\(299\) −0.125379 −0.00725085
\(300\) 1.50001 0.0866029
\(301\) −14.2561 −0.821710
\(302\) 5.19141 0.298732
\(303\) 3.15245 0.181103
\(304\) −8.54692 −0.490199
\(305\) −6.09317 −0.348894
\(306\) −2.58419 −0.147728
\(307\) −9.55211 −0.545168 −0.272584 0.962132i \(-0.587878\pi\)
−0.272584 + 0.962132i \(0.587878\pi\)
\(308\) −1.70446 −0.0971204
\(309\) −4.55183 −0.258945
\(310\) 20.3540 1.15603
\(311\) −20.5365 −1.16452 −0.582259 0.813003i \(-0.697831\pi\)
−0.582259 + 0.813003i \(0.697831\pi\)
\(312\) 0.0787390 0.00445772
\(313\) −23.2423 −1.31373 −0.656866 0.754007i \(-0.728118\pi\)
−0.656866 + 0.754007i \(0.728118\pi\)
\(314\) 1.64185 0.0926550
\(315\) −2.13620 −0.120361
\(316\) 0.855219 0.0481098
\(317\) 3.45907 0.194281 0.0971405 0.995271i \(-0.469030\pi\)
0.0971405 + 0.995271i \(0.469030\pi\)
\(318\) 14.1371 0.792771
\(319\) −1.32735 −0.0743176
\(320\) −2.40821 −0.134623
\(321\) −37.3713 −2.08587
\(322\) −5.09235 −0.283786
\(323\) 42.4396 2.36140
\(324\) −10.2904 −0.571691
\(325\) 0.0335496 0.00186100
\(326\) −16.1985 −0.897152
\(327\) 23.7144 1.31141
\(328\) 10.0542 0.555148
\(329\) 0.637745 0.0351600
\(330\) 4.51847 0.248734
\(331\) 5.11299 0.281035 0.140518 0.990078i \(-0.455123\pi\)
0.140518 + 0.990078i \(0.455123\pi\)
\(332\) −2.00289 −0.109923
\(333\) 2.10613 0.115415
\(334\) −6.30982 −0.345258
\(335\) −7.15785 −0.391075
\(336\) 3.19804 0.174467
\(337\) 5.31525 0.289540 0.144770 0.989465i \(-0.453756\pi\)
0.144770 + 0.989465i \(0.453756\pi\)
\(338\) −12.9982 −0.707011
\(339\) 26.4477 1.43644
\(340\) 11.9579 0.648509
\(341\) 8.45192 0.457697
\(342\) −4.44808 −0.240525
\(343\) −18.9107 −1.02108
\(344\) −8.36404 −0.450959
\(345\) 13.4997 0.726800
\(346\) −4.95476 −0.266370
\(347\) −16.9611 −0.910522 −0.455261 0.890358i \(-0.650454\pi\)
−0.455261 + 0.890358i \(0.650454\pi\)
\(348\) 2.49049 0.133504
\(349\) −16.7962 −0.899079 −0.449540 0.893260i \(-0.648412\pi\)
−0.449540 + 0.893260i \(0.648412\pi\)
\(350\) 1.36264 0.0728362
\(351\) −0.195239 −0.0104211
\(352\) −1.00000 −0.0533002
\(353\) 25.6462 1.36501 0.682505 0.730881i \(-0.260891\pi\)
0.682505 + 0.730881i \(0.260891\pi\)
\(354\) −3.83249 −0.203695
\(355\) 23.7314 1.25953
\(356\) 3.10139 0.164373
\(357\) −15.8798 −0.840449
\(358\) 18.5569 0.980764
\(359\) 22.7650 1.20149 0.600746 0.799440i \(-0.294870\pi\)
0.600746 + 0.799440i \(0.294870\pi\)
\(360\) −1.25330 −0.0660550
\(361\) 54.0498 2.84473
\(362\) 8.53533 0.448607
\(363\) 1.87628 0.0984792
\(364\) 0.0715283 0.00374910
\(365\) −12.7411 −0.666898
\(366\) 4.74731 0.248146
\(367\) 19.5005 1.01792 0.508959 0.860791i \(-0.330030\pi\)
0.508959 + 0.860791i \(0.330030\pi\)
\(368\) −2.98767 −0.155743
\(369\) 5.23249 0.272393
\(370\) −9.74574 −0.506657
\(371\) 12.8425 0.666749
\(372\) −15.8582 −0.822208
\(373\) −0.368520 −0.0190813 −0.00954063 0.999954i \(-0.503037\pi\)
−0.00954063 + 0.999954i \(0.503037\pi\)
\(374\) 4.96548 0.256759
\(375\) 18.9800 0.980125
\(376\) 0.374163 0.0192960
\(377\) 0.0557030 0.00286885
\(378\) −7.92976 −0.407863
\(379\) −27.6329 −1.41940 −0.709702 0.704502i \(-0.751171\pi\)
−0.709702 + 0.704502i \(0.751171\pi\)
\(380\) 20.5827 1.05587
\(381\) −20.7660 −1.06387
\(382\) −5.88789 −0.301251
\(383\) 5.88684 0.300803 0.150402 0.988625i \(-0.451943\pi\)
0.150402 + 0.988625i \(0.451943\pi\)
\(384\) 1.87628 0.0957486
\(385\) 4.10468 0.209194
\(386\) 7.07445 0.360080
\(387\) −4.35290 −0.221271
\(388\) 11.8510 0.601643
\(389\) 21.5887 1.09459 0.547295 0.836940i \(-0.315658\pi\)
0.547295 + 0.836940i \(0.315658\pi\)
\(390\) −0.189620 −0.00960177
\(391\) 14.8352 0.750250
\(392\) −4.09483 −0.206820
\(393\) −29.4056 −1.48332
\(394\) 1.00000 0.0503793
\(395\) −2.05954 −0.103627
\(396\) −0.520431 −0.0261526
\(397\) −8.60327 −0.431786 −0.215893 0.976417i \(-0.569266\pi\)
−0.215893 + 0.976417i \(0.569266\pi\)
\(398\) −6.66336 −0.334004
\(399\) −27.3334 −1.36838
\(400\) 0.799457 0.0399729
\(401\) 20.9056 1.04398 0.521988 0.852953i \(-0.325191\pi\)
0.521988 + 0.852953i \(0.325191\pi\)
\(402\) 5.57682 0.278147
\(403\) −0.354689 −0.0176683
\(404\) 1.68016 0.0835910
\(405\) 24.7815 1.23140
\(406\) 2.26242 0.112282
\(407\) −4.04689 −0.200597
\(408\) −9.31664 −0.461243
\(409\) −4.59003 −0.226962 −0.113481 0.993540i \(-0.536200\pi\)
−0.113481 + 0.993540i \(0.536200\pi\)
\(410\) −24.2125 −1.19577
\(411\) 6.28651 0.310091
\(412\) −2.42599 −0.119520
\(413\) −3.48152 −0.171315
\(414\) −1.55488 −0.0764180
\(415\) 4.82338 0.236770
\(416\) 0.0419654 0.00205753
\(417\) 21.4761 1.05169
\(418\) 8.54692 0.418044
\(419\) 20.5895 1.00586 0.502931 0.864327i \(-0.332255\pi\)
0.502931 + 0.864327i \(0.332255\pi\)
\(420\) −7.70154 −0.375797
\(421\) −0.545293 −0.0265759 −0.0132880 0.999912i \(-0.504230\pi\)
−0.0132880 + 0.999912i \(0.504230\pi\)
\(422\) −1.97095 −0.0959444
\(423\) 0.194726 0.00946790
\(424\) 7.53465 0.365915
\(425\) −3.96969 −0.192558
\(426\) −18.4896 −0.895825
\(427\) 4.31257 0.208700
\(428\) −19.9178 −0.962762
\(429\) −0.0787390 −0.00380155
\(430\) 20.1423 0.971349
\(431\) −22.0236 −1.06084 −0.530419 0.847735i \(-0.677966\pi\)
−0.530419 + 0.847735i \(0.677966\pi\)
\(432\) −4.65237 −0.223837
\(433\) 24.7037 1.18719 0.593593 0.804766i \(-0.297709\pi\)
0.593593 + 0.804766i \(0.297709\pi\)
\(434\) −14.4059 −0.691507
\(435\) −5.99761 −0.287564
\(436\) 12.6390 0.605299
\(437\) 25.5354 1.22152
\(438\) 9.92682 0.474322
\(439\) 6.60749 0.315358 0.157679 0.987490i \(-0.449599\pi\)
0.157679 + 0.987490i \(0.449599\pi\)
\(440\) 2.40821 0.114807
\(441\) −2.13107 −0.101480
\(442\) −0.208379 −0.00991157
\(443\) 37.8275 1.79724 0.898620 0.438729i \(-0.144571\pi\)
0.898620 + 0.438729i \(0.144571\pi\)
\(444\) 7.59310 0.360353
\(445\) −7.46879 −0.354054
\(446\) −11.0359 −0.522563
\(447\) −5.26116 −0.248844
\(448\) 1.70446 0.0805280
\(449\) 34.2765 1.61761 0.808805 0.588078i \(-0.200115\pi\)
0.808805 + 0.588078i \(0.200115\pi\)
\(450\) 0.416062 0.0196134
\(451\) −10.0542 −0.473432
\(452\) 14.0958 0.663011
\(453\) 9.74054 0.457650
\(454\) 9.96423 0.467645
\(455\) −0.172255 −0.00807543
\(456\) −16.0364 −0.750974
\(457\) −34.3365 −1.60619 −0.803096 0.595850i \(-0.796815\pi\)
−0.803096 + 0.595850i \(0.796815\pi\)
\(458\) −14.4346 −0.674485
\(459\) 23.1013 1.07827
\(460\) 7.19493 0.335465
\(461\) −0.343935 −0.0160186 −0.00800932 0.999968i \(-0.502549\pi\)
−0.00800932 + 0.999968i \(0.502549\pi\)
\(462\) −3.19804 −0.148786
\(463\) 6.86964 0.319259 0.159630 0.987177i \(-0.448970\pi\)
0.159630 + 0.987177i \(0.448970\pi\)
\(464\) 1.32735 0.0616209
\(465\) 38.1898 1.77101
\(466\) 16.7948 0.778006
\(467\) −6.00587 −0.277919 −0.138959 0.990298i \(-0.544376\pi\)
−0.138959 + 0.990298i \(0.544376\pi\)
\(468\) 0.0218401 0.00100956
\(469\) 5.06611 0.233931
\(470\) −0.901062 −0.0415629
\(471\) 3.08057 0.141945
\(472\) −2.04260 −0.0940183
\(473\) 8.36404 0.384579
\(474\) 1.60463 0.0737031
\(475\) −6.83290 −0.313515
\(476\) −8.46345 −0.387922
\(477\) 3.92127 0.179542
\(478\) 24.8879 1.13835
\(479\) −32.0469 −1.46426 −0.732130 0.681165i \(-0.761473\pi\)
−0.732130 + 0.681165i \(0.761473\pi\)
\(480\) −4.51847 −0.206239
\(481\) 0.169830 0.00774356
\(482\) −17.5898 −0.801191
\(483\) −9.55469 −0.434753
\(484\) 1.00000 0.0454545
\(485\) −28.5396 −1.29592
\(486\) −5.35066 −0.242711
\(487\) −16.8881 −0.765273 −0.382637 0.923899i \(-0.624984\pi\)
−0.382637 + 0.923899i \(0.624984\pi\)
\(488\) 2.53017 0.114535
\(489\) −30.3929 −1.37442
\(490\) 9.86119 0.445483
\(491\) 31.7080 1.43096 0.715481 0.698632i \(-0.246208\pi\)
0.715481 + 0.698632i \(0.246208\pi\)
\(492\) 18.8644 0.850474
\(493\) −6.59096 −0.296842
\(494\) −0.358675 −0.0161376
\(495\) 1.25330 0.0563319
\(496\) −8.45192 −0.379502
\(497\) −16.7964 −0.753421
\(498\) −3.75799 −0.168400
\(499\) −28.1109 −1.25842 −0.629208 0.777237i \(-0.716621\pi\)
−0.629208 + 0.777237i \(0.716621\pi\)
\(500\) 10.1158 0.452391
\(501\) −11.8390 −0.528927
\(502\) −27.1453 −1.21155
\(503\) −34.6747 −1.54607 −0.773034 0.634365i \(-0.781262\pi\)
−0.773034 + 0.634365i \(0.781262\pi\)
\(504\) 0.887052 0.0395124
\(505\) −4.04617 −0.180052
\(506\) 2.98767 0.132818
\(507\) −24.3883 −1.08312
\(508\) −11.0676 −0.491047
\(509\) −20.0200 −0.887371 −0.443686 0.896182i \(-0.646329\pi\)
−0.443686 + 0.896182i \(0.646329\pi\)
\(510\) 22.4364 0.993500
\(511\) 9.01775 0.398922
\(512\) 1.00000 0.0441942
\(513\) 39.7634 1.75560
\(514\) −24.2591 −1.07002
\(515\) 5.84227 0.257441
\(516\) −15.6933 −0.690859
\(517\) −0.374163 −0.0164557
\(518\) 6.89775 0.303070
\(519\) −9.29653 −0.408072
\(520\) −0.101061 −0.00443184
\(521\) 13.6175 0.596592 0.298296 0.954473i \(-0.403582\pi\)
0.298296 + 0.954473i \(0.403582\pi\)
\(522\) 0.690796 0.0302353
\(523\) 5.47057 0.239211 0.119606 0.992821i \(-0.461837\pi\)
0.119606 + 0.992821i \(0.461837\pi\)
\(524\) −15.6723 −0.684648
\(525\) 2.55670 0.111583
\(526\) −8.84355 −0.385597
\(527\) 41.9679 1.82815
\(528\) −1.87628 −0.0816547
\(529\) −14.0738 −0.611906
\(530\) −18.1450 −0.788168
\(531\) −1.06303 −0.0461317
\(532\) −14.5679 −0.631597
\(533\) 0.421927 0.0182757
\(534\) 5.81908 0.251816
\(535\) 47.9661 2.07376
\(536\) 2.97228 0.128383
\(537\) 34.8180 1.50251
\(538\) −21.9464 −0.946177
\(539\) 4.09483 0.176377
\(540\) 11.2039 0.482138
\(541\) 11.6009 0.498764 0.249382 0.968405i \(-0.419773\pi\)
0.249382 + 0.968405i \(0.419773\pi\)
\(542\) 9.20821 0.395527
\(543\) 16.0147 0.687256
\(544\) −4.96548 −0.212893
\(545\) −30.4374 −1.30379
\(546\) 0.134207 0.00574354
\(547\) 17.0241 0.727896 0.363948 0.931419i \(-0.381429\pi\)
0.363948 + 0.931419i \(0.381429\pi\)
\(548\) 3.35052 0.143127
\(549\) 1.31678 0.0561987
\(550\) −0.799457 −0.0340890
\(551\) −11.3448 −0.483304
\(552\) −5.60571 −0.238595
\(553\) 1.45768 0.0619870
\(554\) −25.4499 −1.08126
\(555\) −18.2858 −0.776187
\(556\) 11.4461 0.485424
\(557\) 27.7694 1.17663 0.588314 0.808633i \(-0.299792\pi\)
0.588314 + 0.808633i \(0.299792\pi\)
\(558\) −4.39864 −0.186209
\(559\) −0.351001 −0.0148457
\(560\) −4.10468 −0.173455
\(561\) 9.31664 0.393349
\(562\) −12.9781 −0.547450
\(563\) −43.5022 −1.83340 −0.916700 0.399575i \(-0.869158\pi\)
−0.916700 + 0.399575i \(0.869158\pi\)
\(564\) 0.702035 0.0295610
\(565\) −33.9456 −1.42810
\(566\) −4.06933 −0.171047
\(567\) −17.5396 −0.736595
\(568\) −9.85440 −0.413481
\(569\) 24.9377 1.04544 0.522722 0.852503i \(-0.324917\pi\)
0.522722 + 0.852503i \(0.324917\pi\)
\(570\) 38.6190 1.61757
\(571\) −0.515714 −0.0215820 −0.0107910 0.999942i \(-0.503435\pi\)
−0.0107910 + 0.999942i \(0.503435\pi\)
\(572\) −0.0419654 −0.00175466
\(573\) −11.0473 −0.461509
\(574\) 17.1369 0.715279
\(575\) −2.38851 −0.0996079
\(576\) 0.520431 0.0216846
\(577\) −15.1581 −0.631039 −0.315520 0.948919i \(-0.602179\pi\)
−0.315520 + 0.948919i \(0.602179\pi\)
\(578\) 7.65603 0.318449
\(579\) 13.2736 0.551634
\(580\) −3.19654 −0.132729
\(581\) −3.41384 −0.141630
\(582\) 22.2358 0.921703
\(583\) −7.53465 −0.312053
\(584\) 5.29069 0.218930
\(585\) −0.0525955 −0.00217456
\(586\) 34.2111 1.41325
\(587\) 2.41795 0.0997997 0.0498998 0.998754i \(-0.484110\pi\)
0.0498998 + 0.998754i \(0.484110\pi\)
\(588\) −7.68305 −0.316844
\(589\) 72.2379 2.97651
\(590\) 4.91900 0.202512
\(591\) 1.87628 0.0771799
\(592\) 4.04689 0.166326
\(593\) 16.6269 0.682787 0.341393 0.939921i \(-0.389101\pi\)
0.341393 + 0.939921i \(0.389101\pi\)
\(594\) 4.65237 0.190889
\(595\) 20.3817 0.835570
\(596\) −2.80403 −0.114858
\(597\) −12.5023 −0.511687
\(598\) −0.125379 −0.00512712
\(599\) 7.77552 0.317699 0.158850 0.987303i \(-0.449221\pi\)
0.158850 + 0.987303i \(0.449221\pi\)
\(600\) 1.50001 0.0612375
\(601\) 5.84421 0.238390 0.119195 0.992871i \(-0.461969\pi\)
0.119195 + 0.992871i \(0.461969\pi\)
\(602\) −14.2561 −0.581037
\(603\) 1.54686 0.0629932
\(604\) 5.19141 0.211235
\(605\) −2.40821 −0.0979075
\(606\) 3.15245 0.128059
\(607\) −12.4887 −0.506902 −0.253451 0.967348i \(-0.581566\pi\)
−0.253451 + 0.967348i \(0.581566\pi\)
\(608\) −8.54692 −0.346623
\(609\) 4.24493 0.172013
\(610\) −6.09317 −0.246705
\(611\) 0.0157019 0.000635232 0
\(612\) −2.58419 −0.104460
\(613\) −0.763653 −0.0308437 −0.0154218 0.999881i \(-0.504909\pi\)
−0.0154218 + 0.999881i \(0.504909\pi\)
\(614\) −9.55211 −0.385492
\(615\) −45.4294 −1.83189
\(616\) −1.70446 −0.0686745
\(617\) 22.4130 0.902315 0.451157 0.892444i \(-0.351011\pi\)
0.451157 + 0.892444i \(0.351011\pi\)
\(618\) −4.55183 −0.183102
\(619\) 18.2027 0.731626 0.365813 0.930688i \(-0.380791\pi\)
0.365813 + 0.930688i \(0.380791\pi\)
\(620\) 20.3540 0.817435
\(621\) 13.8997 0.557777
\(622\) −20.5365 −0.823439
\(623\) 5.28619 0.211787
\(624\) 0.0787390 0.00315208
\(625\) −28.3582 −1.13433
\(626\) −23.2423 −0.928949
\(627\) 16.0364 0.640433
\(628\) 1.64185 0.0655170
\(629\) −20.0948 −0.801231
\(630\) −2.13620 −0.0851084
\(631\) −15.3306 −0.610302 −0.305151 0.952304i \(-0.598707\pi\)
−0.305151 + 0.952304i \(0.598707\pi\)
\(632\) 0.855219 0.0340188
\(633\) −3.69806 −0.146985
\(634\) 3.45907 0.137377
\(635\) 26.6531 1.05770
\(636\) 14.1371 0.560574
\(637\) −0.171841 −0.00680860
\(638\) −1.32735 −0.0525505
\(639\) −5.12853 −0.202882
\(640\) −2.40821 −0.0951927
\(641\) 20.6817 0.816879 0.408439 0.912785i \(-0.366073\pi\)
0.408439 + 0.912785i \(0.366073\pi\)
\(642\) −37.3713 −1.47493
\(643\) −4.09439 −0.161467 −0.0807335 0.996736i \(-0.525726\pi\)
−0.0807335 + 0.996736i \(0.525726\pi\)
\(644\) −5.09235 −0.200667
\(645\) 37.7927 1.48809
\(646\) 42.4396 1.66976
\(647\) 21.3797 0.840523 0.420262 0.907403i \(-0.361938\pi\)
0.420262 + 0.907403i \(0.361938\pi\)
\(648\) −10.2904 −0.404247
\(649\) 2.04260 0.0801791
\(650\) 0.0335496 0.00131592
\(651\) −27.0296 −1.05937
\(652\) −16.1985 −0.634382
\(653\) 15.4836 0.605919 0.302959 0.953003i \(-0.402025\pi\)
0.302959 + 0.953003i \(0.402025\pi\)
\(654\) 23.7144 0.927305
\(655\) 37.7421 1.47471
\(656\) 10.0542 0.392549
\(657\) 2.75344 0.107422
\(658\) 0.637745 0.0248619
\(659\) 36.7103 1.43003 0.715015 0.699109i \(-0.246420\pi\)
0.715015 + 0.699109i \(0.246420\pi\)
\(660\) 4.51847 0.175881
\(661\) −35.6168 −1.38533 −0.692666 0.721258i \(-0.743564\pi\)
−0.692666 + 0.721258i \(0.743564\pi\)
\(662\) 5.11299 0.198722
\(663\) −0.390977 −0.0151843
\(664\) −2.00289 −0.0777273
\(665\) 35.0824 1.36044
\(666\) 2.10613 0.0816107
\(667\) −3.96570 −0.153552
\(668\) −6.30982 −0.244134
\(669\) −20.7064 −0.800554
\(670\) −7.15785 −0.276532
\(671\) −2.53017 −0.0976762
\(672\) 3.19804 0.123367
\(673\) −5.23076 −0.201631 −0.100816 0.994905i \(-0.532145\pi\)
−0.100816 + 0.994905i \(0.532145\pi\)
\(674\) 5.31525 0.204736
\(675\) −3.71937 −0.143159
\(676\) −12.9982 −0.499932
\(677\) −11.7675 −0.452263 −0.226131 0.974097i \(-0.572608\pi\)
−0.226131 + 0.974097i \(0.572608\pi\)
\(678\) 26.4477 1.01572
\(679\) 20.1995 0.775186
\(680\) 11.9579 0.458565
\(681\) 18.6957 0.716421
\(682\) 8.45192 0.323641
\(683\) 28.7929 1.10173 0.550865 0.834594i \(-0.314298\pi\)
0.550865 + 0.834594i \(0.314298\pi\)
\(684\) −4.44808 −0.170077
\(685\) −8.06874 −0.308291
\(686\) −18.9107 −0.722012
\(687\) −27.0834 −1.03330
\(688\) −8.36404 −0.318876
\(689\) 0.316195 0.0120461
\(690\) 13.4997 0.513925
\(691\) −12.2366 −0.465501 −0.232751 0.972536i \(-0.574773\pi\)
−0.232751 + 0.972536i \(0.574773\pi\)
\(692\) −4.95476 −0.188352
\(693\) −0.887052 −0.0336963
\(694\) −16.9611 −0.643836
\(695\) −27.5646 −1.04559
\(696\) 2.49049 0.0944018
\(697\) −49.9237 −1.89100
\(698\) −16.7962 −0.635745
\(699\) 31.5119 1.19189
\(700\) 1.36264 0.0515030
\(701\) 19.6685 0.742871 0.371435 0.928459i \(-0.378866\pi\)
0.371435 + 0.928459i \(0.378866\pi\)
\(702\) −0.195239 −0.00736881
\(703\) −34.5884 −1.30453
\(704\) −1.00000 −0.0376889
\(705\) −1.69065 −0.0636734
\(706\) 25.6462 0.965207
\(707\) 2.86376 0.107703
\(708\) −3.83249 −0.144034
\(709\) −26.7301 −1.00387 −0.501936 0.864905i \(-0.667379\pi\)
−0.501936 + 0.864905i \(0.667379\pi\)
\(710\) 23.7314 0.890625
\(711\) 0.445082 0.0166919
\(712\) 3.10139 0.116230
\(713\) 25.2515 0.945678
\(714\) −15.8798 −0.594287
\(715\) 0.101061 0.00377948
\(716\) 18.5569 0.693505
\(717\) 46.6968 1.74392
\(718\) 22.7650 0.849583
\(719\) 48.6144 1.81301 0.906505 0.422194i \(-0.138740\pi\)
0.906505 + 0.422194i \(0.138740\pi\)
\(720\) −1.25330 −0.0467079
\(721\) −4.13499 −0.153995
\(722\) 54.0498 2.01153
\(723\) −33.0033 −1.22741
\(724\) 8.53533 0.317213
\(725\) 1.06116 0.0394106
\(726\) 1.87628 0.0696353
\(727\) −5.62286 −0.208540 −0.104270 0.994549i \(-0.533251\pi\)
−0.104270 + 0.994549i \(0.533251\pi\)
\(728\) 0.0715283 0.00265101
\(729\) 20.8320 0.771555
\(730\) −12.7411 −0.471568
\(731\) 41.5315 1.53610
\(732\) 4.74731 0.175466
\(733\) −38.1632 −1.40959 −0.704795 0.709411i \(-0.748961\pi\)
−0.704795 + 0.709411i \(0.748961\pi\)
\(734\) 19.5005 0.719777
\(735\) 18.5024 0.682470
\(736\) −2.98767 −0.110127
\(737\) −2.97228 −0.109485
\(738\) 5.23249 0.192611
\(739\) 13.6636 0.502623 0.251311 0.967906i \(-0.419138\pi\)
0.251311 + 0.967906i \(0.419138\pi\)
\(740\) −9.74574 −0.358261
\(741\) −0.672976 −0.0247224
\(742\) 12.8425 0.471463
\(743\) −13.3942 −0.491384 −0.245692 0.969348i \(-0.579015\pi\)
−0.245692 + 0.969348i \(0.579015\pi\)
\(744\) −15.8582 −0.581389
\(745\) 6.75269 0.247400
\(746\) −0.368520 −0.0134925
\(747\) −1.04237 −0.0381382
\(748\) 4.96548 0.181556
\(749\) −33.9490 −1.24047
\(750\) 18.9800 0.693053
\(751\) 26.2938 0.959475 0.479737 0.877412i \(-0.340732\pi\)
0.479737 + 0.877412i \(0.340732\pi\)
\(752\) 0.374163 0.0136443
\(753\) −50.9322 −1.85607
\(754\) 0.0557030 0.00202858
\(755\) −12.5020 −0.454994
\(756\) −7.92976 −0.288403
\(757\) −19.8549 −0.721639 −0.360819 0.932636i \(-0.617503\pi\)
−0.360819 + 0.932636i \(0.617503\pi\)
\(758\) −27.6329 −1.00367
\(759\) 5.60571 0.203474
\(760\) 20.5827 0.746615
\(761\) −7.52743 −0.272869 −0.136435 0.990649i \(-0.543564\pi\)
−0.136435 + 0.990649i \(0.543564\pi\)
\(762\) −20.7660 −0.752272
\(763\) 21.5427 0.779897
\(764\) −5.88789 −0.213016
\(765\) 6.22326 0.225003
\(766\) 5.88684 0.212700
\(767\) −0.0857186 −0.00309512
\(768\) 1.87628 0.0677045
\(769\) 25.1185 0.905798 0.452899 0.891562i \(-0.350390\pi\)
0.452899 + 0.891562i \(0.350390\pi\)
\(770\) 4.10468 0.147923
\(771\) −45.5168 −1.63925
\(772\) 7.07445 0.254615
\(773\) 0.424823 0.0152798 0.00763991 0.999971i \(-0.497568\pi\)
0.00763991 + 0.999971i \(0.497568\pi\)
\(774\) −4.35290 −0.156462
\(775\) −6.75695 −0.242717
\(776\) 11.8510 0.425426
\(777\) 12.9421 0.464296
\(778\) 21.5887 0.773992
\(779\) −85.9321 −3.07884
\(780\) −0.189620 −0.00678947
\(781\) 9.85440 0.352618
\(782\) 14.8352 0.530507
\(783\) −6.17534 −0.220689
\(784\) −4.09483 −0.146244
\(785\) −3.95392 −0.141121
\(786\) −29.4056 −1.04886
\(787\) −22.7385 −0.810541 −0.405271 0.914197i \(-0.632823\pi\)
−0.405271 + 0.914197i \(0.632823\pi\)
\(788\) 1.00000 0.0356235
\(789\) −16.5930 −0.590726
\(790\) −2.05954 −0.0732753
\(791\) 24.0257 0.854255
\(792\) −0.520431 −0.0184927
\(793\) 0.106180 0.00377055
\(794\) −8.60327 −0.305319
\(795\) −34.0451 −1.20746
\(796\) −6.66336 −0.236177
\(797\) 46.9378 1.66262 0.831310 0.555809i \(-0.187591\pi\)
0.831310 + 0.555809i \(0.187591\pi\)
\(798\) −27.3334 −0.967592
\(799\) −1.85790 −0.0657278
\(800\) 0.799457 0.0282651
\(801\) 1.61406 0.0570300
\(802\) 20.9056 0.738202
\(803\) −5.29069 −0.186704
\(804\) 5.57682 0.196679
\(805\) 12.2634 0.432229
\(806\) −0.354689 −0.0124934
\(807\) −41.1776 −1.44952
\(808\) 1.68016 0.0591077
\(809\) −55.3470 −1.94590 −0.972949 0.231019i \(-0.925794\pi\)
−0.972949 + 0.231019i \(0.925794\pi\)
\(810\) 24.7815 0.870734
\(811\) −17.3938 −0.610778 −0.305389 0.952228i \(-0.598786\pi\)
−0.305389 + 0.952228i \(0.598786\pi\)
\(812\) 2.26242 0.0793953
\(813\) 17.2772 0.605938
\(814\) −4.04689 −0.141843
\(815\) 39.0093 1.36644
\(816\) −9.31664 −0.326148
\(817\) 71.4868 2.50101
\(818\) −4.59003 −0.160487
\(819\) 0.0372255 0.00130076
\(820\) −24.2125 −0.845536
\(821\) −47.9222 −1.67250 −0.836248 0.548352i \(-0.815255\pi\)
−0.836248 + 0.548352i \(0.815255\pi\)
\(822\) 6.28651 0.219267
\(823\) −9.34377 −0.325703 −0.162852 0.986651i \(-0.552069\pi\)
−0.162852 + 0.986651i \(0.552069\pi\)
\(824\) −2.42599 −0.0845132
\(825\) −1.50001 −0.0522235
\(826\) −3.48152 −0.121138
\(827\) 14.1179 0.490929 0.245464 0.969406i \(-0.421060\pi\)
0.245464 + 0.969406i \(0.421060\pi\)
\(828\) −1.55488 −0.0540357
\(829\) −50.7623 −1.76305 −0.881525 0.472138i \(-0.843482\pi\)
−0.881525 + 0.472138i \(0.843482\pi\)
\(830\) 4.82338 0.167422
\(831\) −47.7512 −1.65647
\(832\) 0.0419654 0.00145489
\(833\) 20.3328 0.704490
\(834\) 21.4761 0.743658
\(835\) 15.1953 0.525856
\(836\) 8.54692 0.295601
\(837\) 39.3214 1.35915
\(838\) 20.5895 0.711252
\(839\) −26.7496 −0.923499 −0.461750 0.887010i \(-0.652778\pi\)
−0.461750 + 0.887010i \(0.652778\pi\)
\(840\) −7.70154 −0.265728
\(841\) −27.2381 −0.939246
\(842\) −0.545293 −0.0187920
\(843\) −24.3507 −0.838681
\(844\) −1.97095 −0.0678430
\(845\) 31.3024 1.07684
\(846\) 0.194726 0.00669482
\(847\) 1.70446 0.0585658
\(848\) 7.53465 0.258741
\(849\) −7.63520 −0.262039
\(850\) −3.96969 −0.136159
\(851\) −12.0908 −0.414466
\(852\) −18.4896 −0.633444
\(853\) −24.1631 −0.827329 −0.413664 0.910429i \(-0.635751\pi\)
−0.413664 + 0.910429i \(0.635751\pi\)
\(854\) 4.31257 0.147573
\(855\) 10.7119 0.366339
\(856\) −19.9178 −0.680776
\(857\) −34.1833 −1.16768 −0.583839 0.811869i \(-0.698450\pi\)
−0.583839 + 0.811869i \(0.698450\pi\)
\(858\) −0.0787390 −0.00268810
\(859\) −26.3380 −0.898639 −0.449320 0.893371i \(-0.648334\pi\)
−0.449320 + 0.893371i \(0.648334\pi\)
\(860\) 20.1423 0.686848
\(861\) 32.1536 1.09579
\(862\) −22.0236 −0.750126
\(863\) −40.9829 −1.39507 −0.697537 0.716549i \(-0.745721\pi\)
−0.697537 + 0.716549i \(0.745721\pi\)
\(864\) −4.65237 −0.158277
\(865\) 11.9321 0.405703
\(866\) 24.7037 0.839467
\(867\) 14.3649 0.487856
\(868\) −14.4059 −0.488969
\(869\) −0.855219 −0.0290113
\(870\) −5.99761 −0.203338
\(871\) 0.124733 0.00422641
\(872\) 12.6390 0.428011
\(873\) 6.16762 0.208742
\(874\) 25.5354 0.863747
\(875\) 17.2419 0.582882
\(876\) 9.92682 0.335396
\(877\) −51.5191 −1.73968 −0.869839 0.493336i \(-0.835777\pi\)
−0.869839 + 0.493336i \(0.835777\pi\)
\(878\) 6.60749 0.222992
\(879\) 64.1896 2.16506
\(880\) 2.40821 0.0811806
\(881\) −45.3704 −1.52857 −0.764283 0.644881i \(-0.776907\pi\)
−0.764283 + 0.644881i \(0.776907\pi\)
\(882\) −2.13107 −0.0717570
\(883\) −15.0235 −0.505580 −0.252790 0.967521i \(-0.581348\pi\)
−0.252790 + 0.967521i \(0.581348\pi\)
\(884\) −0.208379 −0.00700854
\(885\) 9.22943 0.310244
\(886\) 37.8275 1.27084
\(887\) −37.4267 −1.25666 −0.628332 0.777945i \(-0.716262\pi\)
−0.628332 + 0.777945i \(0.716262\pi\)
\(888\) 7.59310 0.254808
\(889\) −18.8643 −0.632688
\(890\) −7.46879 −0.250354
\(891\) 10.2904 0.344743
\(892\) −11.0359 −0.369508
\(893\) −3.19794 −0.107015
\(894\) −5.26116 −0.175959
\(895\) −44.6889 −1.49379
\(896\) 1.70446 0.0569419
\(897\) −0.235246 −0.00785464
\(898\) 34.2765 1.14382
\(899\) −11.2187 −0.374164
\(900\) 0.416062 0.0138687
\(901\) −37.4132 −1.24641
\(902\) −10.0542 −0.334767
\(903\) −26.7485 −0.890135
\(904\) 14.0958 0.468820
\(905\) −20.5548 −0.683266
\(906\) 9.74054 0.323608
\(907\) −22.8577 −0.758977 −0.379489 0.925196i \(-0.623900\pi\)
−0.379489 + 0.925196i \(0.623900\pi\)
\(908\) 9.96423 0.330675
\(909\) 0.874406 0.0290022
\(910\) −0.172255 −0.00571019
\(911\) −0.0167353 −0.000554466 0 −0.000277233 1.00000i \(-0.500088\pi\)
−0.000277233 1.00000i \(0.500088\pi\)
\(912\) −16.0364 −0.531019
\(913\) 2.00289 0.0662861
\(914\) −34.3365 −1.13575
\(915\) −11.4325 −0.377947
\(916\) −14.4346 −0.476933
\(917\) −26.7128 −0.882133
\(918\) 23.1013 0.762455
\(919\) 29.1329 0.961006 0.480503 0.876993i \(-0.340454\pi\)
0.480503 + 0.876993i \(0.340454\pi\)
\(920\) 7.19493 0.237210
\(921\) −17.9224 −0.590565
\(922\) −0.343935 −0.0113269
\(923\) −0.413544 −0.0136120
\(924\) −3.19804 −0.105208
\(925\) 3.23531 0.106376
\(926\) 6.86964 0.225750
\(927\) −1.26256 −0.0414678
\(928\) 1.32735 0.0435726
\(929\) 9.31463 0.305603 0.152802 0.988257i \(-0.451170\pi\)
0.152802 + 0.988257i \(0.451170\pi\)
\(930\) 38.1898 1.25229
\(931\) 34.9982 1.14702
\(932\) 16.7948 0.550133
\(933\) −38.5322 −1.26149
\(934\) −6.00587 −0.196518
\(935\) −11.9579 −0.391065
\(936\) 0.0218401 0.000713866 0
\(937\) 27.0791 0.884637 0.442318 0.896858i \(-0.354156\pi\)
0.442318 + 0.896858i \(0.354156\pi\)
\(938\) 5.06611 0.165415
\(939\) −43.6091 −1.42313
\(940\) −0.901062 −0.0293894
\(941\) 16.5510 0.539547 0.269774 0.962924i \(-0.413051\pi\)
0.269774 + 0.962924i \(0.413051\pi\)
\(942\) 3.08057 0.100371
\(943\) −30.0385 −0.978188
\(944\) −2.04260 −0.0664810
\(945\) 19.0965 0.621209
\(946\) 8.36404 0.271938
\(947\) −1.62188 −0.0527041 −0.0263520 0.999653i \(-0.508389\pi\)
−0.0263520 + 0.999653i \(0.508389\pi\)
\(948\) 1.60463 0.0521160
\(949\) 0.222026 0.00720728
\(950\) −6.83290 −0.221688
\(951\) 6.49020 0.210459
\(952\) −8.46345 −0.274302
\(953\) 58.7320 1.90251 0.951257 0.308398i \(-0.0997930\pi\)
0.951257 + 0.308398i \(0.0997930\pi\)
\(954\) 3.92127 0.126956
\(955\) 14.1793 0.458830
\(956\) 24.8879 0.804934
\(957\) −2.49049 −0.0805061
\(958\) −32.0469 −1.03539
\(959\) 5.71081 0.184412
\(960\) −4.51847 −0.145833
\(961\) 40.4349 1.30435
\(962\) 0.169830 0.00547552
\(963\) −10.3658 −0.334034
\(964\) −17.5898 −0.566528
\(965\) −17.0367 −0.548432
\(966\) −9.55469 −0.307417
\(967\) −27.3851 −0.880647 −0.440323 0.897839i \(-0.645136\pi\)
−0.440323 + 0.897839i \(0.645136\pi\)
\(968\) 1.00000 0.0321412
\(969\) 79.6286 2.55804
\(970\) −28.5396 −0.916352
\(971\) −15.4104 −0.494542 −0.247271 0.968946i \(-0.579534\pi\)
−0.247271 + 0.968946i \(0.579534\pi\)
\(972\) −5.35066 −0.171622
\(973\) 19.5094 0.625443
\(974\) −16.8881 −0.541130
\(975\) 0.0629484 0.00201596
\(976\) 2.53017 0.0809888
\(977\) 20.7467 0.663747 0.331873 0.943324i \(-0.392319\pi\)
0.331873 + 0.943324i \(0.392319\pi\)
\(978\) −30.3929 −0.971859
\(979\) −3.10139 −0.0991209
\(980\) 9.86119 0.315004
\(981\) 6.57774 0.210011
\(982\) 31.7080 1.01184
\(983\) −10.7658 −0.343377 −0.171688 0.985151i \(-0.554922\pi\)
−0.171688 + 0.985151i \(0.554922\pi\)
\(984\) 18.8644 0.601376
\(985\) −2.40821 −0.0767318
\(986\) −6.59096 −0.209899
\(987\) 1.19659 0.0380878
\(988\) −0.358675 −0.0114110
\(989\) 24.9890 0.794604
\(990\) 1.25330 0.0398326
\(991\) 14.5127 0.461013 0.230506 0.973071i \(-0.425962\pi\)
0.230506 + 0.973071i \(0.425962\pi\)
\(992\) −8.45192 −0.268349
\(993\) 9.59340 0.304437
\(994\) −16.7964 −0.532749
\(995\) 16.0467 0.508716
\(996\) −3.75799 −0.119076
\(997\) −14.9530 −0.473566 −0.236783 0.971563i \(-0.576093\pi\)
−0.236783 + 0.971563i \(0.576093\pi\)
\(998\) −28.1109 −0.889834
\(999\) −18.8276 −0.595680
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 4334.2.a.d.1.15 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4334.2.a.d.1.15 17 1.1 even 1 trivial