Properties

Label 5415.2.a.z.1.3
Level $5415$
Weight $2$
Character 5415.1
Self dual yes
Analytic conductor $43.239$
Analytic rank $0$
Dimension $5$
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] = [5415,2,Mod(1,5415)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5415, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("5415.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5415 = 3 \cdot 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5415.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: \(43.2389926945\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.8797896.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 8x^{3} + 5x^{2} + 13x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 285)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.290697\) of defining polynomial
Character \(\chi\) \(=\) 5415.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.290697 q^{2} +1.00000 q^{3} -1.91550 q^{4} +1.00000 q^{5} +0.290697 q^{6} -0.486575 q^{7} -1.13822 q^{8} +1.00000 q^{9} +0.290697 q^{10} +5.34764 q^{11} -1.91550 q^{12} +2.48657 q^{13} -0.141446 q^{14} +1.00000 q^{15} +3.50011 q^{16} -3.41561 q^{17} +0.290697 q^{18} -1.91550 q^{20} -0.486575 q^{21} +1.55454 q^{22} +6.41238 q^{23} -1.13822 q^{24} +1.00000 q^{25} +0.722840 q^{26} +1.00000 q^{27} +0.932031 q^{28} -2.76624 q^{29} +0.290697 q^{30} -4.83099 q^{31} +3.29392 q^{32} +5.34764 q^{33} -0.992907 q^{34} -0.486575 q^{35} -1.91550 q^{36} +6.31756 q^{37} +2.48657 q^{39} -1.13822 q^{40} -2.58139 q^{41} -0.141446 q^{42} -2.48657 q^{43} -10.2434 q^{44} +1.00000 q^{45} +1.86406 q^{46} +11.1107 q^{47} +3.50011 q^{48} -6.76325 q^{49} +0.290697 q^{50} -3.41561 q^{51} -4.76302 q^{52} -2.99678 q^{53} +0.290697 q^{54} +5.34764 q^{55} +0.553831 q^{56} -0.804139 q^{58} -12.7332 q^{59} -1.91550 q^{60} +11.8548 q^{61} -1.40436 q^{62} -0.486575 q^{63} -6.04269 q^{64} +2.48657 q^{65} +1.55454 q^{66} +15.1754 q^{67} +6.54258 q^{68} +6.41238 q^{69} -0.141446 q^{70} +9.92903 q^{71} -1.13822 q^{72} -11.0128 q^{73} +1.83650 q^{74} +1.00000 q^{75} -2.60202 q^{77} +0.722840 q^{78} -8.13272 q^{79} +3.50011 q^{80} +1.00000 q^{81} -0.750404 q^{82} -5.86106 q^{83} +0.932031 q^{84} -3.41561 q^{85} -0.722840 q^{86} -2.76624 q^{87} -6.08681 q^{88} +6.51043 q^{89} +0.290697 q^{90} -1.20990 q^{91} -12.2829 q^{92} -4.83099 q^{93} +3.22984 q^{94} +3.29392 q^{96} -2.00000 q^{97} -1.96606 q^{98} +5.34764 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + q^{2} + 5 q^{3} + 7 q^{4} + 5 q^{5} + q^{6} + 2 q^{7} + 6 q^{8} + 5 q^{9} + q^{10} + 5 q^{11} + 7 q^{12} + 8 q^{13} + 4 q^{14} + 5 q^{15} + 7 q^{16} + 10 q^{17} + q^{18} + 7 q^{20} + 2 q^{21}+ \cdots + 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.290697 0.205554 0.102777 0.994704i \(-0.467227\pi\)
0.102777 + 0.994704i \(0.467227\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.91550 −0.957748
\(5\) 1.00000 0.447214
\(6\) 0.290697 0.118677
\(7\) −0.486575 −0.183908 −0.0919539 0.995763i \(-0.529311\pi\)
−0.0919539 + 0.995763i \(0.529311\pi\)
\(8\) −1.13822 −0.402423
\(9\) 1.00000 0.333333
\(10\) 0.290697 0.0919265
\(11\) 5.34764 1.61237 0.806187 0.591661i \(-0.201528\pi\)
0.806187 + 0.591661i \(0.201528\pi\)
\(12\) −1.91550 −0.552956
\(13\) 2.48657 0.689652 0.344826 0.938667i \(-0.387938\pi\)
0.344826 + 0.938667i \(0.387938\pi\)
\(14\) −0.141446 −0.0378030
\(15\) 1.00000 0.258199
\(16\) 3.50011 0.875028
\(17\) −3.41561 −0.828406 −0.414203 0.910184i \(-0.635940\pi\)
−0.414203 + 0.910184i \(0.635940\pi\)
\(18\) 0.290697 0.0685180
\(19\) 0 0
\(20\) −1.91550 −0.428318
\(21\) −0.486575 −0.106179
\(22\) 1.55454 0.331430
\(23\) 6.41238 1.33707 0.668537 0.743679i \(-0.266921\pi\)
0.668537 + 0.743679i \(0.266921\pi\)
\(24\) −1.13822 −0.232339
\(25\) 1.00000 0.200000
\(26\) 0.722840 0.141761
\(27\) 1.00000 0.192450
\(28\) 0.932031 0.176137
\(29\) −2.76624 −0.513679 −0.256839 0.966454i \(-0.582681\pi\)
−0.256839 + 0.966454i \(0.582681\pi\)
\(30\) 0.290697 0.0530738
\(31\) −4.83099 −0.867671 −0.433836 0.900992i \(-0.642840\pi\)
−0.433836 + 0.900992i \(0.642840\pi\)
\(32\) 3.29392 0.582288
\(33\) 5.34764 0.930904
\(34\) −0.992907 −0.170282
\(35\) −0.486575 −0.0822461
\(36\) −1.91550 −0.319249
\(37\) 6.31756 1.03860 0.519301 0.854592i \(-0.326192\pi\)
0.519301 + 0.854592i \(0.326192\pi\)
\(38\) 0 0
\(39\) 2.48657 0.398171
\(40\) −1.13822 −0.179969
\(41\) −2.58139 −0.403146 −0.201573 0.979473i \(-0.564605\pi\)
−0.201573 + 0.979473i \(0.564605\pi\)
\(42\) −0.141446 −0.0218256
\(43\) −2.48657 −0.379199 −0.189600 0.981862i \(-0.560719\pi\)
−0.189600 + 0.981862i \(0.560719\pi\)
\(44\) −10.2434 −1.54425
\(45\) 1.00000 0.149071
\(46\) 1.86406 0.274841
\(47\) 11.1107 1.62066 0.810328 0.585976i \(-0.199289\pi\)
0.810328 + 0.585976i \(0.199289\pi\)
\(48\) 3.50011 0.505198
\(49\) −6.76325 −0.966178
\(50\) 0.290697 0.0411108
\(51\) −3.41561 −0.478281
\(52\) −4.76302 −0.660512
\(53\) −2.99678 −0.411639 −0.205820 0.978590i \(-0.565986\pi\)
−0.205820 + 0.978590i \(0.565986\pi\)
\(54\) 0.290697 0.0395589
\(55\) 5.34764 0.721075
\(56\) 0.553831 0.0740087
\(57\) 0 0
\(58\) −0.804139 −0.105589
\(59\) −12.7332 −1.65772 −0.828859 0.559458i \(-0.811009\pi\)
−0.828859 + 0.559458i \(0.811009\pi\)
\(60\) −1.91550 −0.247289
\(61\) 11.8548 1.51786 0.758929 0.651174i \(-0.225723\pi\)
0.758929 + 0.651174i \(0.225723\pi\)
\(62\) −1.40436 −0.178353
\(63\) −0.486575 −0.0613026
\(64\) −6.04269 −0.755336
\(65\) 2.48657 0.308422
\(66\) 1.55454 0.191351
\(67\) 15.1754 1.85397 0.926985 0.375097i \(-0.122391\pi\)
0.926985 + 0.375097i \(0.122391\pi\)
\(68\) 6.54258 0.793404
\(69\) 6.41238 0.771960
\(70\) −0.141446 −0.0169060
\(71\) 9.92903 1.17836 0.589180 0.808002i \(-0.299451\pi\)
0.589180 + 0.808002i \(0.299451\pi\)
\(72\) −1.13822 −0.134141
\(73\) −11.0128 −1.28896 −0.644478 0.764623i \(-0.722925\pi\)
−0.644478 + 0.764623i \(0.722925\pi\)
\(74\) 1.83650 0.213489
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −2.60202 −0.296528
\(78\) 0.722840 0.0818455
\(79\) −8.13272 −0.915002 −0.457501 0.889209i \(-0.651255\pi\)
−0.457501 + 0.889209i \(0.651255\pi\)
\(80\) 3.50011 0.391324
\(81\) 1.00000 0.111111
\(82\) −0.750404 −0.0828683
\(83\) −5.86106 −0.643335 −0.321668 0.946853i \(-0.604243\pi\)
−0.321668 + 0.946853i \(0.604243\pi\)
\(84\) 0.932031 0.101693
\(85\) −3.41561 −0.370475
\(86\) −0.722840 −0.0779459
\(87\) −2.76624 −0.296572
\(88\) −6.08681 −0.648856
\(89\) 6.51043 0.690104 0.345052 0.938584i \(-0.387861\pi\)
0.345052 + 0.938584i \(0.387861\pi\)
\(90\) 0.290697 0.0306422
\(91\) −1.20990 −0.126832
\(92\) −12.2829 −1.28058
\(93\) −4.83099 −0.500950
\(94\) 3.22984 0.333132
\(95\) 0 0
\(96\) 3.29392 0.336184
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −1.96606 −0.198602
\(99\) 5.34764 0.537458
\(100\) −1.91550 −0.191550
\(101\) 1.51665 0.150912 0.0754561 0.997149i \(-0.475959\pi\)
0.0754561 + 0.997149i \(0.475959\pi\)
\(102\) −0.992907 −0.0983125
\(103\) 9.15478 0.902047 0.451023 0.892512i \(-0.351059\pi\)
0.451023 + 0.892512i \(0.351059\pi\)
\(104\) −2.83028 −0.277532
\(105\) −0.486575 −0.0474848
\(106\) −0.871155 −0.0846140
\(107\) −0.138936 −0.0134315 −0.00671574 0.999977i \(-0.502138\pi\)
−0.00671574 + 0.999977i \(0.502138\pi\)
\(108\) −1.91550 −0.184319
\(109\) 15.7299 1.50666 0.753328 0.657645i \(-0.228447\pi\)
0.753328 + 0.657645i \(0.228447\pi\)
\(110\) 1.55454 0.148220
\(111\) 6.31756 0.599637
\(112\) −1.70307 −0.160925
\(113\) −14.8011 −1.39237 −0.696187 0.717860i \(-0.745122\pi\)
−0.696187 + 0.717860i \(0.745122\pi\)
\(114\) 0 0
\(115\) 6.41238 0.597958
\(116\) 5.29873 0.491974
\(117\) 2.48657 0.229884
\(118\) −3.70150 −0.340750
\(119\) 1.66195 0.152350
\(120\) −1.13822 −0.103905
\(121\) 17.5972 1.59975
\(122\) 3.44617 0.312002
\(123\) −2.58139 −0.232756
\(124\) 9.25374 0.831010
\(125\) 1.00000 0.0894427
\(126\) −0.141446 −0.0126010
\(127\) 9.83099 0.872359 0.436180 0.899860i \(-0.356331\pi\)
0.436180 + 0.899860i \(0.356331\pi\)
\(128\) −8.34443 −0.737551
\(129\) −2.48657 −0.218931
\(130\) 0.722840 0.0633973
\(131\) 14.2292 1.24321 0.621605 0.783331i \(-0.286481\pi\)
0.621605 + 0.783331i \(0.286481\pi\)
\(132\) −10.2434 −0.891571
\(133\) 0 0
\(134\) 4.41145 0.381091
\(135\) 1.00000 0.0860663
\(136\) 3.88772 0.333370
\(137\) 6.58439 0.562543 0.281271 0.959628i \(-0.409244\pi\)
0.281271 + 0.959628i \(0.409244\pi\)
\(138\) 1.86406 0.158680
\(139\) 0.233756 0.0198269 0.00991347 0.999951i \(-0.496844\pi\)
0.00991347 + 0.999951i \(0.496844\pi\)
\(140\) 0.932031 0.0787710
\(141\) 11.1107 0.935686
\(142\) 2.88634 0.242216
\(143\) 13.2973 1.11198
\(144\) 3.50011 0.291676
\(145\) −2.76624 −0.229724
\(146\) −3.20140 −0.264950
\(147\) −6.76325 −0.557823
\(148\) −12.1013 −0.994718
\(149\) −2.67299 −0.218980 −0.109490 0.993988i \(-0.534922\pi\)
−0.109490 + 0.993988i \(0.534922\pi\)
\(150\) 0.290697 0.0237353
\(151\) −5.95266 −0.484421 −0.242210 0.970224i \(-0.577872\pi\)
−0.242210 + 0.970224i \(0.577872\pi\)
\(152\) 0 0
\(153\) −3.41561 −0.276135
\(154\) −0.756401 −0.0609525
\(155\) −4.83099 −0.388034
\(156\) −4.76302 −0.381347
\(157\) −7.20848 −0.575299 −0.287650 0.957736i \(-0.592874\pi\)
−0.287650 + 0.957736i \(0.592874\pi\)
\(158\) −2.36416 −0.188082
\(159\) −2.99678 −0.237660
\(160\) 3.29392 0.260407
\(161\) −3.12010 −0.245899
\(162\) 0.290697 0.0228393
\(163\) 14.1756 1.11032 0.555161 0.831743i \(-0.312657\pi\)
0.555161 + 0.831743i \(0.312657\pi\)
\(164\) 4.94465 0.386112
\(165\) 5.34764 0.416313
\(166\) −1.70379 −0.132240
\(167\) −12.5563 −0.971639 −0.485819 0.874059i \(-0.661479\pi\)
−0.485819 + 0.874059i \(0.661479\pi\)
\(168\) 0.553831 0.0427290
\(169\) −6.81695 −0.524381
\(170\) −0.992907 −0.0761525
\(171\) 0 0
\(172\) 4.76302 0.363177
\(173\) −2.18964 −0.166475 −0.0832376 0.996530i \(-0.526526\pi\)
−0.0832376 + 0.996530i \(0.526526\pi\)
\(174\) −0.804139 −0.0609616
\(175\) −0.486575 −0.0367816
\(176\) 18.7173 1.41087
\(177\) −12.7332 −0.957084
\(178\) 1.89256 0.141854
\(179\) −7.50421 −0.560891 −0.280445 0.959870i \(-0.590482\pi\)
−0.280445 + 0.959870i \(0.590482\pi\)
\(180\) −1.91550 −0.142773
\(181\) 12.6652 0.941397 0.470699 0.882294i \(-0.344002\pi\)
0.470699 + 0.882294i \(0.344002\pi\)
\(182\) −0.351716 −0.0260709
\(183\) 11.8548 0.876335
\(184\) −7.29873 −0.538069
\(185\) 6.31756 0.464477
\(186\) −1.40436 −0.102972
\(187\) −18.2654 −1.33570
\(188\) −21.2824 −1.55218
\(189\) −0.486575 −0.0353931
\(190\) 0 0
\(191\) 8.15777 0.590276 0.295138 0.955455i \(-0.404634\pi\)
0.295138 + 0.955455i \(0.404634\pi\)
\(192\) −6.04269 −0.436094
\(193\) 3.36505 0.242221 0.121111 0.992639i \(-0.461354\pi\)
0.121111 + 0.992639i \(0.461354\pi\)
\(194\) −0.581394 −0.0417417
\(195\) 2.48657 0.178067
\(196\) 12.9550 0.925355
\(197\) 3.83399 0.273160 0.136580 0.990629i \(-0.456389\pi\)
0.136580 + 0.990629i \(0.456389\pi\)
\(198\) 1.55454 0.110477
\(199\) 12.6985 0.900173 0.450086 0.892985i \(-0.351393\pi\)
0.450086 + 0.892985i \(0.351393\pi\)
\(200\) −1.13822 −0.0804846
\(201\) 15.1754 1.07039
\(202\) 0.440885 0.0310206
\(203\) 1.34598 0.0944695
\(204\) 6.54258 0.458072
\(205\) −2.58139 −0.180292
\(206\) 2.66127 0.185419
\(207\) 6.41238 0.445692
\(208\) 8.70329 0.603465
\(209\) 0 0
\(210\) −0.141446 −0.00976069
\(211\) 9.15657 0.630364 0.315182 0.949031i \(-0.397934\pi\)
0.315182 + 0.949031i \(0.397934\pi\)
\(212\) 5.74031 0.394246
\(213\) 9.92903 0.680326
\(214\) −0.0403884 −0.00276089
\(215\) −2.48657 −0.169583
\(216\) −1.13822 −0.0774463
\(217\) 2.35064 0.159572
\(218\) 4.57265 0.309699
\(219\) −11.0128 −0.744179
\(220\) −10.2434 −0.690608
\(221\) −8.49316 −0.571312
\(222\) 1.83650 0.123258
\(223\) −10.8975 −0.729753 −0.364876 0.931056i \(-0.618889\pi\)
−0.364876 + 0.931056i \(0.618889\pi\)
\(224\) −1.60274 −0.107087
\(225\) 1.00000 0.0666667
\(226\) −4.30265 −0.286208
\(227\) 24.6351 1.63509 0.817545 0.575864i \(-0.195334\pi\)
0.817545 + 0.575864i \(0.195334\pi\)
\(228\) 0 0
\(229\) 17.1676 1.13447 0.567234 0.823557i \(-0.308013\pi\)
0.567234 + 0.823557i \(0.308013\pi\)
\(230\) 1.86406 0.122913
\(231\) −2.60202 −0.171201
\(232\) 3.14860 0.206716
\(233\) 28.4631 1.86468 0.932341 0.361581i \(-0.117763\pi\)
0.932341 + 0.361581i \(0.117763\pi\)
\(234\) 0.722840 0.0472535
\(235\) 11.1107 0.724780
\(236\) 24.3903 1.58768
\(237\) −8.13272 −0.528277
\(238\) 0.483123 0.0313162
\(239\) 4.07741 0.263746 0.131873 0.991267i \(-0.457901\pi\)
0.131873 + 0.991267i \(0.457901\pi\)
\(240\) 3.50011 0.225931
\(241\) 12.6684 0.816045 0.408023 0.912972i \(-0.366218\pi\)
0.408023 + 0.912972i \(0.366218\pi\)
\(242\) 5.11547 0.328835
\(243\) 1.00000 0.0641500
\(244\) −22.7079 −1.45372
\(245\) −6.76325 −0.432088
\(246\) −0.750404 −0.0478440
\(247\) 0 0
\(248\) 5.49875 0.349171
\(249\) −5.86106 −0.371430
\(250\) 0.290697 0.0183853
\(251\) −11.6241 −0.733706 −0.366853 0.930279i \(-0.619565\pi\)
−0.366853 + 0.930279i \(0.619565\pi\)
\(252\) 0.932031 0.0587124
\(253\) 34.2911 2.15586
\(254\) 2.85784 0.179317
\(255\) −3.41561 −0.213894
\(256\) 9.65968 0.603730
\(257\) −10.5483 −0.657986 −0.328993 0.944332i \(-0.606709\pi\)
−0.328993 + 0.944332i \(0.606709\pi\)
\(258\) −0.722840 −0.0450021
\(259\) −3.07397 −0.191007
\(260\) −4.76302 −0.295390
\(261\) −2.76624 −0.171226
\(262\) 4.13639 0.255547
\(263\) 23.7935 1.46717 0.733585 0.679598i \(-0.237846\pi\)
0.733585 + 0.679598i \(0.237846\pi\)
\(264\) −6.08681 −0.374617
\(265\) −2.99678 −0.184091
\(266\) 0 0
\(267\) 6.51043 0.398432
\(268\) −29.0684 −1.77564
\(269\) 4.22754 0.257757 0.128879 0.991660i \(-0.458862\pi\)
0.128879 + 0.991660i \(0.458862\pi\)
\(270\) 0.290697 0.0176913
\(271\) 31.1471 1.89205 0.946027 0.324089i \(-0.105058\pi\)
0.946027 + 0.324089i \(0.105058\pi\)
\(272\) −11.9550 −0.724879
\(273\) −1.20990 −0.0732267
\(274\) 1.91406 0.115633
\(275\) 5.34764 0.322475
\(276\) −12.2829 −0.739343
\(277\) 1.05393 0.0633243 0.0316621 0.999499i \(-0.489920\pi\)
0.0316621 + 0.999499i \(0.489920\pi\)
\(278\) 0.0679523 0.00407551
\(279\) −4.83099 −0.289224
\(280\) 0.553831 0.0330977
\(281\) −25.6354 −1.52928 −0.764638 0.644459i \(-0.777082\pi\)
−0.764638 + 0.644459i \(0.777082\pi\)
\(282\) 3.22984 0.192334
\(283\) 29.4112 1.74831 0.874157 0.485644i \(-0.161415\pi\)
0.874157 + 0.485644i \(0.161415\pi\)
\(284\) −19.0190 −1.12857
\(285\) 0 0
\(286\) 3.86549 0.228571
\(287\) 1.25604 0.0741417
\(288\) 3.29392 0.194096
\(289\) −5.33363 −0.313743
\(290\) −0.804139 −0.0472207
\(291\) −2.00000 −0.117242
\(292\) 21.0950 1.23449
\(293\) 10.8548 0.634147 0.317073 0.948401i \(-0.397300\pi\)
0.317073 + 0.948401i \(0.397300\pi\)
\(294\) −1.96606 −0.114663
\(295\) −12.7332 −0.741354
\(296\) −7.19080 −0.417957
\(297\) 5.34764 0.310301
\(298\) −0.777031 −0.0450122
\(299\) 15.9449 0.922116
\(300\) −1.91550 −0.110591
\(301\) 1.20990 0.0697377
\(302\) −1.73042 −0.0995746
\(303\) 1.51665 0.0871291
\(304\) 0 0
\(305\) 11.8548 0.678806
\(306\) −0.992907 −0.0567607
\(307\) −33.6891 −1.92274 −0.961368 0.275266i \(-0.911234\pi\)
−0.961368 + 0.275266i \(0.911234\pi\)
\(308\) 4.98417 0.283999
\(309\) 9.15478 0.520797
\(310\) −1.40436 −0.0797620
\(311\) −11.7221 −0.664701 −0.332350 0.943156i \(-0.607842\pi\)
−0.332350 + 0.943156i \(0.607842\pi\)
\(312\) −2.83028 −0.160233
\(313\) 4.47856 0.253143 0.126572 0.991957i \(-0.459603\pi\)
0.126572 + 0.991957i \(0.459603\pi\)
\(314\) −2.09548 −0.118255
\(315\) −0.486575 −0.0274154
\(316\) 15.5782 0.876341
\(317\) −12.3873 −0.695742 −0.347871 0.937542i \(-0.613095\pi\)
−0.347871 + 0.937542i \(0.613095\pi\)
\(318\) −0.871155 −0.0488519
\(319\) −14.7929 −0.828242
\(320\) −6.04269 −0.337797
\(321\) −0.138936 −0.00775467
\(322\) −0.907005 −0.0505454
\(323\) 0 0
\(324\) −1.91550 −0.106416
\(325\) 2.48657 0.137930
\(326\) 4.12082 0.228231
\(327\) 15.7299 0.869868
\(328\) 2.93820 0.162235
\(329\) −5.40616 −0.298051
\(330\) 1.55454 0.0855748
\(331\) −24.5058 −1.34696 −0.673480 0.739206i \(-0.735201\pi\)
−0.673480 + 0.739206i \(0.735201\pi\)
\(332\) 11.2268 0.616153
\(333\) 6.31756 0.346200
\(334\) −3.65009 −0.199724
\(335\) 15.1754 0.829121
\(336\) −1.70307 −0.0929098
\(337\) −10.5008 −0.572013 −0.286006 0.958228i \(-0.592328\pi\)
−0.286006 + 0.958228i \(0.592328\pi\)
\(338\) −1.98167 −0.107788
\(339\) −14.8011 −0.803888
\(340\) 6.54258 0.354821
\(341\) −25.8344 −1.39901
\(342\) 0 0
\(343\) 6.69684 0.361596
\(344\) 2.83028 0.152598
\(345\) 6.41238 0.345231
\(346\) −0.636522 −0.0342196
\(347\) −1.45512 −0.0781151 −0.0390576 0.999237i \(-0.512436\pi\)
−0.0390576 + 0.999237i \(0.512436\pi\)
\(348\) 5.29873 0.284042
\(349\) 23.7332 1.27041 0.635204 0.772345i \(-0.280916\pi\)
0.635204 + 0.772345i \(0.280916\pi\)
\(350\) −0.141446 −0.00756060
\(351\) 2.48657 0.132724
\(352\) 17.6147 0.938866
\(353\) −7.39055 −0.393359 −0.196680 0.980468i \(-0.563016\pi\)
−0.196680 + 0.980468i \(0.563016\pi\)
\(354\) −3.70150 −0.196732
\(355\) 9.92903 0.526978
\(356\) −12.4707 −0.660945
\(357\) 1.66195 0.0879596
\(358\) −2.18145 −0.115293
\(359\) 2.35868 0.124487 0.0622433 0.998061i \(-0.480175\pi\)
0.0622433 + 0.998061i \(0.480175\pi\)
\(360\) −1.13822 −0.0599896
\(361\) 0 0
\(362\) 3.68174 0.193508
\(363\) 17.5972 0.923615
\(364\) 2.31756 0.121473
\(365\) −11.0128 −0.576438
\(366\) 3.44617 0.180134
\(367\) −22.1486 −1.15615 −0.578073 0.815985i \(-0.696195\pi\)
−0.578073 + 0.815985i \(0.696195\pi\)
\(368\) 22.4441 1.16998
\(369\) −2.58139 −0.134382
\(370\) 1.83650 0.0954750
\(371\) 1.45816 0.0757037
\(372\) 9.25374 0.479784
\(373\) −31.1738 −1.61412 −0.807060 0.590469i \(-0.798943\pi\)
−0.807060 + 0.590469i \(0.798943\pi\)
\(374\) −5.30971 −0.274559
\(375\) 1.00000 0.0516398
\(376\) −12.6464 −0.652189
\(377\) −6.87847 −0.354259
\(378\) −0.141446 −0.00727519
\(379\) 26.0349 1.33732 0.668662 0.743567i \(-0.266868\pi\)
0.668662 + 0.743567i \(0.266868\pi\)
\(380\) 0 0
\(381\) 9.83099 0.503657
\(382\) 2.37144 0.121334
\(383\) 2.13294 0.108988 0.0544941 0.998514i \(-0.482645\pi\)
0.0544941 + 0.998514i \(0.482645\pi\)
\(384\) −8.34443 −0.425825
\(385\) −2.60202 −0.132611
\(386\) 0.978209 0.0497895
\(387\) −2.48657 −0.126400
\(388\) 3.83099 0.194489
\(389\) −13.0427 −0.661291 −0.330645 0.943755i \(-0.607266\pi\)
−0.330645 + 0.943755i \(0.607266\pi\)
\(390\) 0.722840 0.0366024
\(391\) −21.9022 −1.10764
\(392\) 7.69808 0.388812
\(393\) 14.2292 0.717768
\(394\) 1.11453 0.0561492
\(395\) −8.13272 −0.409201
\(396\) −10.2434 −0.514749
\(397\) 0.0286428 0.00143754 0.000718770 1.00000i \(-0.499771\pi\)
0.000718770 1.00000i \(0.499771\pi\)
\(398\) 3.69142 0.185034
\(399\) 0 0
\(400\) 3.50011 0.175006
\(401\) 27.3191 1.36425 0.682126 0.731234i \(-0.261055\pi\)
0.682126 + 0.731234i \(0.261055\pi\)
\(402\) 4.41145 0.220023
\(403\) −12.0126 −0.598391
\(404\) −2.90513 −0.144536
\(405\) 1.00000 0.0496904
\(406\) 0.391274 0.0194186
\(407\) 33.7841 1.67461
\(408\) 3.88772 0.192471
\(409\) 6.73917 0.333230 0.166615 0.986022i \(-0.446716\pi\)
0.166615 + 0.986022i \(0.446716\pi\)
\(410\) −0.750404 −0.0370598
\(411\) 6.58439 0.324784
\(412\) −17.5359 −0.863933
\(413\) 6.19564 0.304867
\(414\) 1.86406 0.0916137
\(415\) −5.86106 −0.287708
\(416\) 8.19058 0.401576
\(417\) 0.233756 0.0114471
\(418\) 0 0
\(419\) 26.4076 1.29010 0.645048 0.764142i \(-0.276837\pi\)
0.645048 + 0.764142i \(0.276837\pi\)
\(420\) 0.932031 0.0454785
\(421\) −37.8895 −1.84662 −0.923311 0.384053i \(-0.874528\pi\)
−0.923311 + 0.384053i \(0.874528\pi\)
\(422\) 2.66179 0.129574
\(423\) 11.1107 0.540219
\(424\) 3.41100 0.165653
\(425\) −3.41561 −0.165681
\(426\) 2.88634 0.139844
\(427\) −5.76826 −0.279146
\(428\) 0.266132 0.0128640
\(429\) 13.2973 0.642000
\(430\) −0.722840 −0.0348585
\(431\) 19.2057 0.925106 0.462553 0.886592i \(-0.346933\pi\)
0.462553 + 0.886592i \(0.346933\pi\)
\(432\) 3.50011 0.168399
\(433\) −16.9512 −0.814624 −0.407312 0.913289i \(-0.633534\pi\)
−0.407312 + 0.913289i \(0.633534\pi\)
\(434\) 0.683323 0.0328006
\(435\) −2.76624 −0.132631
\(436\) −30.1306 −1.44300
\(437\) 0 0
\(438\) −3.20140 −0.152969
\(439\) −11.1029 −0.529911 −0.264955 0.964261i \(-0.585357\pi\)
−0.264955 + 0.964261i \(0.585357\pi\)
\(440\) −6.08681 −0.290177
\(441\) −6.76325 −0.322059
\(442\) −2.46894 −0.117435
\(443\) −13.8942 −0.660132 −0.330066 0.943958i \(-0.607071\pi\)
−0.330066 + 0.943958i \(0.607071\pi\)
\(444\) −12.1013 −0.574301
\(445\) 6.51043 0.308624
\(446\) −3.16788 −0.150004
\(447\) −2.67299 −0.126428
\(448\) 2.94022 0.138912
\(449\) −27.8675 −1.31515 −0.657573 0.753391i \(-0.728417\pi\)
−0.657573 + 0.753391i \(0.728417\pi\)
\(450\) 0.290697 0.0137036
\(451\) −13.8044 −0.650022
\(452\) 28.3515 1.33354
\(453\) −5.95266 −0.279680
\(454\) 7.16136 0.336099
\(455\) −1.20990 −0.0567212
\(456\) 0 0
\(457\) −36.5660 −1.71048 −0.855242 0.518229i \(-0.826591\pi\)
−0.855242 + 0.518229i \(0.826591\pi\)
\(458\) 4.99058 0.233194
\(459\) −3.41561 −0.159427
\(460\) −12.2829 −0.572693
\(461\) 21.9180 1.02082 0.510412 0.859930i \(-0.329493\pi\)
0.510412 + 0.859930i \(0.329493\pi\)
\(462\) −0.756401 −0.0351910
\(463\) −14.0255 −0.651821 −0.325910 0.945401i \(-0.605671\pi\)
−0.325910 + 0.945401i \(0.605671\pi\)
\(464\) −9.68216 −0.449483
\(465\) −4.83099 −0.224032
\(466\) 8.27415 0.383293
\(467\) 33.9294 1.57007 0.785033 0.619453i \(-0.212646\pi\)
0.785033 + 0.619453i \(0.212646\pi\)
\(468\) −4.76302 −0.220171
\(469\) −7.38397 −0.340960
\(470\) 3.22984 0.148981
\(471\) −7.20848 −0.332149
\(472\) 14.4932 0.667103
\(473\) −13.2973 −0.611411
\(474\) −2.36416 −0.108589
\(475\) 0 0
\(476\) −3.18345 −0.145913
\(477\) −2.99678 −0.137213
\(478\) 1.18529 0.0542140
\(479\) 31.9497 1.45982 0.729909 0.683544i \(-0.239562\pi\)
0.729909 + 0.683544i \(0.239562\pi\)
\(480\) 3.29392 0.150346
\(481\) 15.7091 0.716273
\(482\) 3.68268 0.167741
\(483\) −3.12010 −0.141970
\(484\) −33.7074 −1.53216
\(485\) −2.00000 −0.0908153
\(486\) 0.290697 0.0131863
\(487\) −37.6229 −1.70486 −0.852428 0.522844i \(-0.824871\pi\)
−0.852428 + 0.522844i \(0.824871\pi\)
\(488\) −13.4935 −0.610820
\(489\) 14.1756 0.641044
\(490\) −1.96606 −0.0888174
\(491\) −40.0842 −1.80897 −0.904487 0.426500i \(-0.859746\pi\)
−0.904487 + 0.426500i \(0.859746\pi\)
\(492\) 4.94465 0.222922
\(493\) 9.44840 0.425535
\(494\) 0 0
\(495\) 5.34764 0.240358
\(496\) −16.9090 −0.759237
\(497\) −4.83121 −0.216710
\(498\) −1.70379 −0.0763489
\(499\) −30.3067 −1.35671 −0.678357 0.734732i \(-0.737308\pi\)
−0.678357 + 0.734732i \(0.737308\pi\)
\(500\) −1.91550 −0.0856635
\(501\) −12.5563 −0.560976
\(502\) −3.37909 −0.150816
\(503\) −17.2879 −0.770827 −0.385414 0.922744i \(-0.625941\pi\)
−0.385414 + 0.922744i \(0.625941\pi\)
\(504\) 0.553831 0.0246696
\(505\) 1.51665 0.0674899
\(506\) 9.96833 0.443146
\(507\) −6.81695 −0.302751
\(508\) −18.8312 −0.835500
\(509\) −5.60825 −0.248581 −0.124291 0.992246i \(-0.539665\pi\)
−0.124291 + 0.992246i \(0.539665\pi\)
\(510\) −0.992907 −0.0439667
\(511\) 5.35857 0.237049
\(512\) 19.4969 0.861650
\(513\) 0 0
\(514\) −3.06637 −0.135252
\(515\) 9.15478 0.403408
\(516\) 4.76302 0.209680
\(517\) 59.4158 2.61310
\(518\) −0.893593 −0.0392622
\(519\) −2.18964 −0.0961145
\(520\) −2.83028 −0.124116
\(521\) −13.7128 −0.600767 −0.300384 0.953818i \(-0.597115\pi\)
−0.300384 + 0.953818i \(0.597115\pi\)
\(522\) −0.804139 −0.0351962
\(523\) 35.2342 1.54068 0.770341 0.637632i \(-0.220086\pi\)
0.770341 + 0.637632i \(0.220086\pi\)
\(524\) −27.2559 −1.19068
\(525\) −0.486575 −0.0212359
\(526\) 6.91670 0.301583
\(527\) 16.5008 0.718785
\(528\) 18.7173 0.814567
\(529\) 18.1187 0.787769
\(530\) −0.871155 −0.0378405
\(531\) −12.7332 −0.552573
\(532\) 0 0
\(533\) −6.41883 −0.278030
\(534\) 1.89256 0.0818992
\(535\) −0.138936 −0.00600674
\(536\) −17.2730 −0.746080
\(537\) −7.50421 −0.323830
\(538\) 1.22893 0.0529830
\(539\) −36.1674 −1.55784
\(540\) −1.91550 −0.0824298
\(541\) 30.2357 1.29994 0.649968 0.759962i \(-0.274782\pi\)
0.649968 + 0.759962i \(0.274782\pi\)
\(542\) 9.05438 0.388919
\(543\) 12.6652 0.543516
\(544\) −11.2507 −0.482371
\(545\) 15.7299 0.673797
\(546\) −0.351716 −0.0150520
\(547\) −34.9531 −1.49449 −0.747244 0.664550i \(-0.768623\pi\)
−0.747244 + 0.664550i \(0.768623\pi\)
\(548\) −12.6124 −0.538774
\(549\) 11.8548 0.505952
\(550\) 1.55454 0.0662860
\(551\) 0 0
\(552\) −7.29873 −0.310654
\(553\) 3.95717 0.168276
\(554\) 0.306373 0.0130166
\(555\) 6.31756 0.268166
\(556\) −0.447759 −0.0189892
\(557\) 3.16279 0.134012 0.0670058 0.997753i \(-0.478655\pi\)
0.0670058 + 0.997753i \(0.478655\pi\)
\(558\) −1.40436 −0.0594511
\(559\) −6.18305 −0.261515
\(560\) −1.70307 −0.0719676
\(561\) −18.2654 −0.771167
\(562\) −7.45213 −0.314349
\(563\) 29.0570 1.22460 0.612302 0.790624i \(-0.290244\pi\)
0.612302 + 0.790624i \(0.290244\pi\)
\(564\) −21.2824 −0.896151
\(565\) −14.8011 −0.622689
\(566\) 8.54975 0.359373
\(567\) −0.486575 −0.0204342
\(568\) −11.3015 −0.474199
\(569\) 31.3398 1.31383 0.656916 0.753964i \(-0.271861\pi\)
0.656916 + 0.753964i \(0.271861\pi\)
\(570\) 0 0
\(571\) 2.56556 0.107365 0.0536827 0.998558i \(-0.482904\pi\)
0.0536827 + 0.998558i \(0.482904\pi\)
\(572\) −25.4709 −1.06499
\(573\) 8.15777 0.340796
\(574\) 0.365128 0.0152401
\(575\) 6.41238 0.267415
\(576\) −6.04269 −0.251779
\(577\) 3.12153 0.129951 0.0649755 0.997887i \(-0.479303\pi\)
0.0649755 + 0.997887i \(0.479303\pi\)
\(578\) −1.55047 −0.0644911
\(579\) 3.36505 0.139847
\(580\) 5.29873 0.220018
\(581\) 2.85184 0.118314
\(582\) −0.581394 −0.0240996
\(583\) −16.0257 −0.663716
\(584\) 12.5351 0.518705
\(585\) 2.48657 0.102807
\(586\) 3.15547 0.130351
\(587\) −9.27970 −0.383014 −0.191507 0.981491i \(-0.561338\pi\)
−0.191507 + 0.981491i \(0.561338\pi\)
\(588\) 12.9550 0.534254
\(589\) 0 0
\(590\) −3.70150 −0.152388
\(591\) 3.83399 0.157709
\(592\) 22.1122 0.908805
\(593\) 34.0447 1.39805 0.699025 0.715097i \(-0.253618\pi\)
0.699025 + 0.715097i \(0.253618\pi\)
\(594\) 1.55454 0.0637837
\(595\) 1.66195 0.0681332
\(596\) 5.12010 0.209728
\(597\) 12.6985 0.519715
\(598\) 4.63513 0.189545
\(599\) −25.0732 −1.02446 −0.512231 0.858848i \(-0.671181\pi\)
−0.512231 + 0.858848i \(0.671181\pi\)
\(600\) −1.13822 −0.0464678
\(601\) 26.0225 1.06148 0.530739 0.847535i \(-0.321914\pi\)
0.530739 + 0.847535i \(0.321914\pi\)
\(602\) 0.351716 0.0143349
\(603\) 15.1754 0.617990
\(604\) 11.4023 0.463953
\(605\) 17.5972 0.715429
\(606\) 0.440885 0.0179097
\(607\) 7.54650 0.306303 0.153151 0.988203i \(-0.451058\pi\)
0.153151 + 0.988203i \(0.451058\pi\)
\(608\) 0 0
\(609\) 1.34598 0.0545420
\(610\) 3.44617 0.139531
\(611\) 27.6275 1.11769
\(612\) 6.54258 0.264468
\(613\) −38.2690 −1.54567 −0.772836 0.634606i \(-0.781162\pi\)
−0.772836 + 0.634606i \(0.781162\pi\)
\(614\) −9.79331 −0.395226
\(615\) −2.58139 −0.104092
\(616\) 2.96169 0.119330
\(617\) −11.6508 −0.469043 −0.234522 0.972111i \(-0.575352\pi\)
−0.234522 + 0.972111i \(0.575352\pi\)
\(618\) 2.66127 0.107052
\(619\) −2.47710 −0.0995630 −0.0497815 0.998760i \(-0.515852\pi\)
−0.0497815 + 0.998760i \(0.515852\pi\)
\(620\) 9.25374 0.371639
\(621\) 6.41238 0.257320
\(622\) −3.40759 −0.136632
\(623\) −3.16781 −0.126916
\(624\) 8.70329 0.348410
\(625\) 1.00000 0.0400000
\(626\) 1.30191 0.0520346
\(627\) 0 0
\(628\) 13.8078 0.550991
\(629\) −21.5783 −0.860384
\(630\) −0.141446 −0.00563534
\(631\) −43.6558 −1.73791 −0.868954 0.494893i \(-0.835207\pi\)
−0.868954 + 0.494893i \(0.835207\pi\)
\(632\) 9.25685 0.368218
\(633\) 9.15657 0.363941
\(634\) −3.60096 −0.143012
\(635\) 9.83099 0.390131
\(636\) 5.74031 0.227618
\(637\) −16.8173 −0.666326
\(638\) −4.30025 −0.170248
\(639\) 9.92903 0.392786
\(640\) −8.34443 −0.329843
\(641\) −24.2326 −0.957128 −0.478564 0.878053i \(-0.658843\pi\)
−0.478564 + 0.878053i \(0.658843\pi\)
\(642\) −0.0403884 −0.00159400
\(643\) 8.31756 0.328013 0.164006 0.986459i \(-0.447558\pi\)
0.164006 + 0.986459i \(0.447558\pi\)
\(644\) 5.97654 0.235509
\(645\) −2.48657 −0.0979088
\(646\) 0 0
\(647\) −10.9164 −0.429170 −0.214585 0.976705i \(-0.568840\pi\)
−0.214585 + 0.976705i \(0.568840\pi\)
\(648\) −1.13822 −0.0447136
\(649\) −68.0924 −2.67286
\(650\) 0.722840 0.0283521
\(651\) 2.35064 0.0921287
\(652\) −27.1534 −1.06341
\(653\) −21.8452 −0.854868 −0.427434 0.904047i \(-0.640582\pi\)
−0.427434 + 0.904047i \(0.640582\pi\)
\(654\) 4.57265 0.178805
\(655\) 14.2292 0.555980
\(656\) −9.03517 −0.352764
\(657\) −11.0128 −0.429652
\(658\) −1.57156 −0.0612657
\(659\) 26.4188 1.02913 0.514566 0.857451i \(-0.327953\pi\)
0.514566 + 0.857451i \(0.327953\pi\)
\(660\) −10.2434 −0.398723
\(661\) −18.6144 −0.724017 −0.362008 0.932175i \(-0.617909\pi\)
−0.362008 + 0.932175i \(0.617909\pi\)
\(662\) −7.12376 −0.276873
\(663\) −8.49316 −0.329847
\(664\) 6.67120 0.258893
\(665\) 0 0
\(666\) 1.83650 0.0711629
\(667\) −17.7382 −0.686827
\(668\) 24.0516 0.930585
\(669\) −10.8975 −0.421323
\(670\) 4.41145 0.170429
\(671\) 63.3954 2.44735
\(672\) −1.60274 −0.0618269
\(673\) −5.98693 −0.230779 −0.115390 0.993320i \(-0.536812\pi\)
−0.115390 + 0.993320i \(0.536812\pi\)
\(674\) −3.05254 −0.117580
\(675\) 1.00000 0.0384900
\(676\) 13.0578 0.502224
\(677\) −33.7287 −1.29630 −0.648150 0.761512i \(-0.724457\pi\)
−0.648150 + 0.761512i \(0.724457\pi\)
\(678\) −4.30265 −0.165242
\(679\) 0.973149 0.0373460
\(680\) 3.88772 0.149087
\(681\) 24.6351 0.944020
\(682\) −7.50998 −0.287572
\(683\) −11.4959 −0.439880 −0.219940 0.975513i \(-0.570586\pi\)
−0.219940 + 0.975513i \(0.570586\pi\)
\(684\) 0 0
\(685\) 6.58439 0.251577
\(686\) 1.94675 0.0743274
\(687\) 17.1676 0.654985
\(688\) −8.70329 −0.331810
\(689\) −7.45171 −0.283888
\(690\) 1.86406 0.0709636
\(691\) 36.3039 1.38107 0.690533 0.723301i \(-0.257376\pi\)
0.690533 + 0.723301i \(0.257376\pi\)
\(692\) 4.19424 0.159441
\(693\) −2.60202 −0.0988427
\(694\) −0.423000 −0.0160569
\(695\) 0.233756 0.00886688
\(696\) 3.14860 0.119348
\(697\) 8.81703 0.333969
\(698\) 6.89917 0.261137
\(699\) 28.4631 1.07657
\(700\) 0.932031 0.0352275
\(701\) 13.5275 0.510925 0.255463 0.966819i \(-0.417772\pi\)
0.255463 + 0.966819i \(0.417772\pi\)
\(702\) 0.722840 0.0272818
\(703\) 0 0
\(704\) −32.3141 −1.21788
\(705\) 11.1107 0.418452
\(706\) −2.14841 −0.0808566
\(707\) −0.737962 −0.0277539
\(708\) 24.3903 0.916645
\(709\) −0.993555 −0.0373137 −0.0186569 0.999826i \(-0.505939\pi\)
−0.0186569 + 0.999826i \(0.505939\pi\)
\(710\) 2.88634 0.108322
\(711\) −8.13272 −0.305001
\(712\) −7.41032 −0.277714
\(713\) −30.9782 −1.16014
\(714\) 0.483123 0.0180804
\(715\) 13.2973 0.497291
\(716\) 14.3743 0.537192
\(717\) 4.07741 0.152274
\(718\) 0.685662 0.0255887
\(719\) −1.40134 −0.0522612 −0.0261306 0.999659i \(-0.508319\pi\)
−0.0261306 + 0.999659i \(0.508319\pi\)
\(720\) 3.50011 0.130441
\(721\) −4.45448 −0.165894
\(722\) 0 0
\(723\) 12.6684 0.471144
\(724\) −24.2601 −0.901621
\(725\) −2.76624 −0.102736
\(726\) 5.11547 0.189853
\(727\) −46.9669 −1.74191 −0.870953 0.491367i \(-0.836497\pi\)
−0.870953 + 0.491367i \(0.836497\pi\)
\(728\) 1.37714 0.0510402
\(729\) 1.00000 0.0370370
\(730\) −3.20140 −0.118489
\(731\) 8.49316 0.314131
\(732\) −22.7079 −0.839308
\(733\) −21.6068 −0.798066 −0.399033 0.916937i \(-0.630654\pi\)
−0.399033 + 0.916937i \(0.630654\pi\)
\(734\) −6.43852 −0.237650
\(735\) −6.76325 −0.249466
\(736\) 21.1219 0.778563
\(737\) 81.1526 2.98929
\(738\) −0.750404 −0.0276228
\(739\) −30.8126 −1.13346 −0.566729 0.823904i \(-0.691791\pi\)
−0.566729 + 0.823904i \(0.691791\pi\)
\(740\) −12.1013 −0.444851
\(741\) 0 0
\(742\) 0.423882 0.0155612
\(743\) −44.4286 −1.62993 −0.814964 0.579512i \(-0.803243\pi\)
−0.814964 + 0.579512i \(0.803243\pi\)
\(744\) 5.49875 0.201594
\(745\) −2.67299 −0.0979308
\(746\) −9.06215 −0.331789
\(747\) −5.86106 −0.214445
\(748\) 34.9873 1.27926
\(749\) 0.0676029 0.00247016
\(750\) 0.290697 0.0106148
\(751\) −14.8787 −0.542931 −0.271466 0.962448i \(-0.587508\pi\)
−0.271466 + 0.962448i \(0.587508\pi\)
\(752\) 38.8886 1.41812
\(753\) −11.6241 −0.423605
\(754\) −1.99955 −0.0728194
\(755\) −5.95266 −0.216640
\(756\) 0.932031 0.0338976
\(757\) 23.4408 0.851972 0.425986 0.904730i \(-0.359927\pi\)
0.425986 + 0.904730i \(0.359927\pi\)
\(758\) 7.56827 0.274892
\(759\) 34.2911 1.24469
\(760\) 0 0
\(761\) 36.3224 1.31669 0.658344 0.752717i \(-0.271257\pi\)
0.658344 + 0.752717i \(0.271257\pi\)
\(762\) 2.85784 0.103529
\(763\) −7.65379 −0.277086
\(764\) −15.6262 −0.565335
\(765\) −3.41561 −0.123492
\(766\) 0.620039 0.0224029
\(767\) −31.6620 −1.14325
\(768\) 9.65968 0.348564
\(769\) 20.8104 0.750441 0.375220 0.926936i \(-0.377567\pi\)
0.375220 + 0.926936i \(0.377567\pi\)
\(770\) −0.756401 −0.0272588
\(771\) −10.5483 −0.379889
\(772\) −6.44573 −0.231987
\(773\) −3.56702 −0.128297 −0.0641484 0.997940i \(-0.520433\pi\)
−0.0641484 + 0.997940i \(0.520433\pi\)
\(774\) −0.722840 −0.0259820
\(775\) −4.83099 −0.173534
\(776\) 2.27645 0.0817197
\(777\) −3.07397 −0.110278
\(778\) −3.79147 −0.135931
\(779\) 0 0
\(780\) −4.76302 −0.170544
\(781\) 53.0969 1.89996
\(782\) −6.36690 −0.227680
\(783\) −2.76624 −0.0988575
\(784\) −23.6721 −0.845433
\(785\) −7.20848 −0.257282
\(786\) 4.13639 0.147540
\(787\) −44.5660 −1.58860 −0.794302 0.607523i \(-0.792163\pi\)
−0.794302 + 0.607523i \(0.792163\pi\)
\(788\) −7.34399 −0.261619
\(789\) 23.7935 0.847071
\(790\) −2.36416 −0.0841129
\(791\) 7.20186 0.256069
\(792\) −6.08681 −0.216285
\(793\) 29.4779 1.04679
\(794\) 0.00832637 0.000295492 0
\(795\) −2.99678 −0.106285
\(796\) −24.3239 −0.862138
\(797\) 44.7396 1.58476 0.792378 0.610030i \(-0.208843\pi\)
0.792378 + 0.610030i \(0.208843\pi\)
\(798\) 0 0
\(799\) −37.9496 −1.34256
\(800\) 3.29392 0.116458
\(801\) 6.51043 0.230035
\(802\) 7.94160 0.280428
\(803\) −58.8927 −2.07828
\(804\) −29.0684 −1.02516
\(805\) −3.12010 −0.109969
\(806\) −3.49203 −0.123002
\(807\) 4.22754 0.148816
\(808\) −1.72628 −0.0607305
\(809\) 8.67442 0.304976 0.152488 0.988305i \(-0.451271\pi\)
0.152488 + 0.988305i \(0.451271\pi\)
\(810\) 0.290697 0.0102141
\(811\) −26.7302 −0.938623 −0.469312 0.883033i \(-0.655498\pi\)
−0.469312 + 0.883033i \(0.655498\pi\)
\(812\) −2.57823 −0.0904780
\(813\) 31.1471 1.09238
\(814\) 9.82093 0.344223
\(815\) 14.1756 0.496551
\(816\) −11.9550 −0.418509
\(817\) 0 0
\(818\) 1.95906 0.0684968
\(819\) −1.20990 −0.0422775
\(820\) 4.94465 0.172675
\(821\) −9.44863 −0.329759 −0.164880 0.986314i \(-0.552724\pi\)
−0.164880 + 0.986314i \(0.552724\pi\)
\(822\) 1.91406 0.0667607
\(823\) −19.4668 −0.678570 −0.339285 0.940684i \(-0.610185\pi\)
−0.339285 + 0.940684i \(0.610185\pi\)
\(824\) −10.4202 −0.363004
\(825\) 5.34764 0.186181
\(826\) 1.80105 0.0626667
\(827\) 37.0018 1.28668 0.643340 0.765581i \(-0.277548\pi\)
0.643340 + 0.765581i \(0.277548\pi\)
\(828\) −12.2829 −0.426860
\(829\) −20.5418 −0.713448 −0.356724 0.934210i \(-0.616106\pi\)
−0.356724 + 0.934210i \(0.616106\pi\)
\(830\) −1.70379 −0.0591396
\(831\) 1.05393 0.0365603
\(832\) −15.0256 −0.520919
\(833\) 23.1006 0.800388
\(834\) 0.0679523 0.00235299
\(835\) −12.5563 −0.434530
\(836\) 0 0
\(837\) −4.83099 −0.166983
\(838\) 7.67661 0.265184
\(839\) 47.7891 1.64986 0.824932 0.565233i \(-0.191214\pi\)
0.824932 + 0.565233i \(0.191214\pi\)
\(840\) 0.553831 0.0191090
\(841\) −21.3479 −0.736134
\(842\) −11.0144 −0.379580
\(843\) −25.6354 −0.882928
\(844\) −17.5394 −0.603730
\(845\) −6.81695 −0.234510
\(846\) 3.22984 0.111044
\(847\) −8.56237 −0.294206
\(848\) −10.4891 −0.360196
\(849\) 29.4112 1.00939
\(850\) −0.992907 −0.0340564
\(851\) 40.5107 1.38869
\(852\) −19.0190 −0.651581
\(853\) 11.4326 0.391446 0.195723 0.980659i \(-0.437295\pi\)
0.195723 + 0.980659i \(0.437295\pi\)
\(854\) −1.67682 −0.0573795
\(855\) 0 0
\(856\) 0.158141 0.00540513
\(857\) −5.57043 −0.190282 −0.0951412 0.995464i \(-0.530330\pi\)
−0.0951412 + 0.995464i \(0.530330\pi\)
\(858\) 3.86549 0.131966
\(859\) −17.0142 −0.580517 −0.290258 0.956948i \(-0.593741\pi\)
−0.290258 + 0.956948i \(0.593741\pi\)
\(860\) 4.76302 0.162418
\(861\) 1.25604 0.0428058
\(862\) 5.58304 0.190159
\(863\) −46.8850 −1.59598 −0.797992 0.602668i \(-0.794104\pi\)
−0.797992 + 0.602668i \(0.794104\pi\)
\(864\) 3.29392 0.112061
\(865\) −2.18964 −0.0744500
\(866\) −4.92768 −0.167449
\(867\) −5.33363 −0.181140
\(868\) −4.50263 −0.152829
\(869\) −43.4908 −1.47532
\(870\) −0.804139 −0.0272629
\(871\) 37.7348 1.27859
\(872\) −17.9042 −0.606313
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) −0.486575 −0.0164492
\(876\) 21.0950 0.712735
\(877\) 29.7844 1.00575 0.502873 0.864360i \(-0.332276\pi\)
0.502873 + 0.864360i \(0.332276\pi\)
\(878\) −3.22757 −0.108925
\(879\) 10.8548 0.366125
\(880\) 18.7173 0.630961
\(881\) −20.4884 −0.690271 −0.345135 0.938553i \(-0.612167\pi\)
−0.345135 + 0.938553i \(0.612167\pi\)
\(882\) −1.96606 −0.0662006
\(883\) 31.4804 1.05940 0.529699 0.848186i \(-0.322305\pi\)
0.529699 + 0.848186i \(0.322305\pi\)
\(884\) 16.2686 0.547173
\(885\) −12.7332 −0.428021
\(886\) −4.03900 −0.135693
\(887\) 33.6103 1.12852 0.564262 0.825596i \(-0.309161\pi\)
0.564262 + 0.825596i \(0.309161\pi\)
\(888\) −7.19080 −0.241307
\(889\) −4.78351 −0.160434
\(890\) 1.89256 0.0634389
\(891\) 5.34764 0.179153
\(892\) 20.8742 0.698919
\(893\) 0 0
\(894\) −0.777031 −0.0259878
\(895\) −7.50421 −0.250838
\(896\) 4.06019 0.135641
\(897\) 15.9449 0.532384
\(898\) −8.10099 −0.270334
\(899\) 13.3637 0.445704
\(900\) −1.91550 −0.0638498
\(901\) 10.2358 0.341004
\(902\) −4.01289 −0.133615
\(903\) 1.20990 0.0402631
\(904\) 16.8470 0.560323
\(905\) 12.6652 0.421006
\(906\) −1.73042 −0.0574894
\(907\) −21.9494 −0.728819 −0.364410 0.931239i \(-0.618729\pi\)
−0.364410 + 0.931239i \(0.618729\pi\)
\(908\) −47.1885 −1.56600
\(909\) 1.51665 0.0503040
\(910\) −0.351716 −0.0116593
\(911\) −21.2865 −0.705253 −0.352626 0.935764i \(-0.614711\pi\)
−0.352626 + 0.935764i \(0.614711\pi\)
\(912\) 0 0
\(913\) −31.3428 −1.03730
\(914\) −10.6296 −0.351597
\(915\) 11.8548 0.391909
\(916\) −32.8845 −1.08653
\(917\) −6.92356 −0.228636
\(918\) −0.992907 −0.0327708
\(919\) −2.61948 −0.0864087 −0.0432043 0.999066i \(-0.513757\pi\)
−0.0432043 + 0.999066i \(0.513757\pi\)
\(920\) −7.29873 −0.240632
\(921\) −33.6891 −1.11009
\(922\) 6.37151 0.209834
\(923\) 24.6893 0.812658
\(924\) 4.98417 0.163967
\(925\) 6.31756 0.207720
\(926\) −4.07718 −0.133984
\(927\) 9.15478 0.300682
\(928\) −9.11178 −0.299109
\(929\) 3.82474 0.125486 0.0627428 0.998030i \(-0.480015\pi\)
0.0627428 + 0.998030i \(0.480015\pi\)
\(930\) −1.40436 −0.0460506
\(931\) 0 0
\(932\) −54.5210 −1.78589
\(933\) −11.7221 −0.383765
\(934\) 9.86319 0.322733
\(935\) −18.2654 −0.597343
\(936\) −2.83028 −0.0925105
\(937\) −7.39167 −0.241475 −0.120738 0.992684i \(-0.538526\pi\)
−0.120738 + 0.992684i \(0.538526\pi\)
\(938\) −2.14650 −0.0700856
\(939\) 4.47856 0.146152
\(940\) −21.2824 −0.694156
\(941\) −48.6115 −1.58469 −0.792344 0.610075i \(-0.791139\pi\)
−0.792344 + 0.610075i \(0.791139\pi\)
\(942\) −2.09548 −0.0682746
\(943\) −16.5529 −0.539036
\(944\) −44.5675 −1.45055
\(945\) −0.486575 −0.0158283
\(946\) −3.86549 −0.125678
\(947\) −36.0207 −1.17051 −0.585257 0.810848i \(-0.699006\pi\)
−0.585257 + 0.810848i \(0.699006\pi\)
\(948\) 15.5782 0.505956
\(949\) −27.3843 −0.888930
\(950\) 0 0
\(951\) −12.3873 −0.401687
\(952\) −1.89167 −0.0613093
\(953\) −11.2121 −0.363196 −0.181598 0.983373i \(-0.558127\pi\)
−0.181598 + 0.983373i \(0.558127\pi\)
\(954\) −0.871155 −0.0282047
\(955\) 8.15777 0.263979
\(956\) −7.81026 −0.252602
\(957\) −14.7929 −0.478186
\(958\) 9.28769 0.300072
\(959\) −3.20380 −0.103456
\(960\) −6.04269 −0.195027
\(961\) −7.66153 −0.247146
\(962\) 4.56659 0.147233
\(963\) −0.138936 −0.00447716
\(964\) −24.2663 −0.781565
\(965\) 3.36505 0.108325
\(966\) −0.907005 −0.0291824
\(967\) −10.0461 −0.323062 −0.161531 0.986868i \(-0.551643\pi\)
−0.161531 + 0.986868i \(0.551643\pi\)
\(968\) −20.0296 −0.643775
\(969\) 0 0
\(970\) −0.581394 −0.0186674
\(971\) 37.8989 1.21623 0.608117 0.793847i \(-0.291925\pi\)
0.608117 + 0.793847i \(0.291925\pi\)
\(972\) −1.91550 −0.0614395
\(973\) −0.113740 −0.00364633
\(974\) −10.9369 −0.350440
\(975\) 2.48657 0.0796341
\(976\) 41.4933 1.32817
\(977\) −44.6435 −1.42827 −0.714135 0.700008i \(-0.753180\pi\)
−0.714135 + 0.700008i \(0.753180\pi\)
\(978\) 4.12082 0.131769
\(979\) 34.8154 1.11271
\(980\) 12.9550 0.413831
\(981\) 15.7299 0.502219
\(982\) −11.6524 −0.371842
\(983\) 12.5091 0.398978 0.199489 0.979900i \(-0.436072\pi\)
0.199489 + 0.979900i \(0.436072\pi\)
\(984\) 2.93820 0.0936665
\(985\) 3.83399 0.122161
\(986\) 2.74662 0.0874703
\(987\) −5.40616 −0.172080
\(988\) 0 0
\(989\) −15.9449 −0.507017
\(990\) 1.55454 0.0494066
\(991\) −50.3007 −1.59785 −0.798927 0.601428i \(-0.794599\pi\)
−0.798927 + 0.601428i \(0.794599\pi\)
\(992\) −15.9129 −0.505235
\(993\) −24.5058 −0.777667
\(994\) −1.40442 −0.0445455
\(995\) 12.6985 0.402569
\(996\) 11.2268 0.355736
\(997\) 12.9375 0.409735 0.204867 0.978790i \(-0.434324\pi\)
0.204867 + 0.978790i \(0.434324\pi\)
\(998\) −8.81007 −0.278878
\(999\) 6.31756 0.199879
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 5415.2.a.z.1.3 5
19.8 odd 6 285.2.i.f.121.3 yes 10
19.12 odd 6 285.2.i.f.106.3 10
19.18 odd 2 5415.2.a.y.1.3 5
57.8 even 6 855.2.k.i.406.3 10
57.50 even 6 855.2.k.i.676.3 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.i.f.106.3 10 19.12 odd 6
285.2.i.f.121.3 yes 10 19.8 odd 6
855.2.k.i.406.3 10 57.8 even 6
855.2.k.i.676.3 10 57.50 even 6
5415.2.a.y.1.3 5 19.18 odd 2
5415.2.a.z.1.3 5 1.1 even 1 trivial