Properties

Label 5070.2.b.x.1351.1
Level $5070$
Weight $2$
Character 5070.1351
Analytic conductor $40.484$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5070,2,Mod(1351,5070)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5070, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5070.1351");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 5x^{4} + 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1351.1
Root \(-1.24698i\) of defining polynomial
Character \(\chi\) \(=\) 5070.1351
Dual form 5070.2.b.x.1351.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -0.801938i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -0.801938i q^{7} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -3.04892i q^{11} -1.00000 q^{12} -0.801938 q^{14} -1.00000i q^{15} +1.00000 q^{16} +1.24698 q^{17} -1.00000i q^{18} -7.40581i q^{19} +1.00000i q^{20} -0.801938i q^{21} -3.04892 q^{22} +0.356896 q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} +0.801938i q^{28} +0.307979 q^{29} -1.00000 q^{30} -3.44504i q^{31} -1.00000i q^{32} -3.04892i q^{33} -1.24698i q^{34} -0.801938 q^{35} -1.00000 q^{36} +7.29590i q^{37} -7.40581 q^{38} +1.00000 q^{40} -3.40581i q^{41} -0.801938 q^{42} -6.54288 q^{43} +3.04892i q^{44} -1.00000i q^{45} -0.356896i q^{46} +0.603875i q^{47} +1.00000 q^{48} +6.35690 q^{49} +1.00000i q^{50} +1.24698 q^{51} -6.29590 q^{53} -1.00000i q^{54} -3.04892 q^{55} +0.801938 q^{56} -7.40581i q^{57} -0.307979i q^{58} +3.11529i q^{59} +1.00000i q^{60} +11.0368 q^{61} -3.44504 q^{62} -0.801938i q^{63} -1.00000 q^{64} -3.04892 q^{66} -5.96077i q^{67} -1.24698 q^{68} +0.356896 q^{69} +0.801938i q^{70} -6.67456i q^{71} +1.00000i q^{72} +10.1371i q^{73} +7.29590 q^{74} -1.00000 q^{75} +7.40581i q^{76} -2.44504 q^{77} -13.6799 q^{79} -1.00000i q^{80} +1.00000 q^{81} -3.40581 q^{82} +3.93900i q^{83} +0.801938i q^{84} -1.24698i q^{85} +6.54288i q^{86} +0.307979 q^{87} +3.04892 q^{88} -8.05861i q^{89} -1.00000 q^{90} -0.356896 q^{92} -3.44504i q^{93} +0.603875 q^{94} -7.40581 q^{95} -1.00000i q^{96} -9.73556i q^{97} -6.35690i q^{98} -3.04892i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{3} - 6 q^{4} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{3} - 6 q^{4} + 6 q^{9} - 6 q^{10} - 6 q^{12} + 4 q^{14} + 6 q^{16} - 2 q^{17} - 6 q^{23} - 6 q^{25} + 6 q^{27} + 12 q^{29} - 6 q^{30} + 4 q^{35} - 6 q^{36} - 18 q^{38} + 6 q^{40} + 4 q^{42} - 2 q^{43} + 6 q^{48} + 30 q^{49} - 2 q^{51} - 10 q^{53} - 4 q^{56} + 10 q^{61} - 20 q^{62} - 6 q^{64} + 2 q^{68} - 6 q^{69} + 16 q^{74} - 6 q^{75} - 14 q^{77} - 34 q^{79} + 6 q^{81} + 6 q^{82} + 12 q^{87} - 6 q^{90} + 6 q^{92} - 14 q^{94} - 18 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times\).

\(n\) \(1691\) \(1861\) \(4057\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) − 1.00000i − 0.447214i
\(6\) − 1.00000i − 0.408248i
\(7\) − 0.801938i − 0.303104i −0.988449 0.151552i \(-0.951573\pi\)
0.988449 0.151552i \(-0.0484271\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) − 3.04892i − 0.919283i −0.888105 0.459642i \(-0.847978\pi\)
0.888105 0.459642i \(-0.152022\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −0.801938 −0.214327
\(15\) − 1.00000i − 0.258199i
\(16\) 1.00000 0.250000
\(17\) 1.24698 0.302437 0.151218 0.988500i \(-0.451680\pi\)
0.151218 + 0.988500i \(0.451680\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) − 7.40581i − 1.69901i −0.527581 0.849505i \(-0.676901\pi\)
0.527581 0.849505i \(-0.323099\pi\)
\(20\) 1.00000i 0.223607i
\(21\) − 0.801938i − 0.174997i
\(22\) −3.04892 −0.650031
\(23\) 0.356896 0.0744179 0.0372090 0.999308i \(-0.488153\pi\)
0.0372090 + 0.999308i \(0.488153\pi\)
\(24\) 1.00000i 0.204124i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0.801938i 0.151552i
\(29\) 0.307979 0.0571902 0.0285951 0.999591i \(-0.490897\pi\)
0.0285951 + 0.999591i \(0.490897\pi\)
\(30\) −1.00000 −0.182574
\(31\) − 3.44504i − 0.618748i −0.950940 0.309374i \(-0.899881\pi\)
0.950940 0.309374i \(-0.100119\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 3.04892i − 0.530748i
\(34\) − 1.24698i − 0.213855i
\(35\) −0.801938 −0.135552
\(36\) −1.00000 −0.166667
\(37\) 7.29590i 1.19944i 0.800211 + 0.599719i \(0.204721\pi\)
−0.800211 + 0.599719i \(0.795279\pi\)
\(38\) −7.40581 −1.20138
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) − 3.40581i − 0.531899i −0.963987 0.265949i \(-0.914315\pi\)
0.963987 0.265949i \(-0.0856854\pi\)
\(42\) −0.801938 −0.123742
\(43\) −6.54288 −0.997779 −0.498890 0.866666i \(-0.666259\pi\)
−0.498890 + 0.866666i \(0.666259\pi\)
\(44\) 3.04892i 0.459642i
\(45\) − 1.00000i − 0.149071i
\(46\) − 0.356896i − 0.0526214i
\(47\) 0.603875i 0.0880843i 0.999030 + 0.0440421i \(0.0140236\pi\)
−0.999030 + 0.0440421i \(0.985976\pi\)
\(48\) 1.00000 0.144338
\(49\) 6.35690 0.908128
\(50\) 1.00000i 0.141421i
\(51\) 1.24698 0.174612
\(52\) 0 0
\(53\) −6.29590 −0.864808 −0.432404 0.901680i \(-0.642335\pi\)
−0.432404 + 0.901680i \(0.642335\pi\)
\(54\) − 1.00000i − 0.136083i
\(55\) −3.04892 −0.411116
\(56\) 0.801938 0.107163
\(57\) − 7.40581i − 0.980924i
\(58\) − 0.307979i − 0.0404396i
\(59\) 3.11529i 0.405577i 0.979223 + 0.202788i \(0.0650003\pi\)
−0.979223 + 0.202788i \(0.935000\pi\)
\(60\) 1.00000i 0.129099i
\(61\) 11.0368 1.41312 0.706561 0.707652i \(-0.250246\pi\)
0.706561 + 0.707652i \(0.250246\pi\)
\(62\) −3.44504 −0.437521
\(63\) − 0.801938i − 0.101035i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −3.04892 −0.375296
\(67\) − 5.96077i − 0.728224i −0.931355 0.364112i \(-0.881373\pi\)
0.931355 0.364112i \(-0.118627\pi\)
\(68\) −1.24698 −0.151218
\(69\) 0.356896 0.0429652
\(70\) 0.801938i 0.0958499i
\(71\) − 6.67456i − 0.792125i −0.918224 0.396062i \(-0.870376\pi\)
0.918224 0.396062i \(-0.129624\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 10.1371i 1.18645i 0.805035 + 0.593227i \(0.202146\pi\)
−0.805035 + 0.593227i \(0.797854\pi\)
\(74\) 7.29590 0.848131
\(75\) −1.00000 −0.115470
\(76\) 7.40581i 0.849505i
\(77\) −2.44504 −0.278638
\(78\) 0 0
\(79\) −13.6799 −1.53911 −0.769557 0.638578i \(-0.779523\pi\)
−0.769557 + 0.638578i \(0.779523\pi\)
\(80\) − 1.00000i − 0.111803i
\(81\) 1.00000 0.111111
\(82\) −3.40581 −0.376109
\(83\) 3.93900i 0.432362i 0.976353 + 0.216181i \(0.0693601\pi\)
−0.976353 + 0.216181i \(0.930640\pi\)
\(84\) 0.801938i 0.0874986i
\(85\) − 1.24698i − 0.135254i
\(86\) 6.54288i 0.705537i
\(87\) 0.307979 0.0330188
\(88\) 3.04892 0.325016
\(89\) − 8.05861i − 0.854211i −0.904202 0.427105i \(-0.859533\pi\)
0.904202 0.427105i \(-0.140467\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) −0.356896 −0.0372090
\(93\) − 3.44504i − 0.357234i
\(94\) 0.603875 0.0622850
\(95\) −7.40581 −0.759820
\(96\) − 1.00000i − 0.102062i
\(97\) − 9.73556i − 0.988497i −0.869321 0.494248i \(-0.835443\pi\)
0.869321 0.494248i \(-0.164557\pi\)
\(98\) − 6.35690i − 0.642143i
\(99\) − 3.04892i − 0.306428i
\(100\) 1.00000 0.100000
\(101\) −7.65817 −0.762016 −0.381008 0.924572i \(-0.624423\pi\)
−0.381008 + 0.924572i \(0.624423\pi\)
\(102\) − 1.24698i − 0.123469i
\(103\) −2.32975 −0.229557 −0.114778 0.993391i \(-0.536616\pi\)
−0.114778 + 0.993391i \(0.536616\pi\)
\(104\) 0 0
\(105\) −0.801938 −0.0782611
\(106\) 6.29590i 0.611512i
\(107\) 3.77479 0.364923 0.182461 0.983213i \(-0.441594\pi\)
0.182461 + 0.983213i \(0.441594\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 1.05861i 0.101396i 0.998714 + 0.0506980i \(0.0161446\pi\)
−0.998714 + 0.0506980i \(0.983855\pi\)
\(110\) 3.04892i 0.290703i
\(111\) 7.29590i 0.692496i
\(112\) − 0.801938i − 0.0757760i
\(113\) 11.0368 1.03826 0.519129 0.854696i \(-0.326256\pi\)
0.519129 + 0.854696i \(0.326256\pi\)
\(114\) −7.40581 −0.693618
\(115\) − 0.356896i − 0.0332807i
\(116\) −0.307979 −0.0285951
\(117\) 0 0
\(118\) 3.11529 0.286786
\(119\) − 1.00000i − 0.0916698i
\(120\) 1.00000 0.0912871
\(121\) 1.70410 0.154918
\(122\) − 11.0368i − 0.999228i
\(123\) − 3.40581i − 0.307092i
\(124\) 3.44504i 0.309374i
\(125\) 1.00000i 0.0894427i
\(126\) −0.801938 −0.0714423
\(127\) −15.9541 −1.41569 −0.707847 0.706366i \(-0.750333\pi\)
−0.707847 + 0.706366i \(0.750333\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −6.54288 −0.576068
\(130\) 0 0
\(131\) −17.0911 −1.49326 −0.746629 0.665240i \(-0.768329\pi\)
−0.746629 + 0.665240i \(0.768329\pi\)
\(132\) 3.04892i 0.265374i
\(133\) −5.93900 −0.514977
\(134\) −5.96077 −0.514932
\(135\) − 1.00000i − 0.0860663i
\(136\) 1.24698i 0.106928i
\(137\) − 16.4373i − 1.40433i −0.712014 0.702165i \(-0.752217\pi\)
0.712014 0.702165i \(-0.247783\pi\)
\(138\) − 0.356896i − 0.0303810i
\(139\) −8.10992 −0.687874 −0.343937 0.938993i \(-0.611761\pi\)
−0.343937 + 0.938993i \(0.611761\pi\)
\(140\) 0.801938 0.0677761
\(141\) 0.603875i 0.0508555i
\(142\) −6.67456 −0.560117
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) − 0.307979i − 0.0255762i
\(146\) 10.1371 0.838949
\(147\) 6.35690 0.524308
\(148\) − 7.29590i − 0.599719i
\(149\) − 11.0532i − 0.905516i −0.891633 0.452758i \(-0.850440\pi\)
0.891633 0.452758i \(-0.149560\pi\)
\(150\) 1.00000i 0.0816497i
\(151\) 6.89546i 0.561145i 0.959833 + 0.280572i \(0.0905242\pi\)
−0.959833 + 0.280572i \(0.909476\pi\)
\(152\) 7.40581 0.600691
\(153\) 1.24698 0.100812
\(154\) 2.44504i 0.197027i
\(155\) −3.44504 −0.276712
\(156\) 0 0
\(157\) −20.8388 −1.66312 −0.831558 0.555439i \(-0.812550\pi\)
−0.831558 + 0.555439i \(0.812550\pi\)
\(158\) 13.6799i 1.08832i
\(159\) −6.29590 −0.499297
\(160\) −1.00000 −0.0790569
\(161\) − 0.286208i − 0.0225564i
\(162\) − 1.00000i − 0.0785674i
\(163\) − 4.71917i − 0.369634i −0.982773 0.184817i \(-0.940831\pi\)
0.982773 0.184817i \(-0.0591692\pi\)
\(164\) 3.40581i 0.265949i
\(165\) −3.04892 −0.237358
\(166\) 3.93900 0.305726
\(167\) − 14.5308i − 1.12443i −0.826992 0.562213i \(-0.809950\pi\)
0.826992 0.562213i \(-0.190050\pi\)
\(168\) 0.801938 0.0618708
\(169\) 0 0
\(170\) −1.24698 −0.0956390
\(171\) − 7.40581i − 0.566337i
\(172\) 6.54288 0.498890
\(173\) −11.4523 −0.870705 −0.435353 0.900260i \(-0.643376\pi\)
−0.435353 + 0.900260i \(0.643376\pi\)
\(174\) − 0.307979i − 0.0233478i
\(175\) 0.801938i 0.0606208i
\(176\) − 3.04892i − 0.229821i
\(177\) 3.11529i 0.234160i
\(178\) −8.05861 −0.604018
\(179\) 4.32975 0.323621 0.161810 0.986822i \(-0.448267\pi\)
0.161810 + 0.986822i \(0.448267\pi\)
\(180\) 1.00000i 0.0745356i
\(181\) 24.8321 1.84575 0.922877 0.385096i \(-0.125832\pi\)
0.922877 + 0.385096i \(0.125832\pi\)
\(182\) 0 0
\(183\) 11.0368 0.815866
\(184\) 0.356896i 0.0263107i
\(185\) 7.29590 0.536405
\(186\) −3.44504 −0.252603
\(187\) − 3.80194i − 0.278025i
\(188\) − 0.603875i − 0.0440421i
\(189\) − 0.801938i − 0.0583324i
\(190\) 7.40581i 0.537274i
\(191\) 22.1933 1.60585 0.802925 0.596081i \(-0.203276\pi\)
0.802925 + 0.596081i \(0.203276\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 25.4306i 1.83053i 0.402850 + 0.915266i \(0.368020\pi\)
−0.402850 + 0.915266i \(0.631980\pi\)
\(194\) −9.73556 −0.698973
\(195\) 0 0
\(196\) −6.35690 −0.454064
\(197\) 10.6310i 0.757429i 0.925514 + 0.378714i \(0.123634\pi\)
−0.925514 + 0.378714i \(0.876366\pi\)
\(198\) −3.04892 −0.216677
\(199\) −11.0465 −0.783067 −0.391534 0.920164i \(-0.628055\pi\)
−0.391534 + 0.920164i \(0.628055\pi\)
\(200\) − 1.00000i − 0.0707107i
\(201\) − 5.96077i − 0.420440i
\(202\) 7.65817i 0.538827i
\(203\) − 0.246980i − 0.0173346i
\(204\) −1.24698 −0.0873060
\(205\) −3.40581 −0.237872
\(206\) 2.32975i 0.162321i
\(207\) 0.356896 0.0248060
\(208\) 0 0
\(209\) −22.5797 −1.56187
\(210\) 0.801938i 0.0553390i
\(211\) −21.3491 −1.46973 −0.734867 0.678211i \(-0.762756\pi\)
−0.734867 + 0.678211i \(0.762756\pi\)
\(212\) 6.29590 0.432404
\(213\) − 6.67456i − 0.457334i
\(214\) − 3.77479i − 0.258039i
\(215\) 6.54288i 0.446220i
\(216\) 1.00000i 0.0680414i
\(217\) −2.76271 −0.187545
\(218\) 1.05861 0.0716978
\(219\) 10.1371i 0.684999i
\(220\) 3.04892 0.205558
\(221\) 0 0
\(222\) 7.29590 0.489669
\(223\) 12.7603i 0.854494i 0.904135 + 0.427247i \(0.140517\pi\)
−0.904135 + 0.427247i \(0.859483\pi\)
\(224\) −0.801938 −0.0535817
\(225\) −1.00000 −0.0666667
\(226\) − 11.0368i − 0.734159i
\(227\) − 14.1836i − 0.941398i −0.882294 0.470699i \(-0.844002\pi\)
0.882294 0.470699i \(-0.155998\pi\)
\(228\) 7.40581i 0.490462i
\(229\) − 4.71140i − 0.311338i −0.987809 0.155669i \(-0.950247\pi\)
0.987809 0.155669i \(-0.0497533\pi\)
\(230\) −0.356896 −0.0235330
\(231\) −2.44504 −0.160872
\(232\) 0.307979i 0.0202198i
\(233\) −0.0556221 −0.00364393 −0.00182196 0.999998i \(-0.500580\pi\)
−0.00182196 + 0.999998i \(0.500580\pi\)
\(234\) 0 0
\(235\) 0.603875 0.0393925
\(236\) − 3.11529i − 0.202788i
\(237\) −13.6799 −0.888608
\(238\) −1.00000 −0.0648204
\(239\) − 21.0489i − 1.36154i −0.732497 0.680771i \(-0.761645\pi\)
0.732497 0.680771i \(-0.238355\pi\)
\(240\) − 1.00000i − 0.0645497i
\(241\) − 12.0030i − 0.773180i −0.922252 0.386590i \(-0.873653\pi\)
0.922252 0.386590i \(-0.126347\pi\)
\(242\) − 1.70410i − 0.109544i
\(243\) 1.00000 0.0641500
\(244\) −11.0368 −0.706561
\(245\) − 6.35690i − 0.406127i
\(246\) −3.40581 −0.217147
\(247\) 0 0
\(248\) 3.44504 0.218760
\(249\) 3.93900i 0.249624i
\(250\) 1.00000 0.0632456
\(251\) 2.25906 0.142591 0.0712953 0.997455i \(-0.477287\pi\)
0.0712953 + 0.997455i \(0.477287\pi\)
\(252\) 0.801938i 0.0505173i
\(253\) − 1.08815i − 0.0684112i
\(254\) 15.9541i 1.00105i
\(255\) − 1.24698i − 0.0780889i
\(256\) 1.00000 0.0625000
\(257\) 17.0043 1.06070 0.530350 0.847779i \(-0.322061\pi\)
0.530350 + 0.847779i \(0.322061\pi\)
\(258\) 6.54288i 0.407342i
\(259\) 5.85086 0.363554
\(260\) 0 0
\(261\) 0.307979 0.0190634
\(262\) 17.0911i 1.05589i
\(263\) −28.5773 −1.76215 −0.881077 0.472974i \(-0.843181\pi\)
−0.881077 + 0.472974i \(0.843181\pi\)
\(264\) 3.04892 0.187648
\(265\) 6.29590i 0.386754i
\(266\) 5.93900i 0.364144i
\(267\) − 8.05861i − 0.493179i
\(268\) 5.96077i 0.364112i
\(269\) −3.28083 −0.200036 −0.100018 0.994986i \(-0.531890\pi\)
−0.100018 + 0.994986i \(0.531890\pi\)
\(270\) −1.00000 −0.0608581
\(271\) − 8.70948i − 0.529063i −0.964377 0.264532i \(-0.914783\pi\)
0.964377 0.264532i \(-0.0852173\pi\)
\(272\) 1.24698 0.0756092
\(273\) 0 0
\(274\) −16.4373 −0.993012
\(275\) 3.04892i 0.183857i
\(276\) −0.356896 −0.0214826
\(277\) 8.91185 0.535461 0.267731 0.963494i \(-0.413726\pi\)
0.267731 + 0.963494i \(0.413726\pi\)
\(278\) 8.10992i 0.486400i
\(279\) − 3.44504i − 0.206249i
\(280\) − 0.801938i − 0.0479249i
\(281\) − 1.90648i − 0.113731i −0.998382 0.0568654i \(-0.981889\pi\)
0.998382 0.0568654i \(-0.0181106\pi\)
\(282\) 0.603875 0.0359603
\(283\) 14.9245 0.887171 0.443586 0.896232i \(-0.353706\pi\)
0.443586 + 0.896232i \(0.353706\pi\)
\(284\) 6.67456i 0.396062i
\(285\) −7.40581 −0.438683
\(286\) 0 0
\(287\) −2.73125 −0.161221
\(288\) − 1.00000i − 0.0589256i
\(289\) −15.4450 −0.908532
\(290\) −0.307979 −0.0180851
\(291\) − 9.73556i − 0.570709i
\(292\) − 10.1371i − 0.593227i
\(293\) − 12.8901i − 0.753047i −0.926407 0.376523i \(-0.877119\pi\)
0.926407 0.376523i \(-0.122881\pi\)
\(294\) − 6.35690i − 0.370742i
\(295\) 3.11529 0.181379
\(296\) −7.29590 −0.424065
\(297\) − 3.04892i − 0.176916i
\(298\) −11.0532 −0.640296
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) 5.24698i 0.302431i
\(302\) 6.89546 0.396789
\(303\) −7.65817 −0.439950
\(304\) − 7.40581i − 0.424753i
\(305\) − 11.0368i − 0.631967i
\(306\) − 1.24698i − 0.0712851i
\(307\) 19.2892i 1.10089i 0.834871 + 0.550446i \(0.185542\pi\)
−0.834871 + 0.550446i \(0.814458\pi\)
\(308\) 2.44504 0.139319
\(309\) −2.32975 −0.132535
\(310\) 3.44504i 0.195665i
\(311\) −1.23729 −0.0701603 −0.0350802 0.999385i \(-0.511169\pi\)
−0.0350802 + 0.999385i \(0.511169\pi\)
\(312\) 0 0
\(313\) 8.09113 0.457338 0.228669 0.973504i \(-0.426563\pi\)
0.228669 + 0.973504i \(0.426563\pi\)
\(314\) 20.8388i 1.17600i
\(315\) −0.801938 −0.0451841
\(316\) 13.6799 0.769557
\(317\) 21.0368i 1.18155i 0.806838 + 0.590773i \(0.201177\pi\)
−0.806838 + 0.590773i \(0.798823\pi\)
\(318\) 6.29590i 0.353056i
\(319\) − 0.939001i − 0.0525740i
\(320\) 1.00000i 0.0559017i
\(321\) 3.77479 0.210688
\(322\) −0.286208 −0.0159498
\(323\) − 9.23490i − 0.513843i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −4.71917 −0.261371
\(327\) 1.05861i 0.0585410i
\(328\) 3.40581 0.188055
\(329\) 0.484271 0.0266987
\(330\) 3.04892i 0.167837i
\(331\) 29.4252i 1.61735i 0.588252 + 0.808677i \(0.299816\pi\)
−0.588252 + 0.808677i \(0.700184\pi\)
\(332\) − 3.93900i − 0.216181i
\(333\) 7.29590i 0.399813i
\(334\) −14.5308 −0.795090
\(335\) −5.96077 −0.325672
\(336\) − 0.801938i − 0.0437493i
\(337\) 17.5483 0.955914 0.477957 0.878383i \(-0.341377\pi\)
0.477957 + 0.878383i \(0.341377\pi\)
\(338\) 0 0
\(339\) 11.0368 0.599439
\(340\) 1.24698i 0.0676270i
\(341\) −10.5036 −0.568804
\(342\) −7.40581 −0.400461
\(343\) − 10.7114i − 0.578361i
\(344\) − 6.54288i − 0.352768i
\(345\) − 0.356896i − 0.0192146i
\(346\) 11.4523i 0.615681i
\(347\) 1.27844 0.0686302 0.0343151 0.999411i \(-0.489075\pi\)
0.0343151 + 0.999411i \(0.489075\pi\)
\(348\) −0.307979 −0.0165094
\(349\) 18.0911i 0.968396i 0.874958 + 0.484198i \(0.160889\pi\)
−0.874958 + 0.484198i \(0.839111\pi\)
\(350\) 0.801938 0.0428654
\(351\) 0 0
\(352\) −3.04892 −0.162508
\(353\) 18.9071i 1.00632i 0.864193 + 0.503161i \(0.167830\pi\)
−0.864193 + 0.503161i \(0.832170\pi\)
\(354\) 3.11529 0.165576
\(355\) −6.67456 −0.354249
\(356\) 8.05861i 0.427105i
\(357\) − 1.00000i − 0.0529256i
\(358\) − 4.32975i − 0.228834i
\(359\) 32.7308i 1.72746i 0.503951 + 0.863732i \(0.331879\pi\)
−0.503951 + 0.863732i \(0.668121\pi\)
\(360\) 1.00000 0.0527046
\(361\) −35.8461 −1.88664
\(362\) − 24.8321i − 1.30514i
\(363\) 1.70410 0.0894422
\(364\) 0 0
\(365\) 10.1371 0.530598
\(366\) − 11.0368i − 0.576905i
\(367\) −3.10321 −0.161986 −0.0809932 0.996715i \(-0.525809\pi\)
−0.0809932 + 0.996715i \(0.525809\pi\)
\(368\) 0.356896 0.0186045
\(369\) − 3.40581i − 0.177300i
\(370\) − 7.29590i − 0.379296i
\(371\) 5.04892i 0.262127i
\(372\) 3.44504i 0.178617i
\(373\) 13.7366 0.711256 0.355628 0.934628i \(-0.384267\pi\)
0.355628 + 0.934628i \(0.384267\pi\)
\(374\) −3.80194 −0.196594
\(375\) 1.00000i 0.0516398i
\(376\) −0.603875 −0.0311425
\(377\) 0 0
\(378\) −0.801938 −0.0412472
\(379\) − 34.0097i − 1.74696i −0.486860 0.873480i \(-0.661858\pi\)
0.486860 0.873480i \(-0.338142\pi\)
\(380\) 7.40581 0.379910
\(381\) −15.9541 −0.817352
\(382\) − 22.1933i − 1.13551i
\(383\) − 38.6902i − 1.97698i −0.151293 0.988489i \(-0.548344\pi\)
0.151293 0.988489i \(-0.451656\pi\)
\(384\) 1.00000i 0.0510310i
\(385\) 2.44504i 0.124611i
\(386\) 25.4306 1.29438
\(387\) −6.54288 −0.332593
\(388\) 9.73556i 0.494248i
\(389\) 11.2567 0.570736 0.285368 0.958418i \(-0.407884\pi\)
0.285368 + 0.958418i \(0.407884\pi\)
\(390\) 0 0
\(391\) 0.445042 0.0225067
\(392\) 6.35690i 0.321072i
\(393\) −17.0911 −0.862133
\(394\) 10.6310 0.535583
\(395\) 13.6799i 0.688312i
\(396\) 3.04892i 0.153214i
\(397\) − 23.4403i − 1.17643i −0.808704 0.588216i \(-0.799830\pi\)
0.808704 0.588216i \(-0.200170\pi\)
\(398\) 11.0465i 0.553712i
\(399\) −5.93900 −0.297322
\(400\) −1.00000 −0.0500000
\(401\) − 37.6950i − 1.88240i −0.337852 0.941199i \(-0.609700\pi\)
0.337852 0.941199i \(-0.390300\pi\)
\(402\) −5.96077 −0.297296
\(403\) 0 0
\(404\) 7.65817 0.381008
\(405\) − 1.00000i − 0.0496904i
\(406\) −0.246980 −0.0122574
\(407\) 22.2446 1.10262
\(408\) 1.24698i 0.0617347i
\(409\) 1.19567i 0.0591220i 0.999563 + 0.0295610i \(0.00941094\pi\)
−0.999563 + 0.0295610i \(0.990589\pi\)
\(410\) 3.40581i 0.168201i
\(411\) − 16.4373i − 0.810791i
\(412\) 2.32975 0.114778
\(413\) 2.49827 0.122932
\(414\) − 0.356896i − 0.0175405i
\(415\) 3.93900 0.193358
\(416\) 0 0
\(417\) −8.10992 −0.397144
\(418\) 22.5797i 1.10441i
\(419\) 1.77048 0.0864935 0.0432468 0.999064i \(-0.486230\pi\)
0.0432468 + 0.999064i \(0.486230\pi\)
\(420\) 0.801938 0.0391306
\(421\) − 1.80194i − 0.0878211i −0.999035 0.0439105i \(-0.986018\pi\)
0.999035 0.0439105i \(-0.0139817\pi\)
\(422\) 21.3491i 1.03926i
\(423\) 0.603875i 0.0293614i
\(424\) − 6.29590i − 0.305756i
\(425\) −1.24698 −0.0604874
\(426\) −6.67456 −0.323384
\(427\) − 8.85086i − 0.428323i
\(428\) −3.77479 −0.182461
\(429\) 0 0
\(430\) 6.54288 0.315526
\(431\) 0.567040i 0.0273133i 0.999907 + 0.0136567i \(0.00434719\pi\)
−0.999907 + 0.0136567i \(0.995653\pi\)
\(432\) 1.00000 0.0481125
\(433\) −17.5743 −0.844569 −0.422284 0.906463i \(-0.638772\pi\)
−0.422284 + 0.906463i \(0.638772\pi\)
\(434\) 2.76271i 0.132614i
\(435\) − 0.307979i − 0.0147664i
\(436\) − 1.05861i − 0.0506980i
\(437\) − 2.64310i − 0.126437i
\(438\) 10.1371 0.484368
\(439\) −3.00969 −0.143645 −0.0718223 0.997417i \(-0.522881\pi\)
−0.0718223 + 0.997417i \(0.522881\pi\)
\(440\) − 3.04892i − 0.145351i
\(441\) 6.35690 0.302709
\(442\) 0 0
\(443\) 38.4499 1.82681 0.913406 0.407050i \(-0.133442\pi\)
0.913406 + 0.407050i \(0.133442\pi\)
\(444\) − 7.29590i − 0.346248i
\(445\) −8.05861 −0.382015
\(446\) 12.7603 0.604218
\(447\) − 11.0532i − 0.522800i
\(448\) 0.801938i 0.0378880i
\(449\) 26.9909i 1.27378i 0.770955 + 0.636890i \(0.219779\pi\)
−0.770955 + 0.636890i \(0.780221\pi\)
\(450\) 1.00000i 0.0471405i
\(451\) −10.3840 −0.488965
\(452\) −11.0368 −0.519129
\(453\) 6.89546i 0.323977i
\(454\) −14.1836 −0.665669
\(455\) 0 0
\(456\) 7.40581 0.346809
\(457\) − 19.2664i − 0.901242i −0.892715 0.450621i \(-0.851203\pi\)
0.892715 0.450621i \(-0.148797\pi\)
\(458\) −4.71140 −0.220149
\(459\) 1.24698 0.0582040
\(460\) 0.356896i 0.0166404i
\(461\) 18.5942i 0.866018i 0.901390 + 0.433009i \(0.142548\pi\)
−0.901390 + 0.433009i \(0.857452\pi\)
\(462\) 2.44504i 0.113754i
\(463\) 5.79417i 0.269278i 0.990895 + 0.134639i \(0.0429874\pi\)
−0.990895 + 0.134639i \(0.957013\pi\)
\(464\) 0.307979 0.0142975
\(465\) −3.44504 −0.159760
\(466\) 0.0556221i 0.00257664i
\(467\) −11.2457 −0.520387 −0.260193 0.965557i \(-0.583786\pi\)
−0.260193 + 0.965557i \(0.583786\pi\)
\(468\) 0 0
\(469\) −4.78017 −0.220728
\(470\) − 0.603875i − 0.0278547i
\(471\) −20.8388 −0.960200
\(472\) −3.11529 −0.143393
\(473\) 19.9487i 0.917242i
\(474\) 13.6799i 0.628340i
\(475\) 7.40581i 0.339802i
\(476\) 1.00000i 0.0458349i
\(477\) −6.29590 −0.288269
\(478\) −21.0489 −0.962755
\(479\) − 35.2452i − 1.61039i −0.593008 0.805197i \(-0.702060\pi\)
0.593008 0.805197i \(-0.297940\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) −12.0030 −0.546721
\(483\) − 0.286208i − 0.0130229i
\(484\) −1.70410 −0.0774592
\(485\) −9.73556 −0.442069
\(486\) − 1.00000i − 0.0453609i
\(487\) − 8.49827i − 0.385093i −0.981288 0.192547i \(-0.938325\pi\)
0.981288 0.192547i \(-0.0616747\pi\)
\(488\) 11.0368i 0.499614i
\(489\) − 4.71917i − 0.213408i
\(490\) −6.35690 −0.287175
\(491\) 25.5254 1.15195 0.575973 0.817469i \(-0.304623\pi\)
0.575973 + 0.817469i \(0.304623\pi\)
\(492\) 3.40581i 0.153546i
\(493\) 0.384043 0.0172964
\(494\) 0 0
\(495\) −3.04892 −0.137039
\(496\) − 3.44504i − 0.154687i
\(497\) −5.35258 −0.240096
\(498\) 3.93900 0.176511
\(499\) 29.4577i 1.31871i 0.751832 + 0.659354i \(0.229170\pi\)
−0.751832 + 0.659354i \(0.770830\pi\)
\(500\) − 1.00000i − 0.0447214i
\(501\) − 14.5308i − 0.649188i
\(502\) − 2.25906i − 0.100827i
\(503\) 14.4480 0.644206 0.322103 0.946705i \(-0.395610\pi\)
0.322103 + 0.946705i \(0.395610\pi\)
\(504\) 0.801938 0.0357211
\(505\) 7.65817i 0.340784i
\(506\) −1.08815 −0.0483740
\(507\) 0 0
\(508\) 15.9541 0.707847
\(509\) − 4.54288i − 0.201359i −0.994919 0.100680i \(-0.967898\pi\)
0.994919 0.100680i \(-0.0321017\pi\)
\(510\) −1.24698 −0.0552172
\(511\) 8.12929 0.359619
\(512\) − 1.00000i − 0.0441942i
\(513\) − 7.40581i − 0.326975i
\(514\) − 17.0043i − 0.750028i
\(515\) 2.32975i 0.102661i
\(516\) 6.54288 0.288034
\(517\) 1.84117 0.0809744
\(518\) − 5.85086i − 0.257072i
\(519\) −11.4523 −0.502702
\(520\) 0 0
\(521\) −30.0911 −1.31832 −0.659158 0.752004i \(-0.729087\pi\)
−0.659158 + 0.752004i \(0.729087\pi\)
\(522\) − 0.307979i − 0.0134799i
\(523\) −3.18359 −0.139209 −0.0696043 0.997575i \(-0.522174\pi\)
−0.0696043 + 0.997575i \(0.522174\pi\)
\(524\) 17.0911 0.746629
\(525\) 0.801938i 0.0349994i
\(526\) 28.5773i 1.24603i
\(527\) − 4.29590i − 0.187132i
\(528\) − 3.04892i − 0.132687i
\(529\) −22.8726 −0.994462
\(530\) 6.29590 0.273476
\(531\) 3.11529i 0.135192i
\(532\) 5.93900 0.257488
\(533\) 0 0
\(534\) −8.05861 −0.348730
\(535\) − 3.77479i − 0.163198i
\(536\) 5.96077 0.257466
\(537\) 4.32975 0.186842
\(538\) 3.28083i 0.141447i
\(539\) − 19.3817i − 0.834827i
\(540\) 1.00000i 0.0430331i
\(541\) 21.2825i 0.915006i 0.889208 + 0.457503i \(0.151256\pi\)
−0.889208 + 0.457503i \(0.848744\pi\)
\(542\) −8.70948 −0.374104
\(543\) 24.8321 1.06565
\(544\) − 1.24698i − 0.0534638i
\(545\) 1.05861 0.0453457
\(546\) 0 0
\(547\) 27.5545 1.17814 0.589072 0.808080i \(-0.299493\pi\)
0.589072 + 0.808080i \(0.299493\pi\)
\(548\) 16.4373i 0.702165i
\(549\) 11.0368 0.471041
\(550\) 3.04892 0.130006
\(551\) − 2.28083i − 0.0971667i
\(552\) 0.356896i 0.0151905i
\(553\) 10.9705i 0.466511i
\(554\) − 8.91185i − 0.378628i
\(555\) 7.29590 0.309694
\(556\) 8.10992 0.343937
\(557\) − 3.88040i − 0.164418i −0.996615 0.0822088i \(-0.973803\pi\)
0.996615 0.0822088i \(-0.0261974\pi\)
\(558\) −3.44504 −0.145840
\(559\) 0 0
\(560\) −0.801938 −0.0338881
\(561\) − 3.80194i − 0.160518i
\(562\) −1.90648 −0.0804199
\(563\) 38.6286 1.62800 0.814001 0.580863i \(-0.197285\pi\)
0.814001 + 0.580863i \(0.197285\pi\)
\(564\) − 0.603875i − 0.0254277i
\(565\) − 11.0368i − 0.464323i
\(566\) − 14.9245i − 0.627325i
\(567\) − 0.801938i − 0.0336782i
\(568\) 6.67456 0.280058
\(569\) 27.6746 1.16018 0.580089 0.814553i \(-0.303018\pi\)
0.580089 + 0.814553i \(0.303018\pi\)
\(570\) 7.40581i 0.310195i
\(571\) 10.9065 0.456422 0.228211 0.973612i \(-0.426712\pi\)
0.228211 + 0.973612i \(0.426712\pi\)
\(572\) 0 0
\(573\) 22.1933 0.927137
\(574\) 2.73125i 0.114000i
\(575\) −0.356896 −0.0148836
\(576\) −1.00000 −0.0416667
\(577\) − 29.9758i − 1.24791i −0.781460 0.623955i \(-0.785525\pi\)
0.781460 0.623955i \(-0.214475\pi\)
\(578\) 15.4450i 0.642429i
\(579\) 25.4306i 1.05686i
\(580\) 0.307979i 0.0127881i
\(581\) 3.15883 0.131051
\(582\) −9.73556 −0.403552
\(583\) 19.1957i 0.795003i
\(584\) −10.1371 −0.419475
\(585\) 0 0
\(586\) −12.8901 −0.532484
\(587\) − 32.1551i − 1.32718i −0.748095 0.663592i \(-0.769031\pi\)
0.748095 0.663592i \(-0.230969\pi\)
\(588\) −6.35690 −0.262154
\(589\) −25.5133 −1.05126
\(590\) − 3.11529i − 0.128255i
\(591\) 10.6310i 0.437302i
\(592\) 7.29590i 0.299860i
\(593\) − 3.88876i − 0.159692i −0.996807 0.0798460i \(-0.974557\pi\)
0.996807 0.0798460i \(-0.0254429\pi\)
\(594\) −3.04892 −0.125099
\(595\) −1.00000 −0.0409960
\(596\) 11.0532i 0.452758i
\(597\) −11.0465 −0.452104
\(598\) 0 0
\(599\) −37.2301 −1.52118 −0.760591 0.649232i \(-0.775090\pi\)
−0.760591 + 0.649232i \(0.775090\pi\)
\(600\) − 1.00000i − 0.0408248i
\(601\) −4.04593 −0.165037 −0.0825185 0.996590i \(-0.526296\pi\)
−0.0825185 + 0.996590i \(0.526296\pi\)
\(602\) 5.24698 0.213851
\(603\) − 5.96077i − 0.242741i
\(604\) − 6.89546i − 0.280572i
\(605\) − 1.70410i − 0.0692816i
\(606\) 7.65817i 0.311092i
\(607\) −6.36658 −0.258412 −0.129206 0.991618i \(-0.541243\pi\)
−0.129206 + 0.991618i \(0.541243\pi\)
\(608\) −7.40581 −0.300345
\(609\) − 0.246980i − 0.0100081i
\(610\) −11.0368 −0.446868
\(611\) 0 0
\(612\) −1.24698 −0.0504062
\(613\) 42.2519i 1.70654i 0.521471 + 0.853269i \(0.325383\pi\)
−0.521471 + 0.853269i \(0.674617\pi\)
\(614\) 19.2892 0.778448
\(615\) −3.40581 −0.137336
\(616\) − 2.44504i − 0.0985135i
\(617\) − 0.978230i − 0.0393820i −0.999806 0.0196910i \(-0.993732\pi\)
0.999806 0.0196910i \(-0.00626825\pi\)
\(618\) 2.32975i 0.0937162i
\(619\) 19.0392i 0.765251i 0.923903 + 0.382626i \(0.124980\pi\)
−0.923903 + 0.382626i \(0.875020\pi\)
\(620\) 3.44504 0.138356
\(621\) 0.356896 0.0143217
\(622\) 1.23729i 0.0496108i
\(623\) −6.46250 −0.258915
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) − 8.09113i − 0.323387i
\(627\) −22.5797 −0.901747
\(628\) 20.8388 0.831558
\(629\) 9.09783i 0.362754i
\(630\) 0.801938i 0.0319500i
\(631\) 37.2664i 1.48355i 0.670649 + 0.741775i \(0.266016\pi\)
−0.670649 + 0.741775i \(0.733984\pi\)
\(632\) − 13.6799i − 0.544159i
\(633\) −21.3491 −0.848552
\(634\) 21.0368 0.835479
\(635\) 15.9541i 0.633118i
\(636\) 6.29590 0.249649
\(637\) 0 0
\(638\) −0.939001 −0.0371754
\(639\) − 6.67456i − 0.264042i
\(640\) 1.00000 0.0395285
\(641\) 29.7017 1.17315 0.586574 0.809896i \(-0.300477\pi\)
0.586574 + 0.809896i \(0.300477\pi\)
\(642\) − 3.77479i − 0.148979i
\(643\) 21.5907i 0.851455i 0.904851 + 0.425728i \(0.139982\pi\)
−0.904851 + 0.425728i \(0.860018\pi\)
\(644\) 0.286208i 0.0112782i
\(645\) 6.54288i 0.257626i
\(646\) −9.23490 −0.363342
\(647\) 29.3623 1.15435 0.577175 0.816620i \(-0.304155\pi\)
0.577175 + 0.816620i \(0.304155\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 9.49827 0.372840
\(650\) 0 0
\(651\) −2.76271 −0.108279
\(652\) 4.71917i 0.184817i
\(653\) 33.4470 1.30888 0.654440 0.756114i \(-0.272904\pi\)
0.654440 + 0.756114i \(0.272904\pi\)
\(654\) 1.05861 0.0413948
\(655\) 17.0911i 0.667806i
\(656\) − 3.40581i − 0.132975i
\(657\) 10.1371i 0.395485i
\(658\) − 0.484271i − 0.0188788i
\(659\) 39.3715 1.53370 0.766848 0.641829i \(-0.221824\pi\)
0.766848 + 0.641829i \(0.221824\pi\)
\(660\) 3.04892 0.118679
\(661\) − 10.1274i − 0.393909i −0.980413 0.196955i \(-0.936895\pi\)
0.980413 0.196955i \(-0.0631052\pi\)
\(662\) 29.4252 1.14364
\(663\) 0 0
\(664\) −3.93900 −0.152863
\(665\) 5.93900i 0.230305i
\(666\) 7.29590 0.282710
\(667\) 0.109916 0.00425597
\(668\) 14.5308i 0.562213i
\(669\) 12.7603i 0.493342i
\(670\) 5.96077i 0.230285i
\(671\) − 33.6504i − 1.29906i
\(672\) −0.801938 −0.0309354
\(673\) 10.1400 0.390870 0.195435 0.980717i \(-0.437388\pi\)
0.195435 + 0.980717i \(0.437388\pi\)
\(674\) − 17.5483i − 0.675933i
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 22.9952 0.883778 0.441889 0.897070i \(-0.354309\pi\)
0.441889 + 0.897070i \(0.354309\pi\)
\(678\) − 11.0368i − 0.423867i
\(679\) −7.80731 −0.299617
\(680\) 1.24698 0.0478195
\(681\) − 14.1836i − 0.543516i
\(682\) 10.5036i 0.402205i
\(683\) − 4.10800i − 0.157188i −0.996907 0.0785941i \(-0.974957\pi\)
0.996907 0.0785941i \(-0.0250431\pi\)
\(684\) 7.40581i 0.283168i
\(685\) −16.4373 −0.628036
\(686\) −10.7114 −0.408963
\(687\) − 4.71140i − 0.179751i
\(688\) −6.54288 −0.249445
\(689\) 0 0
\(690\) −0.356896 −0.0135868
\(691\) − 30.1618i − 1.14741i −0.819062 0.573705i \(-0.805506\pi\)
0.819062 0.573705i \(-0.194494\pi\)
\(692\) 11.4523 0.435353
\(693\) −2.44504 −0.0928795
\(694\) − 1.27844i − 0.0485289i
\(695\) 8.10992i 0.307627i
\(696\) 0.307979i 0.0116739i
\(697\) − 4.24698i − 0.160866i
\(698\) 18.0911 0.684759
\(699\) −0.0556221 −0.00210382
\(700\) − 0.801938i − 0.0303104i
\(701\) 37.5066 1.41661 0.708303 0.705909i \(-0.249461\pi\)
0.708303 + 0.705909i \(0.249461\pi\)
\(702\) 0 0
\(703\) 54.0320 2.03786
\(704\) 3.04892i 0.114910i
\(705\) 0.603875 0.0227433
\(706\) 18.9071 0.711577
\(707\) 6.14138i 0.230970i
\(708\) − 3.11529i − 0.117080i
\(709\) − 4.20211i − 0.157814i −0.996882 0.0789068i \(-0.974857\pi\)
0.996882 0.0789068i \(-0.0251430\pi\)
\(710\) 6.67456i 0.250492i
\(711\) −13.6799 −0.513038
\(712\) 8.05861 0.302009
\(713\) − 1.22952i − 0.0460459i
\(714\) −1.00000 −0.0374241
\(715\) 0 0
\(716\) −4.32975 −0.161810
\(717\) − 21.0489i − 0.786086i
\(718\) 32.7308 1.22150
\(719\) 28.3532 1.05740 0.528698 0.848810i \(-0.322681\pi\)
0.528698 + 0.848810i \(0.322681\pi\)
\(720\) − 1.00000i − 0.0372678i
\(721\) 1.86831i 0.0695796i
\(722\) 35.8461i 1.33405i
\(723\) − 12.0030i − 0.446396i
\(724\) −24.8321 −0.922877
\(725\) −0.307979 −0.0114380
\(726\) − 1.70410i − 0.0632452i
\(727\) −10.2892 −0.381605 −0.190803 0.981628i \(-0.561109\pi\)
−0.190803 + 0.981628i \(0.561109\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) − 10.1371i − 0.375190i
\(731\) −8.15883 −0.301765
\(732\) −11.0368 −0.407933
\(733\) 48.8963i 1.80603i 0.429613 + 0.903013i \(0.358650\pi\)
−0.429613 + 0.903013i \(0.641350\pi\)
\(734\) 3.10321i 0.114542i
\(735\) − 6.35690i − 0.234478i
\(736\) − 0.356896i − 0.0131554i
\(737\) −18.1739 −0.669444
\(738\) −3.40581 −0.125370
\(739\) − 23.7496i − 0.873642i −0.899549 0.436821i \(-0.856104\pi\)
0.899549 0.436821i \(-0.143896\pi\)
\(740\) −7.29590 −0.268203
\(741\) 0 0
\(742\) 5.04892 0.185352
\(743\) 51.5153i 1.88991i 0.327198 + 0.944956i \(0.393896\pi\)
−0.327198 + 0.944956i \(0.606104\pi\)
\(744\) 3.44504 0.126301
\(745\) −11.0532 −0.404959
\(746\) − 13.7366i − 0.502934i
\(747\) 3.93900i 0.144121i
\(748\) 3.80194i 0.139013i
\(749\) − 3.02715i − 0.110610i
\(750\) 1.00000 0.0365148
\(751\) −43.8939 −1.60171 −0.800856 0.598857i \(-0.795622\pi\)
−0.800856 + 0.598857i \(0.795622\pi\)
\(752\) 0.603875i 0.0220211i
\(753\) 2.25906 0.0823248
\(754\) 0 0
\(755\) 6.89546 0.250952
\(756\) 0.801938i 0.0291662i
\(757\) −18.6136 −0.676521 −0.338261 0.941052i \(-0.609839\pi\)
−0.338261 + 0.941052i \(0.609839\pi\)
\(758\) −34.0097 −1.23529
\(759\) − 1.08815i − 0.0394972i
\(760\) − 7.40581i − 0.268637i
\(761\) − 44.8364i − 1.62532i −0.582740 0.812659i \(-0.698019\pi\)
0.582740 0.812659i \(-0.301981\pi\)
\(762\) 15.9541i 0.577955i
\(763\) 0.848936 0.0307336
\(764\) −22.1933 −0.802925
\(765\) − 1.24698i − 0.0450846i
\(766\) −38.6902 −1.39793
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) − 39.5303i − 1.42550i −0.701418 0.712750i \(-0.747449\pi\)
0.701418 0.712750i \(-0.252551\pi\)
\(770\) 2.44504 0.0881132
\(771\) 17.0043 0.612395
\(772\) − 25.4306i − 0.915266i
\(773\) − 49.8665i − 1.79357i −0.442463 0.896787i \(-0.645895\pi\)
0.442463 0.896787i \(-0.354105\pi\)
\(774\) 6.54288i 0.235179i
\(775\) 3.44504i 0.123750i
\(776\) 9.73556 0.349486
\(777\) 5.85086 0.209898
\(778\) − 11.2567i − 0.403571i
\(779\) −25.2228 −0.903701
\(780\) 0 0
\(781\) −20.3502 −0.728187
\(782\) − 0.445042i − 0.0159147i
\(783\) 0.307979 0.0110063
\(784\) 6.35690 0.227032
\(785\) 20.8388i 0.743768i
\(786\) 17.0911i 0.609620i
\(787\) − 26.0398i − 0.928219i −0.885778 0.464110i \(-0.846374\pi\)
0.885778 0.464110i \(-0.153626\pi\)
\(788\) − 10.6310i − 0.378714i
\(789\) −28.5773 −1.01738
\(790\) 13.6799 0.486710
\(791\) − 8.85086i − 0.314700i
\(792\) 3.04892 0.108339
\(793\) 0 0
\(794\) −23.4403 −0.831864
\(795\) 6.29590i 0.223292i
\(796\) 11.0465 0.391534
\(797\) 23.9855 0.849611 0.424805 0.905285i \(-0.360343\pi\)
0.424805 + 0.905285i \(0.360343\pi\)
\(798\) 5.93900i 0.210238i
\(799\) 0.753020i 0.0266399i
\(800\) 1.00000i 0.0353553i
\(801\) − 8.05861i − 0.284737i
\(802\) −37.6950 −1.33106
\(803\) 30.9071 1.09069
\(804\) 5.96077i 0.210220i
\(805\) −0.286208 −0.0100875
\(806\) 0 0
\(807\) −3.28083 −0.115491
\(808\) − 7.65817i − 0.269413i
\(809\) 52.4825 1.84519 0.922593 0.385776i \(-0.126066\pi\)
0.922593 + 0.385776i \(0.126066\pi\)
\(810\) −1.00000 −0.0351364
\(811\) − 3.63666i − 0.127700i −0.997959 0.0638502i \(-0.979662\pi\)
0.997959 0.0638502i \(-0.0203380\pi\)
\(812\) 0.246980i 0.00866728i
\(813\) − 8.70948i − 0.305455i
\(814\) − 22.2446i − 0.779672i
\(815\) −4.71917 −0.165305
\(816\) 1.24698 0.0436530
\(817\) 48.4553i 1.69524i
\(818\) 1.19567 0.0418056
\(819\) 0 0
\(820\) 3.40581 0.118936
\(821\) − 7.98984i − 0.278847i −0.990233 0.139424i \(-0.955475\pi\)
0.990233 0.139424i \(-0.0445250\pi\)
\(822\) −16.4373 −0.573316
\(823\) −28.1395 −0.980880 −0.490440 0.871475i \(-0.663164\pi\)
−0.490440 + 0.871475i \(0.663164\pi\)
\(824\) − 2.32975i − 0.0811606i
\(825\) 3.04892i 0.106150i
\(826\) − 2.49827i − 0.0869260i
\(827\) − 30.9256i − 1.07539i −0.843140 0.537694i \(-0.819296\pi\)
0.843140 0.537694i \(-0.180704\pi\)
\(828\) −0.356896 −0.0124030
\(829\) 31.9172 1.10853 0.554266 0.832340i \(-0.312999\pi\)
0.554266 + 0.832340i \(0.312999\pi\)
\(830\) − 3.93900i − 0.136725i
\(831\) 8.91185 0.309149
\(832\) 0 0
\(833\) 7.92692 0.274651
\(834\) 8.10992i 0.280823i
\(835\) −14.5308 −0.502859
\(836\) 22.5797 0.780936
\(837\) − 3.44504i − 0.119078i
\(838\) − 1.77048i − 0.0611602i
\(839\) 24.9866i 0.862633i 0.902201 + 0.431316i \(0.141951\pi\)
−0.902201 + 0.431316i \(0.858049\pi\)
\(840\) − 0.801938i − 0.0276695i
\(841\) −28.9051 −0.996729
\(842\) −1.80194 −0.0620989
\(843\) − 1.90648i − 0.0656626i
\(844\) 21.3491 0.734867
\(845\) 0 0
\(846\) 0.603875 0.0207617
\(847\) − 1.36658i − 0.0469564i
\(848\) −6.29590 −0.216202
\(849\) 14.9245 0.512208
\(850\) 1.24698i 0.0427710i
\(851\) 2.60388i 0.0892597i
\(852\) 6.67456i 0.228667i
\(853\) 35.4416i 1.21350i 0.794894 + 0.606748i \(0.207526\pi\)
−0.794894 + 0.606748i \(0.792474\pi\)
\(854\) −8.85086 −0.302870
\(855\) −7.40581 −0.253273
\(856\) 3.77479i 0.129020i
\(857\) 14.6541 0.500575 0.250288 0.968172i \(-0.419475\pi\)
0.250288 + 0.968172i \(0.419475\pi\)
\(858\) 0 0
\(859\) 46.3153 1.58026 0.790128 0.612941i \(-0.210014\pi\)
0.790128 + 0.612941i \(0.210014\pi\)
\(860\) − 6.54288i − 0.223110i
\(861\) −2.73125 −0.0930808
\(862\) 0.567040 0.0193134
\(863\) 40.3642i 1.37401i 0.726651 + 0.687007i \(0.241076\pi\)
−0.726651 + 0.687007i \(0.758924\pi\)
\(864\) − 1.00000i − 0.0340207i
\(865\) 11.4523i 0.389391i
\(866\) 17.5743i 0.597200i
\(867\) −15.4450 −0.524541
\(868\) 2.76271 0.0937725
\(869\) 41.7090i 1.41488i
\(870\) −0.307979 −0.0104414
\(871\) 0 0
\(872\) −1.05861 −0.0358489
\(873\) − 9.73556i − 0.329499i
\(874\) −2.64310 −0.0894043
\(875\) 0.801938 0.0271104
\(876\) − 10.1371i − 0.342500i
\(877\) 35.0103i 1.18221i 0.806593 + 0.591107i \(0.201309\pi\)
−0.806593 + 0.591107i \(0.798691\pi\)
\(878\) 3.00969i 0.101572i
\(879\) − 12.8901i − 0.434772i
\(880\) −3.04892 −0.102779
\(881\) −23.8689 −0.804164 −0.402082 0.915604i \(-0.631713\pi\)
−0.402082 + 0.915604i \(0.631713\pi\)
\(882\) − 6.35690i − 0.214048i
\(883\) −26.1806 −0.881047 −0.440524 0.897741i \(-0.645207\pi\)
−0.440524 + 0.897741i \(0.645207\pi\)
\(884\) 0 0
\(885\) 3.11529 0.104719
\(886\) − 38.4499i − 1.29175i
\(887\) 25.8625 0.868377 0.434188 0.900822i \(-0.357035\pi\)
0.434188 + 0.900822i \(0.357035\pi\)
\(888\) −7.29590 −0.244834
\(889\) 12.7942i 0.429103i
\(890\) 8.05861i 0.270125i
\(891\) − 3.04892i − 0.102143i
\(892\) − 12.7603i − 0.427247i
\(893\) 4.47219 0.149656
\(894\) −11.0532 −0.369675
\(895\) − 4.32975i − 0.144728i
\(896\) 0.801938 0.0267909
\(897\) 0 0
\(898\) 26.9909 0.900698
\(899\) − 1.06100i − 0.0353863i
\(900\) 1.00000 0.0333333
\(901\) −7.85086 −0.261550
\(902\) 10.3840i 0.345751i
\(903\) 5.24698i 0.174609i
\(904\) 11.0368i 0.367080i
\(905\) − 24.8321i − 0.825446i
\(906\) 6.89546 0.229086
\(907\) −12.3980 −0.411670 −0.205835 0.978587i \(-0.565991\pi\)
−0.205835 + 0.978587i \(0.565991\pi\)
\(908\) 14.1836i 0.470699i
\(909\) −7.65817 −0.254005
\(910\) 0 0
\(911\) 5.34183 0.176983 0.0884914 0.996077i \(-0.471795\pi\)
0.0884914 + 0.996077i \(0.471795\pi\)
\(912\) − 7.40581i − 0.245231i
\(913\) 12.0097 0.397463
\(914\) −19.2664 −0.637274
\(915\) − 11.0368i − 0.364867i
\(916\) 4.71140i 0.155669i
\(917\) 13.7060i 0.452613i
\(918\) − 1.24698i − 0.0411565i
\(919\) 32.3327 1.06656 0.533279 0.845939i \(-0.320960\pi\)
0.533279 + 0.845939i \(0.320960\pi\)
\(920\) 0.356896 0.0117665
\(921\) 19.2892i 0.635600i
\(922\) 18.5942 0.612367
\(923\) 0 0
\(924\) 2.44504 0.0804360
\(925\) − 7.29590i − 0.239888i
\(926\) 5.79417 0.190408
\(927\) −2.32975 −0.0765190
\(928\) − 0.307979i − 0.0101099i
\(929\) − 19.3381i − 0.634463i −0.948348 0.317231i \(-0.897247\pi\)
0.948348 0.317231i \(-0.102753\pi\)
\(930\) 3.44504i 0.112967i
\(931\) − 47.0780i − 1.54292i
\(932\) 0.0556221 0.00182196
\(933\) −1.23729 −0.0405071
\(934\) 11.2457i 0.367969i
\(935\) −3.80194 −0.124337
\(936\) 0 0
\(937\) −40.4271 −1.32070 −0.660348 0.750960i \(-0.729591\pi\)
−0.660348 + 0.750960i \(0.729591\pi\)
\(938\) 4.78017i 0.156078i
\(939\) 8.09113 0.264044
\(940\) −0.603875 −0.0196962
\(941\) 9.38537i 0.305954i 0.988230 + 0.152977i \(0.0488861\pi\)
−0.988230 + 0.152977i \(0.951114\pi\)
\(942\) 20.8388i 0.678964i
\(943\) − 1.21552i − 0.0395828i
\(944\) 3.11529i 0.101394i
\(945\) −0.801938 −0.0260870
\(946\) 19.9487 0.648588
\(947\) − 24.4252i − 0.793712i −0.917881 0.396856i \(-0.870101\pi\)
0.917881 0.396856i \(-0.129899\pi\)
\(948\) 13.6799 0.444304
\(949\) 0 0
\(950\) 7.40581 0.240276
\(951\) 21.0368i 0.682166i
\(952\) 1.00000 0.0324102
\(953\) −57.1657 −1.85178 −0.925889 0.377797i \(-0.876682\pi\)
−0.925889 + 0.377797i \(0.876682\pi\)
\(954\) 6.29590i 0.203837i
\(955\) − 22.1933i − 0.718158i
\(956\) 21.0489i 0.680771i
\(957\) − 0.939001i − 0.0303536i
\(958\) −35.2452 −1.13872
\(959\) −13.1817 −0.425658
\(960\) 1.00000i 0.0322749i
\(961\) 19.1317 0.617151
\(962\) 0 0
\(963\) 3.77479 0.121641
\(964\) 12.0030i 0.386590i
\(965\) 25.4306 0.818639
\(966\) −0.286208 −0.00920860
\(967\) 5.76271i 0.185316i 0.995698 + 0.0926581i \(0.0295364\pi\)
−0.995698 + 0.0926581i \(0.970464\pi\)
\(968\) 1.70410i 0.0547719i
\(969\) − 9.23490i − 0.296668i
\(970\) 9.73556i 0.312590i
\(971\) −1.72694 −0.0554201 −0.0277100 0.999616i \(-0.508822\pi\)
−0.0277100 + 0.999616i \(0.508822\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 6.50365i 0.208497i
\(974\) −8.49827 −0.272302
\(975\) 0 0
\(976\) 11.0368 0.353281
\(977\) − 26.4747i − 0.847001i −0.905896 0.423500i \(-0.860801\pi\)
0.905896 0.423500i \(-0.139199\pi\)
\(978\) −4.71917 −0.150902
\(979\) −24.5700 −0.785261
\(980\) 6.35690i 0.203064i
\(981\) 1.05861i 0.0337987i
\(982\) − 25.5254i − 0.814549i
\(983\) 6.81727i 0.217437i 0.994073 + 0.108719i \(0.0346747\pi\)
−0.994073 + 0.108719i \(0.965325\pi\)
\(984\) 3.40581 0.108573
\(985\) 10.6310 0.338733
\(986\) − 0.384043i − 0.0122304i
\(987\) 0.484271 0.0154145
\(988\) 0 0
\(989\) −2.33513 −0.0742527
\(990\) 3.04892i 0.0969010i
\(991\) 8.31203 0.264040 0.132020 0.991247i \(-0.457854\pi\)
0.132020 + 0.991247i \(0.457854\pi\)
\(992\) −3.44504 −0.109380
\(993\) 29.4252i 0.933780i
\(994\) 5.35258i 0.169774i
\(995\) 11.0465i 0.350198i
\(996\) − 3.93900i − 0.124812i
\(997\) 49.6002 1.57085 0.785426 0.618955i \(-0.212444\pi\)
0.785426 + 0.618955i \(0.212444\pi\)
\(998\) 29.4577 0.932468
\(999\) 7.29590i 0.230832i
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 5070.2.b.x.1351.1 6
13.5 odd 4 5070.2.a.bn.1.3 3
13.8 odd 4 5070.2.a.by.1.1 yes 3
13.12 even 2 inner 5070.2.b.x.1351.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bn.1.3 3 13.5 odd 4
5070.2.a.by.1.1 yes 3 13.8 odd 4
5070.2.b.x.1351.1 6 1.1 even 1 trivial
5070.2.b.x.1351.6 6 13.12 even 2 inner